Quantum Information with Continuous Variables

TL;DR: In this article, the authors present the Deutsch-Jozsa algorithm for continuous variables, and a deterministic version of it is used for quantum information processing with continuous variables.

Abstract: Preface. About the Editors. Part I: Quantum Computing. 1. Quantum computing with qubits S.L. Braunstein, A.K. Pati. 2. Quantum computation over continuous variables S. Lloyd, S.L. Braunstein. 3. Error correction for continuous quantum variables S.L. Braunstein. 4. Deutsch-Jozsa algorithm for continuous variables A.K. Pati, S.L. Braunstein. 5. Hybrid quantum computing S. Lloyd. 6. Efficient classical simulation of continuous variable quantum information processes S.D. Bartlett, B.C. Sanders, S.L. Braunstein, K. Nemoto. Part II: Quantum Entanglement. 7. Introduction to entanglement-based protocols S.L. Braunstein, A.K. Pati. 8. Teleportation of continuous uantum variables S.L. Braunstein, H.J. Kimble. 9. Experimental realization of continuous variable teleportation A. Furusawa, H.J. Kimble. 10. Dense coding for continuous variables S.L. Braunstein, H.J. Kimble. 11. Multipartite Greenberger-Horne-Zeilinger paradoxes for continuous variables S. Massar, S. Pironio. 12. Multipartite entanglement for continuous variables P. van Loock, S.L. Braunstein. 13. Inseparability criterion for continuous variable systems Lu-Ming Duan, G. Giedke, J.I. Cirac, P. Zoller. 14. Separability criterion for Gaussian states R. Simon. 15. Distillability and entanglement purification for Gaussian states G. Giedke, Lu-Ming Duan, J.I. Cirac, P. Zoller. 16. Entanglement purification via entanglement swapping S. Parke, S. Bose, M.B. Plenio. 17. Bound entanglement for continuous variables is a rare phenomenon P. Horodecki, J.I. Cirac, M. Lewenstein. Part III: Continuous Variable Optical-Atomic Interfacing. 18. Atomic continuous variable processing and light-atoms quantum interface A. Kuzmich, E.S. Polzik. Part IV: Limits on Quantum Information and Cryptography. 19. Limitations on discrete quantum information and cryptography S.L. Braunstein, A.K. Pati. 20. Quantum cloning with continuous variables N.J. Cerf. 21. Quantum key distribution with continuous variables in optics T.C. Ralph. 22. Secure quantum key distribution using squeezed states D. Gottesman, J. Preskill. 23. Experimental demonstration of dense coding and quantum cryptography with continuous variables Kunchi Peng, Qing Pan, Jing Zhang, Changde Xie. 24. Quantum solitons in optical fibres: basic requisites for experimental quantum communication G. Leuchs, Ch. Silberhorn, E. Konig, P.K. Lam, A. Sizmann, N. Korolkova. Index.

Summary

Introduction

• This article reviews the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field.
• Experiments with Continuous Quantum Variables 565 A. Generation of squeezed-state EPR entanglement 565 1. Broadband entanglement via optical parametric amplification 565 2.
• Furthermore, the exploitation of quantum effects may even prove beneficial for various kinds of information processing and communication.
• Quantum computation over continuous variables, discussed in Sec. VI, is a more subtle issue than the in some sense straightforward continuous-variable extensions of quantum communication protocols.

II. CONTINUOUS VARIABLES IN QUANTUM OPTICS

• For the transition from classical to quantum mechanics, the position and momentum observables of the particles turn into noncommuting Hermitian operators in the Hamiltonian.
• In quantum optics, the quantized electromagnetic modes correspond to quantum harmonic oscillators.
• The modes’ quadratures play the roles of the oscillators’ position and momentum operators obeying an analogous Heisenberg uncertainty relation.

A. The quadratures of the quantized field

• From the Hamiltonian of a quantum harmonic oscillator expressed in terms of dimensionless creation and annihilation operators and representing a single mode k, Ĥk= k âk †âk+.
• In Eq. 2 , the authors see that up to normalization factors the position and the momentum are the real and imaginary parts of the annihilation operator.
• Let us now define the dimensionless pair of conjugate variables, X̂k k2 x̂k =.
• A broadband rather than a single-mode description of homodyne detection can be found in the work of Braunstein and Crouch 1991 , who also investigate the influence of a quantized local oscillator.
• In the following, the authors shall refer mainly to the conjugate pair of quadratures x̂k and p̂k position and momentum, i.e., =0 and = /2 . 25 Just as for position and momentum eigenstates, the quadrature eigenstates are mutually related to each other by a Fourier transformation, x = 1 − e−2ixp p dp , 26 Rev. Mod.

B. Phase-space representations

• The Wigner function is particularly suitable as a “quantum phase-space distribution” for describing the effects on the quadrature observables that may arise from quantum theory and classical statistics.
• On the other hand, in contrast to a classical probability distribution, the Wigner function can become negative.
• For any symmetrized operator Leonhardt, 1997 , the so-called Weyl correspondence Weyl, 1950 , Tr ̂S x̂np̂m = W x,p xnpmdxdp , 32 provides a rule for calculating quantum-mechanical expectation values in a classical-like fashion according to Eq. 31 .
• Such a classical-like formulation of quantum optics in terms of quasiprobability distributions is not unique.
• Hence the Wigner function, though not always positive definite, appears to be a good compromise in describing quantum states in terms of quantum phase-space variables such as single-mode quadratures.

C. Gaussian states

• Multimode Gaussian states may represent optical quantum states that are potentially useful for quantum communication or computation purposes.
• They are efficiently producible in the laboratory, available on demand in an unconditional fashion.
• For Gaussian states of the form of Eq. 35 , the Wigner function is completely determined by the second-moment correlation matrix.
• This matrix equation means that the matrix sum on the left-hand side has only non-negative eigenvalues.
• For any N, Eq. 41 becomes exactly the Heisenberg uncertainty relation of Eq. 12 for each individual mode, if V N is diagonal.

D. Linear optics

• In passive optical devices such as beam splitters and phase shifters, the photon number is preserved and the modes’ annihilation operators are transformed only linearly.
• This means that any mixing between optical modes described by a unitary matrix can be implemented with linear optics.
• Conversely, however, any such network can be described by the linear transformation in Eq. 47 .
• The action of an ideal phase-free beam-splitter operation on two modes can be expressed in the Heisenberg picture by Eq. 45 .
• The input operators are changed, whereas the input states remain invariant.

E. Nonlinear optics

• An important tool of many quantum communication protocols is entanglement, and the essential ingredient in the generation of continuous-variable entanglement is squeezed light.
• In general, squeezing refers to the reduction of quantum fluctuations in one observable below the standard quantum limit the minimal noise level of the vacuum state at the expense of an increased uncertainty of the conjugate variable.
• 52 This Hamiltonian describes the amplification of the signal mode â at half the pump frequency in an interaction picture without explicit time dependence due to the free evolution .
• Here the authors have chosen vacuum-state inputs and replaced the initial quadratures by those of the vacuum labeled by a superscript 0 .
• A two-mode squeezed state, produced by the nondegenerate optical parametric amplifier interaction, is equivalent to two single-mode squeezed states with perpendicular squeezing directions and produced via the degenerate optical parametric amplifier interaction or alternatively via a 3 interaction; see Sec. VII combined at a beam splitter van Loock et al., 2000 .

F. Polarization and spin representations

• The field of quantum information with continuous variables grew out of the analysis of quadraturesqueezed optical states.
• Here the authors briefly discuss a useful alternative encoding for continuous quantum information in terms of collective spinlike variables.
• Let â+ t −z /c and â− t−z /c be the annihilation operators for circularly polarized beams of light propagating along the positive z axis.
• Now suppose the authors restrict their states to those for which Ŝx is near its maximum value.
• In a similar manner, the authors shall consider states with near maximal polarization along the negative z axis, i.e., states corresponding to small variations about F ,−F .

G. Necessity of phase reference

• The quantum information of continuous variables is stored as phase-space distributions.
• Thus quantum information with continuous variables involves experiments that typically are going to require a phase reference.
• In quantum-optics experiments, for example, this local oscillator is just a strong laser beam that is shared amongst the various parties such as sender and receiver in the laboratory.
• Nemoto and Braunstein 2003 noted a flaw in the argument of Rudolph and Sanders: although choosing Pr as uniform seems eminently reasonable, if is truly unobservable as they presume and as is generally accepted Mølmer, 1997 , then Rudolph and Sanders’s choice of Pr is untestable.
• The implication of this is that states of the form of Eq. 77 actually form an equivalence class—any member of which may be chosen to represent the class.

III. CONTINUOUS-VARIABLE ENTANGLEMENT

• Historically, the notion of entanglement Verschränkung first appeared explicitly in the literature in 1935, long before the dawn of the relatively young field of quantum information, and without any reference to discrete-variable qubit states.
• The notion of quantum entanglement itself came to light in a continuous-variable setting.
• It can be thought of as the limiting case of a regularized, properly normalized version in which the positions and momenta are correlated only to some finite extent given by a Gaussian width.
• In the second line of Eq. 85 , the authors have used the disentangling theorem of Collett 1988 .
• The form in Eq. 85 reveals that the two modes of the twomode squeezed vacuum state are also quantum correlated in photon number and phase.

1. Pure states

• Bipartite entanglement, the entanglement of a pair of systems shared by two parties, is easy to handle for pure states.
• The Schmidt coefficients cn are real and non-negative and satisfy ncn 2 =1.
• Note that any pure two-mode Gaussian state can be transformed into the canonical two-mode squeezed-state form via local linear unitary Bogoliubov transformations and hence its entanglement can be quantified as in Eq. 94 .
• In addition, the partial von Neumann Rev. Mod. Phys., Vol. 77, No. 2, April 2005 entropy of a pure two-mode Gaussian state, corresponding to the entropy of an arbitrary single-mode Gaussian state, can be also directly computed Agarwal, 1971 .
• The main features of purestate bipartite entanglement will be summarized by entangled ⇔.

2. Mixed states and inseparability criteria

• The definition of pure-state entanglement via the nonfactorizability of the total state vector is generalized to mixed states through nonseparability or inseparability of the total density operator.
• Expressing partial transposition of a bipartite Gaussian system by a 1 where A B means the blockdiagonal matrix with the matrices A and B as diagonal entries, and A and B are, respectively, 2N 2N and 2M 2M square matrices applicable to N modes at a’s 2Separable states also exhibit correlations, but those are purely classical.
• Separability conditions similar to those above can also be derived in terms of the polarization Stokes operators from Eq. 73 Bowen et al., 2002; Korolkova et al., 2002; Korolkova and Loudon, 2005 .
• Apart from the qubit case Wootters, 1998 , for symmetric two-mode Gaussian states given by a correlation matrix in Eq. 104 with a =b, the entanglement of formation can be calculated via the total variances in Eq. 110 Giedke, Wolf, et al., 2003 .

B. Multipartite entanglement

• Multipartite entanglement, the entanglement shared by more than two parties, is a subtle issue even for pure states.
• In that case, for pure multiparty states, a Schmidt decomposition does not exist in general.
• The total state vector then cannot be written as a single sum over orthonormal basis states.
• Let us first consider discretevariable multipartite entanglement.

1. Discrete variables

• There is one very important representative of multipartite entanglement that does have the form of a multiparty Schmidt decomposition, namely, the Greenberger-Horne-Zeilinger GHZ state Greenberger et al., 1990 , GHZ = 1 2 000 + 111 , 121 here given as a three-qubit state.
• They yield the maximum violations of multiparty inequalities imposed by local realistic theories Mermin, 1990; Klyshko, 1993; Gisin and Bechmann-Pasquinucci, 1998 .
• Further, their entanglement heavily relies on all parties, and, if examined pairwise, they do not contain simple bipartite entanglement see below .

2. Genuine multipartite entanglement

• The term genuine multipartite entanglement refers to states in which none of the parties can be separated from any other party in a mixture of product states.
• In general, multiparty inseparability criteria cannot be formulated in such a compact form as for two parties.
• In order to verify genuine N-party entanglement, one has to rule out any possible partially separable form.
• In principle, this can be done by considering all possible bipartite splittings or groupings and, for instance, applying the npt criterion.

3. Separability properties of Gaussian states

• As for the continuous-variable case, the criteria of Giedke, Kraus, et al. 2001c determine to which of five possible classes of fully and partially separable, and fully inseparable states a three-party three-mode Gaussian state belongs.
• In class 2, the authors have the one-mode biseparable states, in which only one particular mode is separable from the remaining pair of modes.
• A state of the form i îi,kl ̂i,mn also leads to negative partial transpose with respect to any of the four modes when the two pairs k , l and m ,n are each entangled.
• But the npt criterion is still sufficient for inseparability between any two pairs.

4. Generating entanglement

• The authors show how to generate genuine multipartite continuous-variable entanglement of arbitrarily many modes from finitely squeezed sources.
• For the quantum circuit here, a beam-splitter operation as described by Eq. 49 is a suitable substitute for the generalized controlled-NOT gate.
• 141 We see that in order to produce the minimum-energy N-mode state, the single r1 squeezer is, in terms of the squeezing factor, N−1 times as squeezed as each r2 squeezer.the authors.
• Apart from its purity, taking into account its total symmetry, the presence of any kind of partial entanglement proves the genuine N-partite entanglement of the state van Loock, 2002 .
• The separability properties of mixed versions of the three-mode state in Eq. 138 with N=3, having the same correlation matrix contaminated by some noise, Vnoisy 3 =V 3 +%1 /4, are more subtle.

5. Measuring entanglement

• In addition to the generation of entangled states, an important task is the measurement of multiparty entanglement, i.e., projection onto the basis of maximally entangled multiparty states.
• For creating continuous-variable entanglement, the authors replaced the controlled-NOT gates in Fig. 1 by appropriate beam-splitter operations.
• In other words, a projection onto the continuous-variable GHZ basis can be performed by applying an inverse N splitter followed by a Fourier transform of one mode and by subsequently measuring the positions of all modes van Loock, 2002 .
• This is in contrast to Bell- and GHZ-state analyzers for photonic qubits Lütkenhaus et al., 1999; Vaidman and Yoran, 1999; van Loock and Lütkenhaus, 2004 .
• The capabilities of such purely continuous-variable-based schemes, which rely exclusively on Gaussian operations such as beam splitting, homodyne detection, and squeezing, are not unlimited.

C. Bound entanglement

• There are two big issues related to composite mixed quantum states: separability and distillability.
• The former was the subject of the previous sections.
• The reduction criterion is sufficient for distillability, but it has been shown not to be a necessary condition Shor et al., 2001 .
• Hence the set of Gaussian states is fully explored, consisting only of npt distillable, positive partial-transpose entangled undistillable , and separable states.
• Explicit examples were constructed by Werner and Wolf 2001 .

1. Traditional EPR-type approach

• A quantum optical continuous-variable experiment more reminiscent of the original EPR paradox and distinct from tests of Bell inequalities was carried out by Ou et al.
• The smaller the deviation of x̂1 est from the true values x̂1, the better x̂1 may be determined at a distance by detecting x̂2.
• The minimal inferred variances are also referred to as conditional variances Varcond x̂ and Varcond p̂ , a measure of the noise degrading the otherwise perfect correlations between the two modes.
• In fact, the EPR condition in Eq. 146 is satisfied with the two-mode squeezed vacuum state for any nonzero squeezing r 0, since its correlation matrix in Eq. 117 using Eq. 145 yields the conditional variances Varcond x̂ = Varcond p̂ = 1 4 cosh 2r .
• Hence the two formally distinct approaches of EPR and Bell nonlocality both lead to criteria generally stricter than those for inseparability.

2. Phase-space approach

• Following Bell 1987 , an always positive Wigner function can serve as the hidden-variable probability distribution with respect to measurements corresponding to any linear combination of x̂ and p̂.
• For their analysis using photon number parity measurements, Banaszek and Wodkiewicz exploited the fact that the Wigner function is proportional to the quantum expectation value of a displaced parity operator Royer, 1977; Banaszek and Wódkiewicz, 1998 .
• The nonlocality test then simply relies on this set of two-valued measurements for each different setting.
• A similar analysis, using Eq. 148 , reveals the nonlocal N-party correlations of the multipartite entangled Gaussian N-mode state with correlation matrix in Eq. 138 van Loock and Braunstein, 2001a .
• This is different from the qubit GHZ states which show an exponential increase of the violations as the number of parties grows Mermin, 1990; Klyshko, 1993; Gisin and Bechmann-Pasquinucci, 1998 .

3. Pseudospin approach

• Alternatively and distinct from the phase-space approach of Banaszek and Wódkiewicz 1998 , one can also reveal the nonlocality of the continuous-variable states by introducing a pseudospin operator Further theoretical work on quantum nonlocality tests using homodyne-type continuous-variable measurements were published recently by Banaszek et al. 2002 and by Wenger et al. 2003 .
• Similarly, one may consider the N-party N-mode eigenstates of the parityspin operator for different orientations a ·s .
• B.3. The complete measurement of an N-mode Gaussian state is accomplished by determining the 2N 2N second-moment correlation matrix.
• The advantage of all these continuous-variable inseparability criteria is that, though still relying upon the rigorous definition of entanglement in terms of states as given in Secs. III.A and III.B, they can be easily checked via efficient homodyne detections of the quadrature operator statistics.

IV. QUANTUM COMMUNICATION WITH CONTINUOUS VARIABLES

• When using the term quantum communication, the authors refer to any protocol in which the participants’ ability to communicate is enhanced due to the exploitation of quantum features such as nonorthogonality or entanglement.
• By contrast, there are quantum subroutines Lütkenhaus, 2002 that run entirely on the quantum level.
• The coherent superposition of the signal then turns into an incoherent mixture, a process called decoherence.
• Apart from the above-mentioned quantum communication scenarios in which Alice and Bob benefit from using quantum resources, there are also fundamental results of quantum communication on the restrictions imposed by quantum theory on classical communication via quantum states.
• The continuous-variable quantum communication schemes presented below include entanglement-based protocols for quantum teleportation Sec. IV.A and super dense coding Sec. IV.B .

A. Quantum teleportation

• Quantum teleportation, in general, is the reliable transfer of quantum information through a classical communication channel using shared entanglement.
• Hence, in quantum teleportation, the state remains completely unknown to both Alice and Bob throughout the entire teleportation process.
• According to this analogy, the authors call Alice’s quadrature measurements for the teleportation of the state of a single-mode “Bell detection.”.
• As a result of the Bell detection and the entanglement between Alice’s EPR mode and Bob’s EPR mode, suitable phase-space displacements of Bob’s mode convert it into a replica of Alice’s unknown input mode a perfect replica for infinite squeezing .
• The authors shall now turn to the continuous-variable protocol for quantum teleportation in more detail.

1. Teleportation protocol

• The simplest formalism to describe continuousvariable quantum teleportation is based on the Heisenberg representation.
• This product of the measurement accuracies contains the intrinsic quantum limit, the Heisenberg uncertainty of the mode to be detected, plus an additional unit of vacuum noise due to the detection.
• The imperfection of the entanglement resource is expressed by the distortion operator, where the factors 1− n/2 are the Schmidt coefficients of the finitely squeezed, only for the nonmaximally entangled twomode squeezed vacuum state in Eq. 85 .
• Thus the teleported ensemble state, averaged over all measurement results for an ensemble of input states, becomes ̂tel = d2 tel tel , 172 corresponding to Wtel 2 in Eq. 168 .
• Another alternative formulation of nonideal continuous-variable quantum teleportation was proposed by Vukics et al. 2002 , utilizing the coherent-state basis.

2. Teleportation criteria

• The teleportation scheme with Alice and Bob is complete without any further measurement.
• The teleported state remains unknown to both Alice and Bob and need not be demolished in a detection by Bob as a final step.
• The corresponding best average fidelity if the set of input states contains all possible quantum states in a d-dimensional Hilbert space is Fav =2/ 1+d Barnum, 1998 .
• In fact, applied to the two-party teleportation scenario, the entanglement from only one squeezed state makes possible quantum teleportation with Fav 1/2 for any nonzero squeezing van Loock and Braunstein, 2000a .
• The boundary between classical and quantum teleportation defined by the criteria of Ralph and Lam 1998 differs from that in Eq. 175 in terms of fidelity.

3. Entanglement swapping

• In three optical teleportation experiments in Innsbruck Bouwmeester et al., 1997 , Rome Boschi et al., 1998 , and Pasadena Furusawa et al., 1998 , the nonorthogonal input states to be teleported were singlephoton polarization states Bouwmeester et al., 1997; Boschi et al., 1998 and coherent states Furusawa et al., 1998 .
• All these investigations have referred exclusively to discrete-variable systems.
• Van Loock and Braunstein 2000b verified entanglement swapping through a second quantum teleportation process utilizing the entangled output state.
• 179 Up to a phase-space displacement, the resulting ensemble state is the same as the projected displaced twomode squeezed state for a single shot of the continuousvariable Bell measurement.
• For any nonzero squeezing and hence entanglement in both input states, r 0 and r 0, entanglement swapping occurs, i.e., R 0.

B. Dense coding

• Dense coding aims to use shared entanglement to increase the capacity of a communication channel Bennett and Wiesner, 1997 .
• Relative to quantum teleportation, in dense coding the roles played by the quantum and classical channels are interchanged.
• It was shown that by utilizing the entanglement of a two-mode squeezed state one can always beat coherent communication based on coherent states; Braunstein and Kimble, 2000 .
• The continuous-variable scheme attains a capacity approaching in the limit of large squeezing twice that theoretically achievable in the absence of entanglement Braunstein and Kimble, 2000 .
• Before the authors Rev. Mod. Phys., Vol. 77, No. 2, April 2005 discuss how dense coding can be implemented with continuous variables, let us review the ideas behind quantifying information for communication.

1. Information: A measure

• In classical information theory one constructs a measure of information that tries to capture the “surprise” attached to receiving a particular message.
• Thus messages that are common occurrences are assumed to contain very little useful information, whereas rare messages are deemed to be valuable and to contain more information.
• This concept suggests that the underlying symbols or letters or alphabet used to transmit the message are themselves unimportant, but not the probabilities of these symbols or messages.
• In addition to this conceptual framework, it turns out that the measure of information is essentially unique if one takes it to be additive for independent messages.
• 180 This result is the unique measure of average information per letter.

2. Mutual information

• In order to quantify the information in a communication channel the authors must introduce a measure of information corresponding to the amount of information accessible to the receiver which contains information about the message sent.
• The idea behind this equation is that the sum I A +I B accounts for the joint information in both alphabets, but double counts that part which is mutual to both alphabets.
• By subtracting the correct expression for the joint information I A ,B the authors are left solely with the information that is common or mutual.

3. Classical communication

• The authors are now in a position to apply their expression for the mutual information to quantify the information received through a communication channel that contains information or that is mutual or common to the information actually sent.
• For simplicity, the authors shall suppose that the channel has no memory, so that each signal sent is independent of earlier or later channel usage.
• The authors may characterize the channel by the conditional probabilities pb a, for the probability of observing letter b in Bob’s alphabet, given that Alice sent letter a.
• The joint probability is therefore given by pab = pb apa.
• 184 If the authors optimize this expression over Alice’s alphabet, they can determine the maximum achievable throughput per usage.

4. Classical communication via quantum states

• Ultimately, Alice must use some physical carrier to represent the letters she sends.
• When Bob is presented with a state ̂a representing letter a from Alice’s alphabet, he will find instead letter b from his own alphabet with a conditional probability given by Eq. 160 , from which one may compute the mutual information I A :B using Eq. 184 .
• The authors note that either bound is independent of Bob’s measurement strategy, so an achievable upper bound will allow us to determine the channel capacity for transmitting classical information using quantum states.
• For this constraint, the maximum entropy may be interpreted as the channel capacity achieved when Alice uses an alphabet of number states distributed according to a thermal distribution Yuen and Ozawa, 1993; Caves and Drummond, 1994 .
• Figure 4 b shows the channel with the same constraint operating with an input alphabet of coherent states and a heterodyne detection.

5. Dense coding

• In dense coding, Alice and Bob communicate via two channels, however, Alice only needs to modulate one of them.
• As shown in Fig. 5, signal modulation is performed only on Alice’s mode, with the second mode treated as an overall shared resource by Alice and Bob.
• It is worth noting that this dense-coding scheme does not always beat the optimal single-channel capacity.
• A fairer comparison is against single-mode coherentstate communication with heterodyne detection.
• In the limit of strong down conversion and using continuousvariable entanglement, much higher efficiency should be achievable.

C. Quantum error correction

• Let us now proceed with a few remarks on an alternative method for the reliable transmission of quantum information, which is not based on shared entanglement such as quantum teleportation combined with entanglement distillation the latter is the subject of Sec. IV.E .
• In a quantum error-correction scheme used for communication purposes, quantum states are sent directly through a potentially noisy channel after encoding them into a larger system that contains additional auxiliary subsystems.
• It later turned out that continuous-variable quantum error-correction codes also can be implemented using only linear optics and resources of squeezed light Braunstein, 1998b .
• The advantage of this variation over the codes described above is that the codes from Gottesman et al.
• 2001 allow effective protection against small “diffusive” errors, which are closer to typical realistic loss mechanisms.

D. Quantum cryptography

• The authors give an overview of the various proposals of continuous-variable quantum cryptography or quantum key distribution .
• The authors further discuss absolute theoretical security and verification of the experimental security of continuous-variable quantum key distribution.
• Finally, the authors conclude this section with a few remarks on quantum secret sharing with continuous variables.

1. Entanglement-based versus prepare and measure

• For qubit-based quantum cryptography there have been two basic schemes.
• The protocols without entanglement may be termed “prepare and measure” schemes, in which Alice randomly prepares a sequence of nonorthogonal states to be sent to Bob and Bob measures these states in a randomly chosen basis.
• Thus a first test of secure quantum key distribution is to check for optimal entanglement witnesses observables that detect entanglement , given a set of local operations and a corresponding classical distribution P A ,B Curty et al., 2004 .
• Even if there are no such violations, a suitable entanglement witness to prove the presence of quantum correlations may still be found.
• In the continuous-variable case, a particularly practical witness is given by the Duan criterion in Eq. 110 or Eq. 114 , based solely upon efficient homodyne detection.

2. Early ideas and recent progress

• The schemes that do not rely on entanglement are mostly based on alphabets involving nonorthogonal coherent states as the signal states.
• He did, however, study, two kinds of eavesdropper attack: man-in-the-middle or intercept-resend attacks measuring a single quadrature; and quantum-tap attacks using a beam splitter, after which again only a single quadrature is measured.
• 2003 that this protocol is, in principle, secure for any value of the line transmission rate.
• Another promising method for beating the 3-dB loss limit is based on a postselection procedure Silberhorn, Ralph, et al., 2002 .

3. Absolute theoretical security

• From the single-wave-packet noncollective attacks considered above there has been great progress recently for continuous-variable quantum cryptography.
• A detailed proof of absolute theoretical security for one scheme Gottesman and Preskill, 2001; Gottesman et al., 2001 stand out.
• Then given provable bounds to the quantity of information the eavesdropper can have about the key, classical error-correction codes and classical privacy amplification are used to reduce this quantity by any desired amount.
• The remaining issues appear to be as follows: 1 Reanalysis of this proof in a broadband context.
• In cw operation the signal switching limitations must be accounted for in addition to limitations in the detection process.

4. Verifying experimental security

• The authors have seen that there are already approaches to theoretical proofs for absolute security.
• To find out, it makes most sense to take seriously the position of devil’s advocate in the laboratory and work towards serious eavesdropping scenarios in order to put the intended ideally secure schemes through their paces.
• This translation is considered in Sec. V on the cloning of continuous-variable quantum states.
• If true this would give freedom in the approaches taken to implement any final scheme.
• Questions still remain about the translation of theoretical protocols into real implementations and whether new loopholes will not be created during this phase.

5. Quantum secret sharing

• Quantum secret sharing can be thought of as a multiparty generalization of quantum cryptography in which a message is not only protected against potential eavesdroppers, but can only be retrieved from several people who collaborate.
• In the former case, a key can be established between all participants, and using the key requires that all participants work together.
• An eavesdropper would introduce errors and could be detected.
• In fact, their use is equivalent to the two-party senderreceiver scenario when all participants except for the sender team up and share information about local momentum measurements to yield a total “receiver momentum.”.
• Continuous-variable secret sharing of quantum information was proposed by Tyc and Sanders 2002 .

E. Entanglement distillation

• In order to transfer quantum information reliably in the presence of loss, quantum teleportation must be combined with entanglement distillation protocols.
• Entanglement swapping may serve as an entanglement concentration protocol capable of turning two copies of a nonmaximally entangled state into one maximally entangled copy with nonzero probability Bose et al., 1999 .
• As for the general distillation of entanglement including purification, a continuous-variable protocol was proposed by Duan, Giedke, et al. 2000b based on local Rev. Mod.
• Moreover, in this scheme, the distilled entangled states end up in a finite-dimensional Hilbert space.
• Through this kind of protocol, proposed by Browne et al. 2003 and by Eisert et al.

F. Quantum memory

• A way of storing continuous quantum information is a crucial component of a fully integrated technology.
• If the ensemble is highly polarized, then small excursions of the collective spin away from some fixed axis will mimic the phasespace structure of a harmonic oscillator.
• Thus only second-order transitions can produce any effect, leading to an effective Hamiltonian Brune et al., 1992 Ĥeff ŜzF̂z.
• It has been noted that this yields a quantum nondemo- lition probe of F̂z of the atomic sample Happer and Mathur, 1967 .
• A beam-splitter-like coupling may be obtained, for example, between F̂y and Ŝz when the atomic sample is highly polarized along the x axis and for a strongly x-polarized optical beam.

V. QUANTUM CLONING WITH CONTINUOUS VARIABLES

• The authors investigate the consequences of the famous quantum no-cloning theorem, independently found by Wootters and Zurek 1982 and by Dieks 1982 , for continuous quantum variables.
• As mentioned in Sec. IV.D, a potential application of continuousvariable quantum cloning is to implement eavesdropping strategies for continuous-variable quantum cryptography.

A. Local universal cloning

• The authors now consider the possibility of approximately copying an unknown quantum state at a given location using a particular sequence of unitary transformations a quantum circuit .
• Entanglement as a potentially nonlocal resource is therefore not necessarily needed, but it might be an ingredient at the intermediate steps of the cloning circuit.

1. Beyond no-cloning

• The no-cloning theorem, originally derived for qubits, in general forbids exact copying of unknown nonorthogonal or simply arbitrary quantum states Dieks, 1982; Wooters and Zurek, 1982 .
• The first papers that went “beyond the no-cloning theorem” and considered the possibility of approximately copying nonorthogonal Rev. Mod.
• Based on these results, a cloning experiment has been proposed for qubits encoded as singlephoton states Simon et al., 2000 , and two other optical qubit cloning experiments have already been realized Martini and Mussi, 2000; Huang et al., 2001 .
• For this optimum cloning fidelity, the authors use the superscript “univ” to indicate that any d-dimensional quantum state is universally copied with the same fidelity.

2. Universal cloners

• A universal cloner is capable of optimally copying arbitrary quantum states with the same fidelity independent of the particular input state.
• It follows directly from the covariant form of the above density operators that the fidelity of the information transfer is input-state independent.
• 208 8It is pointed out by Werner 1998 that the “constructive” approach the coupling of the input system with an apparatus or “ancilla” described by a unitary transformation, and then tracing out the ancilla consists of completely positive tracepreserving CPTP operations.
• Then, on average, with a small overlap between the original input state and the random state, the two output clones have a cloning fidelity of 1/2 Braunstein, Bužek, and Hillery, 2001 .
• These boundaries, the optimal cloning fidelities, can in fact be attained by means of a single family of quantum circuits.

1. Fidelity bounds for Gaussian cloners

• In the first papers that considered continuous-variable cloning, the set of input states to be copied was re- stricted to Gaussian states Cerf and Iblisdir, 2000a; Cerf et al., 2000 .
• In fact, the resulting fidelities do not depend on the particular values of 0 and 0.
• In fact, analogously to the qubit case Bruß, Ekert, and Macchiavello, 1998 , the optimal measurement optimal state estimate turns out to be the optimal N→ cloner, and hence ̄clon N , = ̄meas N =1/2N.
• This requires knowledge about the input state’s squeezing, making the cloner state dependent when applied to all Gaussian states.
• For arbitrary coherent states, the optimal Gaussian cloner adds an excess noise ̄clon N ,M to the input state without changing its mean amplitude; the coherent-state copies are all in the same mixed state.

2. An optical cloning circuit for coherent states

• So far, the authors have only discussed the fidelity boundaries for the N→M coherent-state cloner.
• In the case of coherent states, implementing an N→M symmetric cloning transformation that attains Eq. 210 only requires a phaseinsensitive linear amplifier and a series of beam splitters Braunstein, Cerf, et al., 2001; Fiurášek, 2001 .
• The optimal duplication can be implemented in two steps via two canonical transformations, 10The optimal cloning via minimizing excess noise variances, as discussed here, obviously only refers to Gaussian cloners based solely on Gaussian operations.
• This one extra unit is indeed the optimal minimal amount according to Eq. 218 for N=1 and M=2.

VI. QUANTUM COMPUTATION WITH CONTINUOUS VARIABLES

• The authors now consider the necessary and sufficient conditions for constructing a universal quantum computer using continuous variables.
• As an example, it is shown how a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using linear optics, squeezers, and at least one further nonlinear optical element such as the Kerr effect.

A. Universal quantum computation

• At first sight therefore it might seem that quantum computation over continuous variables would be an ill-defined concept.
• Ĥ, x̂, and p̂ imply that the addition of ±Ĥ to the set of operations that can be applied allows the construction of Hamiltonians of the form aĤ+bx̂+cp̂+d.
• Since any polynomial of order M+1 can be constructed from monomials of order M+1 and lower, by applying linear operations and a single nonlinear operation a finite number of times one can construct polynomials of arbitrary order in x̂ and p̂ to any desired degree of accuracy.
• Let us now turn to more than one variable, e.g., the case of an interferometer in which many modes of the electromagnetic field interact.
• Note that in contrast to the case of qubits, in which a nonlinear coupling between qubits is required to perform universal quantum computation, in the continuous case only single variable nonlinearities are required, along with linear couplings between the variables.

B. Extension of the Gottesman-Knill theorem

• Quantum mechanics allows for information processing that could not be performed classically.
• This result helps us understand what algorithms performed by a continuousvariable quantum computer may be efficiently simulated by a conventional classical computer.
• The states used in continuousvariable experiments are approximations to the idealized computational basis.
• The authors define the SUM gate as the continuous-variable analog of the controlled-NOT gate.
• An efficient classical simulation involves simulating the statistics of linear combinations of Pauli operator generators.

VII. EXPERIMENTS WITH CONTINUOUS QUANTUM VARIABLES

• The authors discuss some experiments based on continuous quantum variables.
• These include the generation of squeezed-state EPR entanglement via optical parametric amplification and via the Kerr effect.
• Qualitatively different manifestations of continuous-variable entanglement are that between two atomic ensembles, created in an experiment in Copenhagen Julsgaard et al., 2001 , and that between more than two optical modes, experimentally generated and verified in Tokyo for three modes Aoki et al., 2003 .
• Quantum teleportation of coherent states has been achieved already in Pasadena Furusawa et al., 1998; Zhang et al., 2003 and in Canberra Bowen, Treps, et al., 2003 .
• The authors shall briefly show how to describe these experiments in a realistic broadband fashion.

A. Generation of squeezed-state EPR entanglement

• The generation of discrete-variable qubit entanglement can be achieved experimentally via weak down conversion producing polarization-entangled single pho- tons.
• Since successful postselected events occur very rarely, one has to cope with very low efficiency in these single-photon schemes.
• In the continuous-variable setting, the generation of entanglement, for instance, occurring every inverse bandwidth time at the output of an optical parametric amplifier, is more efficient than in the single-photon schemes.
• When making an entangled two-mode squeezed state, one need not exclude the vacuum contribution that originates from the down-conversion source via postselection.
• The first experiment to produce continuous-variable broadband EPR correlations of this kind was performed by Ou et al.

1. Broadband entanglement via optical parametric amplification

• A broadband entangled state is generated either directly by nondegenerate optical parametric amplification in a cavity also called nondegenerate parametric down conversion or by combining at a beam splitter two independently squeezed fields produced via degenerate down conversion.
• Just as for the two discrete modes in Eqs. 88 and 89 , this can be easily seen in the “continuum” representation Caves and Schumaker, 1985 of the quadrature operators, Rev. Mod.
• Here, the annihilation and creation operators now no longer dimensionless, but each in units of root time, t satisfy the commutation relation b̂ , b̂† = − .
• As a result of the nonlinear optical interaction, a pump photon at frequency 2 0 can be annihilated to create two photons at the frequencies 0±' and, conversely, two photons can be annihilated to create a pump photon.
• In general, Eq. 256 may define arbitrary squeezing spectra of two statistically identical but independent broadband squeezed states.

2. Kerr effect and linear interference

• The first light-squeezing experiment was published in 1985 Slusher et al., 1985 .
• Squeezed light was generated via four-wave mixing.
• Using stochastic equations to describe the classical propagation plus the evolution of quantum noise in a fiber, Carter et al. 1987 proposed the squeezing of quantum fiber solitons.
• This state corresponds to a photon number squeezed state with sub-Poissonian statistics, as opposed to the ordinary quadrature squeezed state.
• In fact, the fiber Kerr nonlinearity is so small that the radius of curvature of the “banana” state is far larger than its length.

B. Generation of long-lived atomic entanglement

• B.4, the authors discussed how to make entanglement from sources of nonclassical light such as squeezed states using a network of beam splitters.
• Otherwise, if all the input modes are in a vacuum or coherent state, the output state that emerges from the beam splitters will always remain separable.
• The creation of this entanglement between material objects is an important step towards storing quantum information in an optical communication protocol and proves the feasibility of using light-atom quantum interfaces in a similar approach.
• Upon repeating this procedure with a different light pulse, but now measuring the total y spin in a quantum nondemolition fashion which will not change the previously measured value of the total z spin , both the total z spin and the total y spin may be precisely determined.
• The generated entangled state was maintained for more than 0.5 ms.

C. Generation of genuine multipartite entanglement

• The authors have seen that a particularly efficient way to generate entanglement between electromagnetic modes is to let squeezed light beams interfere using linear optics.
• The generation of tripartite entanglement, the entanglement between three optical modes, only requires combining three input modes at two beam splitters, where at least one of these input modes is in a squeezed state see Sec. III.
• It is sufficient to detect a set of suitable linear combinations of the quadratures see Sec. III.E .
• These must contain the positions and momenta of all modes involved.
• For verification, the variances of the entangled state’s relative positions and total momentum were measured.

D. Quantum teleportation of coherent states

• In the discrete-variable teleportation experiments in Innsbruck Bouwmeester et al., 1997 and in Rome Boschi et al., 1998 , the teleported states were single-photon polarization states.
• As for the transverse structure and the polarization of the input field, the authors as- Rev. Mod.
• Using the broadband EPR state of Eq. 258 for her Bell detection, Alice combines mode 1 with the unknown input field at a 50:50 beam splitter.
• P̂v † ' ei't , 267 assuming a noiseless, classical local oscillator and with hel ' representing the detectors’ responses within their electronic bandwidths 'el: hel ' =1 for '$ 'el and zero otherwise.
• In a second coherent-state teleportation experiment in Pasadena Zhang et al., 2003 , a fidelity of F=0.61±0.02 was achieved, which is a slight improvement over the first experiment.

E. Experimental dense coding

• As described in Sec. IV.B, rather than the reliable transfer of quantum information through a classical channel via quantum teleportation, dense coding aims at transmitting classical information more efficiently using a quantum channel.
• In a dense coding scheme, the amount of classical information transmitted from Alice to Bob is increased when Alice sends her half of a preshared entangled state through a quantum channel to Bob.
• In the dense-coding experiment of Li, Pan, et al. 2002 , bright EPR beams were employed, similar to the entanglement created by Silberhorn et al.
• The noise background in the signal channel, measured by Li, Pan, et al., without exploiting the correlations with the other EPR beam, was about 4.4 dB above the corresponding vacuum limit.
• In the next section, the authors shall turn to a non-entanglement-based continuousvariable quantum key distribution protocol, experimentally demonstrated by the Grangier group Grosshans et al., 2003 .

F. Experimental quantum key distribution

• In Sec. IV.D, the authors gave an overview of the various proposals for continuous-variable quantum key distribution.
• As for experimental progress, a BB84-like entanglement-free quantum cryptography scheme was implemented by Hirano et al.
• In the experiment of Grosshans et al. 2003 , the mutual information between all participants, Alice, Bob, and Eve, was experimentally determined for different values of the line transmission, in particular, including losses of 3.1 dB.
• Phys., Vol. 77, No. 2, April 2005 information-theoretic condition in Eq. 203 , which is sufficient for secure key extraction using privacy amplification and error-correction techniques.
• The signal pulses in the experiment of Grosshans et al.

G. Demonstration of a quantum memory effect

• The creation of long-lived atomic entanglement, as described in Sec. VII.B, is a first step towards storing optical quantum information in atomic states for extended periods and hence implementing light-atom quantum interfaces.
• The atom-light interaction employed in the experiment is again based on coupling of the quantum nondemolition type between the atomic spin and the polarization state of light, as described in Sec. IV.F.
• For this last step, one can exploit the fact that Ŝy out in Eq. 272 is more sensitive to F̂z in due to the squeezing of Ŝy in .
• For Gaussian signal states, this corresponds to reproducing values of the output variances sufficiently close to those of the input variances, similar to the verification of high-fidelity quantum teleportation.
• The experiment by Schori et al. 2002 , however, represented a significant step towards full quantum memory, because it was shown that long-lived atomic spin ensembles may serve as storage for optical quantum information sensitive enough to store fields containing just a few photons.

VIII. CONCLUDING REMARKS

• The field of quantum information has typically concerned itself with the manipulation of discrete systems Rev. Mod.
• For some tasks continuous-variable quantum computers are nonetheless more efficient.
• Similarly, true quantum coding would require a non-Gaussian decoding step at the receiving end.
• The experiments accomplished so far in continuousvariable quantum information reflect the observations of the preceding paragraphs.
• Beyond this border, techniques from the more traditional single-photon-based discretevariable domain will have to be incorporated into the continuous-variable approaches.

ACKNOWLEDGMENTS

• S.L.B. currently holds a Wolfson-Royal Society Research Merit Award.
• This work was funded in part under Project No. QUICOV as part of the IST-FET-QJPC program.
• He acknowledges the financial support of the DFG under the Emmy-Noether program.

Figures

FIG. 6. Implementation of a 1→2 continuous-variable cloning machine. LA stands for linear amplifier, and BS represents a phase-free 50:50 beam splitter.

FIG. 2. Teleportation of a single mode of the electromagnetic field. Alice and Bob share the entangled state of modes 1 and 2. Alice combines the mode “in” to be teleported with her half of the Einstein-Podolsky-Rosen EPR state at a beam splitter. The homodyne detectors Dx and Dp yield classical photocurrents for the quadratures xu and pv, respectively. Bob performs phase-space displacements of his half of the EPR state depending on Alice’s classical results. The entangled state EPR, eventually shared by Alice and Bob, may be produced by combining two single-mode squeezed states at a beam splitter or directly via a nonlinear two-mode squeezing interaction.

FIG. 4. Communication channel with a optimal number-state alphabet; b coherent-state alphabet; and c squeezed-state alphabet.

FIG. 1. Quantum circuit for generating the N-qubit Greenberger-Horne-Zeilinger GHZ state. The gates unitary transformations are a Hadamard gate “H” and pairwiseacting controlled-NOT gates.

FIG. 3. Verification of quantum teleportation. The verifier “Victor” is independent of Alice and Bob. Victor prepares the input states which are known to him, but unknown to Alice and Bob. After a supposed quantum teleportation from Alice to Bob, the teleported states are given back to Victor. Due to his knowledge of the input states, Victor can compare the teleported states with the input states.

FIG. 5. Schematic of dense-coding scheme: An EPR pair is created and one-half sent to Bob. The other half is given to Alice, who modulates it by performing a phase-space translation by the amount . This half is now passed on to Bob, who recombines the two halves and performs a pair of homodyne measurements on the sum and difference fields. These signals 1R and 2I will closely mimic for a sufficiently strongly entangled state.

Full text

Quantum information with continuous variables
Samuel L. Braunstein
Computer Science, University of York, York YO10 5DD, United Kingdom
Peter van Loock
National Institute of Informatics (NII), Tokyo 101-8430, Japan and Institute of Theoretical
Physics, Institute of Optics, Information and Photonics (Max-Planck Forschungsgruppe),
Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
共Published 29 June 2005兲
Quantum information is a rapidly advancing area of interdisciplinary research. It may lead to
real-world applications for communication and computation unavailable without the exploitation of
quantum properties such as nonorthogonality or entanglement. This article reviews the progress in
quantum information based on continuous quantum variables, with emphasis on quantum optical
implementations in terms of the quadrature amplitudes of the electromagnetic field.
CONTENTS
I. Introduction 513
II. Continuous Variables in Quantum Optics 516
A. The quadratures of the quantized field 516
B. Phase-space representations 518
C. Gaussian states 519
D. Linear optics 519
E. Nonlinear optics 520
F. Polarization and spin representations 522
G. Necessity of phase reference 523
III. Continuous-Variable Entanglement 523
A. Bipartite entanglement 525
1. Pure states 525
2. Mixed states and inseparability criteria 526
B. Multipartite entanglement 529
1. Discrete variables 529
2. Genuine multipartite entanglement 530
3. Separability properties of Gaussian states 530
4. Generating entanglement 531
5. Measuring entanglement 533
C. Bound entanglement 534
D. Nonlocality 534
1. Traditional EPR-type approach 535
2. Phase-space approach 536
3. Pseudospin approach 536
E. Verifying entanglement experimentally 537
IV. Quantum Communication with Continuous Variables 538
A. Quantum teleportation 540
1. Teleportation protocol 541
2. Teleportation criteria 543
3. Entanglement swapping 546
B. Dense coding 546
1. Information: A measure 547
2. Mutual information 547
3. Classical communication 547
4. Classical communication via quantum states 547
5. Dense coding 548
C. Quantum error correction 550
D. Quantum cryptography 550
1. Entanglement-based versus prepare and
measure 550
2. Early ideas and recent progress 551
3. Absolute theoretical security 552
4. Verifying experimental security 553
5. Quantum secret sharing 553
E. Entanglement distillation 554
F. Quantum memory 555
V. Quantum Cloning with Continuous Variables 555
A. Local universal cloning 555
1. Beyond no-cloning 555
2. Universal cloners 556
B. Local cloning of Gaussian states 557
1. Fidelity bounds for Gaussian cloners 557
2. An optical cloning circuit for coherent
states 558
C. Telecloning 559
VI. Quantum Computation with Continuous Variables 560
A. Universal quantum computation 560
B. Extension of the Gottesman-Knill theorem 563
VII. Experiments with Continuous Quantum Variables 565
A. Generation of squeezed-state EPR entanglement 565
1. Broadband entanglement via optical
parametric amplification 565
2. Kerr effect and linear interference 567
B. Generation of long-lived atomic entanglement 568
C. Generation of genuine multipartite entanglement 569
D. Quantum teleportation of coherent states 569
E. Experimental dense coding 570
F. Experimental quantum key distribution 571
G. Demonstration of a quantum memory effect 572
VIII. Concluding Remarks 572
Acknowledgments 573
References 573
I. INTRODUCTION
Quantum information is a relatively young branch of
physics. One of its goals is to interpret the concepts of
quantum physics from an information-theoretic point of
view. This may lead to a deeper understanding of quan-
REVIEWS OF MODERN PHYSICS, VOLUME 77, APRIL 2005
0034-6861/2005/77共2兲/513共65兲/$50.00 ©2005 The American Physical Society513

tum theory. Conversely, information and computation
are intrinsically physical concepts, since they rely on
physical systems in which information is stored and by
means of which information is processed or transmitted.
Hence physical concepts, and at a more fundamental
level quantum physical concepts, must be incorporated
in a theory of information and computation. Further-
more, the exploitation of quantum effects may even
prove beneficial for various kinds of information pro-
cessing and communication. The most prominent ex-
amples of this are quantum computation and quantum
key distribution. Quantum computation means in par-
ticular cases, in principle, computation faster than any
known classical computation. Quantum key distribution
makes possible, in principle, unconditionally secure
communication as opposed to communication based on
classical key distribution.
From a conceptual point of view, it is illuminating to
consider continuous quantum variables in quantum in-
formation theory. This includes the extension of quan-
tum communication protocols from discrete to continu-
ous variables and hence from finite to infinite
dimensions. For instance, the original discrete-variable
quantum teleportation protocol for qubits and other
finite-dimensional systems 共Bennett et al., 1993兲 was
soon after its publication translated into the continuous-
variable setting 共Vaidman, 1994兲. The main motivation
for dealing with continuous variables in quantum infor-
mation, however, originated in a more practical observa-
tion: efficient implementation of the essential steps in
quantum communication protocols, namely, preparing,
unitarily manipulating, and measuring 共entangled兲 quan-
tum states, is achievable in quantum optics utilizing con-
tinuous quadrature amplitudes of the quantized electro-
magnetic field. For example, the tools for measuring a
quadrature with near-unit efficiency or for displacing an
optical mode in phase space are provided by homodyne-
detection and feedforward techniques, respectively.
Continuous-variable entanglement can be efficiently
produced using squeezed light 关in which the squeezing
of a quadrature’s quantum fluctuations is due to a non-
linear optical interaction 共Walls and Milburn, 1994兲兴 and
linear optics.
A valuable feature of quantum optical implementa-
tions based upon continuous variables, related to their
high efficiency, is their unconditionalness. Quantum re-
sources such as entangled states emerge from the non-
linear optical interaction of a laser with a crystal 共supple-
mented if necessary by some linear optics兲 in an
unconditional fashion, i.e., every inverse bandwidth
time. This unconditionalness is hard to obtain in
discrete-variable qubit-based implementations using
single-photon states. In that case, the desired prepara-
tion due to the nonlinear optical interaction depends on
particular 共coincidence兲 measurement results ruling out
the unwanted 共in particular, vacuum兲 contributions in
the outgoing state vector. However, the unconditional-
ness of the continuous-variable implementations has its
price: it is at the expense of the quality of the entangle-
ment of the prepared states. This entanglement and
hence any entanglement-based quantum protocol is al-
ways imperfect, the degree of imperfection depending
on the amount of squeezing of the laser light involved.
Good quality and performance require large squeezing
which is technologically demanding, but to a certain ex-
tent 关about 10 dB 共Wu et al., 1986兲兴 already state of the
art. Of course, in continuous-variable protocols that do
not rely on entanglement, for instance, coherent-state-
based quantum key distribution, these imperfections do
not occur.
To summarize, in the most commonly used optical ap-
proaches, the continuous-variable implementations al-
ways work pretty well 共and hence efficiently and uncon-
ditionally兲, but never perfectly. Their discrete-variable
counterparts only work sometimes 共conditioned upon
rare successful events兲, but they succeed, in principle,
perfectly. A similar tradeoff occurs when optical quan-
tum states are sent through noisy channels 共optical fi-
bers兲, for example, in a realistic quantum key distribu-
tion scenario. Subject to losses, the continuous-variable
states accumulate noise and emerge at the receiver as
contaminated versions of the sender’s input states. The
discrete-variable quantum information encoded in
single-photon states is reliably conveyed for each photon
that is not absorbed during transmission.
Due to the recent results of Knill, Laflamme, and Mil-
burn 共Knill et al., 2001兲, it is now known that efficient
quantum information processing is possible, in principle,
solely by means of linear optics. Their scheme is formu-
lated in a discrete-variable setting in which the quantum
information is encoded in single-photon states. Apart
from entangled auxiliary photon states, generated off-
line without restriction to linear optics, conditional dy-
namics 共
feedforward兲 is the essential ingredient in mak-
ing this approach work. Universal quantum gates such as
a controlled-
NOT gate can, in principle, be built using
this scheme without need of any Kerr-type nonlinear op-
tical interaction 共corresponding to an interaction Hamil-
tonian quartic in the optical modes’ annihilation and
creation operators兲. This Kerr-type interaction would be
hard to obtain on the level of single photons. However,
the off-line generation of the complicated auxiliary
states needed in the Knill-Laflamme-Milburn scheme
seems impractical too.
Similarly, in the continuous-variable setting, when it
comes to more advanced quantum information proto-
cols, such as universal quantum computation or, in a
communication scenario, entanglement distillation, it
turns out that tools more sophisticated than mere
Gaussian operations are needed. In fact, the Gaussian
operations are effectively those described by interaction
Hamiltonians at most quadratic in the optical modes’
annihilation and creation operators, thus leading to lin-
ear input-output relations as in beam-splitter or squeez-
ing transformations. Gaussian operations, mapping
Gaussian states onto Gaussian states, also include ho-
modyne detections and phase-space displacements. In
contrast, the non-Gaussian operations required for ad-
vanced continuous-variable quantum communication 共in
particular, long-distance communication based on en-
514
S. L. Braunstein and P. van Loock: Quantum information with continuous variables
Rev. Mod. Phys., Vol. 77, No. 2, April 2005

tanglement distillation and swapping, quantum memory,
and teleportation兲 are due either to at least cubic non-
linear optical interactions or to conditional transforma-
tions depending on non-Gaussian measurements such as
photon counting. It seems that, at this very sophisticated
level, the difficulties and requirements of the discrete-
and continuous-variable implementations are analogous.
In this review, our aim is to highlight the strengths of
the continuous-variable approaches to quantum infor-
mation processing. Therefore we focus on those proto-
cols that are based on Gaussian states and their feasible
manipulation through Gaussian operations. This leads to
continuous-variable proposals for the implementation of
the simplest quantum communication protocols, such as
quantum teleportation and quantum key distribution,
and includes the efficient generation and detection of
continuous-variable entanglement.
Before dealing with quantum communication and
computation, in Sec. II, we first introduce continuous
quantum variables within the framework of quantum
optics. The discussions about the quadratures of quan-
tized electromagnetic modes, about phase-space repre-
sentations, and about Gaussian states include the nota-
tions and conventions that we use throughout this
article. We conclude Sec. II with a few remarks on linear
and nonlinear optics, on alternative polarization and
spin representations, and on the necessity of a phase
reference in continuous-variable implementations. The
notion of entanglement, indispensable in many quantum
protocols, is described in Sec. III in the context of con-
tinuous variables. We discuss pure and mixed entangled
states, entanglement between two 共bipartite兲 and be-
tween many 共multipartite兲 parties, and so-called bound
共undistillable兲 entanglement. The generation, measure-
ment, and verification 共both theoretical and experimen-
tal兲 of continuous-variable entanglement are here of par-
ticular interest. As for the properties of the continuous-
variable entangled states related with their
inseparability, we explain how the nonlocal character of
these states is revealed. This involves, for instance, vio-
lations of Bell-type inequalities imposed by local real-
ism. Such violations, however, cannot occur when the
measurements considered are exclusively of continuous-
variable type. This is due to the strict positivity of the
Wigner function of the Gaussian continuous-variable en-
tangled states, which allows for a hidden-variable de-
scription in terms of the quadrature observables.
In Sec. IV, we describe the conceptually and practi-
cally most important quantum communication protocols
formulated in terms of continuous variables and thus
utilizing the continuous-variable 共entangled兲 states.
These schemes include quantum teleportation and en-
tanglement swapping 共teleportation of entanglement兲,
quantum 共super兲dense coding, quantum error correc-
tion, quantum cryptography, and entanglement distilla-
tion. Since quantum teleportation based on nonmaxi-
mum continuous-variable entanglement, using finitely
squeezed two-mode squeezed states, is always imperfect,
teleportation criteria are needed both for the theoretical
and for the experimental verification. As is known from
classical communication, light, propagating at high
speed and offering a broad range of different frequen-
cies, is an ideal carrier for the transmission of informa-
tion. This applies to quantum communication as well.
However, light is less suited for the storage of informa-
tion. In order to store quantum information, for in-
stance, at the intermediate stations in a quantum re-
peater, atoms are more appropriate media than light.
Significantly, as another motivation to deal with continu-
ous variables, a feasible light-atom interface can be built
via free-space interaction of light with an atomic en-
semble based on the alternative polarization and spin-
type variables. No strong cavity QED coupling is needed
as with single photons. The concepts of this transfer of
quantum information from light to atoms and vice versa,
as the essential ingredients of a quantum memory, are
discussed in Sec. IV.F
Section V is devoted to quantum cloning with con-
tinuous variables. One of the most fundamental 共and
historically one of the first兲 “laws” of quantum informa-
tion theory is the so-called no-cloning theorem 共Dieks,
1982; Wootters and Zurek, 1982兲. It forbids the exact
copying of arbitrary quantum states. However, arbitrary
quantum states can be copied approximately, and the
resemblance 共in mathematical terms, the overlap or fi-
delity兲 between the clones may attain an optimal value
independent of the original states. Such optimal cloning
can be accomplished locally by sending the original
states 共together with some auxiliary system兲 through a
local unitary quantum circuit. Optimal cloning of Gauss-
ian continuous-variable states appears to be more inter-
esting than that of general continuous-variable states,
because the latter can be mimicked by a simple coin
toss. We describe a non-entanglement-based implemen-
tation for the optimal local cloning of Gaussian
continuous-variable states. In addition, for Gaussian
continuous-variable states, an optical implementation
exists of optimal cloning at a distance 共telecloning兲.In
this case, the optimality requires entanglement. The cor-
responding multiparty entanglement is again producible
with nonlinear optics 共squeezed light兲 and linear optics
共beam splitters兲.
Quantum computation over continuous variables, dis-
cussed in Sec. VI, is a more subtle issue than the in some
sense straightforward continuous-variable extensions of
quantum communication protocols. At first sight, con-
tinuous variables do not appear well suited for the pro-
cessing of digital information in a computation. On the
other hand, a continuous-variable quantum state having
an infinite-dimensional spectrum of eigenstates contains
a vast amount of quantum information. Hence it might
be promising to adjust the continuous-variable states
theoretically to the task of computation 共for instance, by
discretization兲 and yet to exploit their continuous-
variable character experimentally in efficient 共optical兲
implementations. We explain in Sec. VI why universal
quantum computation over continuous variables re-
quires Hamiltonians at least cubic in the position and
momentum 共quadrature兲 operators. Similarly, any quan-
tum circuit that consists exclusively of unitary gates from
515
S. L. Braunstein and P. van Loock: Quantum information with continuous variables
Rev. Mod. Phys., Vol. 77, No. 2, April 2005

the continuous-variable Clifford group can be efficiently
simulated by purely classical means. This is a
continuous-variable extension of the discrete-variable
Gottesman-Knill theorem in which the Clifford group
elements include gates such as the Hadamard 共in the
continuous-variable case, Fourier兲 transform or the con-
trolled
NOT 共CNOT兲. The theorem applies, for example,
to quantum teleportation which is fully describable by
CNOT’s and Hadamard 共or Fourier兲 transforms of some
eigenstates supplemented by measurements in that
eigenbasis and spin or phase flip operations 共or phase-
space displacements兲.
Before some concluding remarks in Sec. VIII, we
present some of the experimental approaches to squeez-
ing of light and squeezed-state entanglement generation
in Sec. VII.A. Both quadratic and quartic optical nonlin-
earities are suitable for this, namely, parametric down
conversion and the Kerr effect, respectively. Quantum
teleportation experiments that have been performed al-
ready based on continuous-variable squeezed-state en-
tanglement are described in Sec. VII.D. In Sec. VII, we
further discuss experiments with long-lived atomic en-
tanglement, with genuine multipartite entanglement of
optical modes, experimental dense coding, experimental
quantum key distribution, and the demonstration of a
quantum memory effect.
II. CONTINUOUS VARIABLES IN QUANTUM OPTICS
For the transition from classical to quantum mechan-
ics, the position and momentum observables of the par-
ticles turn into noncommuting Hermitian operators in
the Hamiltonian. In quantum optics, the quantized elec-
tromagnetic modes correspond to quantum harmonic
oscillators. The modes’ quadratures play the roles of the
oscillators’ position and momentum operators obeying
an analogous Heisenberg uncertainty relation.
A. The quadratures of the quantized field
From the Hamiltonian of a quantum harmonic oscil-
lator expressed in terms of 共dimensionless兲 creation and
annihilation operators and representing a single mode k,
H
ˆ
k
=ប
␻
k
共
a
ˆ
k
†
a
ˆ
k
+
1
2
兲
, we obtain the well-known form writ-
ten in terms of “position” and “momentum” operators
共unit mass兲,
H
ˆ
k
=
1
2
共p
ˆ
k
2
+
␻
k
2
x
ˆ
k
2
兲, 共1兲
with
a
ˆ
k
=
1
冑
2ប
␻
k
共
␻
k
x
ˆ
k
+ ip
ˆ
k
兲, 共2兲
a
ˆ
k
†
=
1
冑
2ប
␻
k
共
␻
k
x
ˆ
k
− ip
ˆ
k
兲, 共3兲
or, conversely,
x
ˆ
k
=
冑
ប
2
␻
k
共a
ˆ
k
+ a
ˆ
k
†
兲, 共4兲
p
ˆ
k
=−i
冑
ប
␻
k
2
共a
ˆ
k
− a
ˆ
k
†
兲. 共5兲
Here, we have used the well-known commutation rela-
tion for position and momentum,
关x
ˆ
k
,p
ˆ
k
⬘
兴 = iប
␦
kk
⬘
, 共6兲
which is consistent with the bosonic commutation rela-
tions 关a
ˆ
k
,a
ˆ
k
⬘
†
兴=
␦
kk
⬘
, 关a
ˆ
k
,a
ˆ
k
⬘
兴=0.InEq.共2兲, we see that up
to normalization factors the position and the momentum
are the real and imaginary parts of the annihilation op-
erator. Let us now define the dimensionless pair of con-
jugate variables,
X
ˆ
k
⬅
冑
␻
k
2ប
x
ˆ
k
=Rea
ˆ
k
, P
ˆ
k
⬅
1
冑
2ប
␻
k
p
ˆ
k
=Ima
ˆ
k
. 共 7兲
Their commutation relation is then
关X
ˆ
k
,P
ˆ
k
⬘
兴 =
i
2
␦
kk
⬘
. 共8兲
In other words, the dimensionless position and momen-
tum operators, X
ˆ
k
and P
ˆ
k
, are defined as if we set ប
=1/2. These operators represent the quadratures of a
single mode k, in classical terms corresponding to the
real and imaginary parts of the oscillator’s complex am-
plitude. In the following, by using 共X
ˆ
,P
ˆ
兲 or equivalently
共x
ˆ
,p
ˆ
兲, we shall always refer to these dimensionless
quadratures as playing the roles of position and momen-
tum. Hence 共x
ˆ
,p
ˆ
兲 will also stand for a conjugate pair of
dimensionless quadratures.
The Heisenberg uncertainty relation, expressed in
terms of the variances of two arbitrary noncommuting
observables A
ˆ
and B
ˆ
for an arbitrary given quantum
state,
具共⌬A
ˆ
兲
2
典⬅Š共A
ˆ
− 具A
ˆ
典兲
2
‹ = 具A
ˆ
2
典 − 具A
ˆ
典
2
,
具共⌬B
ˆ
兲
2
典⬅Š共B
ˆ
− 具B
ˆ
典兲
2
‹ = 具B
ˆ
2
典 − 具B
ˆ
典
2
, 共9兲
becomes
具共⌬A
ˆ
兲
2
典具共⌬B
ˆ
兲
2
典 艌
1
4
兩具关A
ˆ
,B
ˆ
兴典兩
2
. 共10兲
Inserting Eq. 共8兲 into Eq. 共10兲 yields the uncertainty re-
lation for a pair of conjugate quadrature observables of
a single mode k,
x
ˆ
k
= 共a
ˆ
k
+ a
ˆ
k
†
兲/2, p
ˆ
k
= 共a
ˆ
k
− a
ˆ
k
†
兲/2i, 共11兲
namely,
具共⌬x
ˆ
k
兲
2
典具共⌬p
ˆ
k
兲
2
典 艌
1
4
兩具关x
ˆ
k
,p
ˆ
k
兴典兩
2
=
1
16
. 共12兲
Thus, in our units, the quadrature variance for a vacuum
or coherent state of a single mode is 1 / 4. Let us further
516
S. L. Braunstein and P. van Loock: Quantum information with continuous variables
Rev. Mod. Phys., Vol. 77, No. 2, April 2005

illuminate the meaning of the quadratures by looking at
a single frequency mode of the electric field 共for a single
polarization兲,
E
ˆ
k
共r,t兲 = E
0
关a
ˆ
k
e
i共k·r−
␻
k
t兲
+ a
ˆ
k
†
e
−i共k·r−
␻
k
t兲
兴. 共13兲
The constant E
0
contains all the dimensional prefactors.
By using Eq. 共11兲, we can rewrite the mode as
E
ˆ
k
共r,t兲 =2E
0
关x
ˆ
k
cos共
␻
k
t − k · r兲 + p
ˆ
k
sin共
␻
k
t − k · r兲兴.
共14兲
Clearly, the position and momentum operators x
ˆ
k
and p
ˆ
k
represent the in-phase and out-of-phase components of
the electric-field amplitude of the single mode k with
respect to a 共classical兲 reference wave ⬀cos共
␻
k
t−k·r兲.
The choice of the phase of this wave is arbitrary, of
course, and a more general reference wave would lead
us to the single-mode description
E
ˆ
k
共r,t兲 =2E
0
关x
ˆ
k
共⌰兲
cos共
␻
k
t − k · r − ⌰兲
+ p
ˆ
k
共⌰兲
sin共
␻
k
t − k · r − ⌰兲兴, 共15兲
with the more general quadratures
x
ˆ
k
共⌰兲
= 共a
ˆ
k
e
−i⌰
+ a
ˆ
k
†
e
+i⌰
兲/2, 共16兲
p
ˆ
k
共⌰兲
= 共a
ˆ
k
e
−i⌰
− a
ˆ
k
†
e
+i⌰
兲/2i. 共17兲
These new quadratures can be obtained from x
ˆ
k
and p
ˆ
k
via the rotation
冉
x
ˆ
k
共⌰兲
p
ˆ
k
共⌰兲
冊
=
冉
cos ⌰ sin ⌰
− sin ⌰ cos ⌰
冊
冉
x
ˆ
k
p
ˆ
k
冊
. 共18兲
Since this is a unitary transformation, we again end up
with a pair of conjugate observables fulfilling the com-
mutation relation 共8兲. Furthermore, because p
ˆ
k
共⌰兲
=x
ˆ
k
共⌰+
␲
/2兲
, the whole continuum of quadratures is cov-
ered by x
ˆ
k
共⌰兲
with ⌰ 苸 关0,
␲
兲. This continuum of observ-
ables is indeed measurable by relatively simple means.
Such a so-called homodyne detection works as follows.
A photodetector measuring an electromagnetic mode
converts the photons into electrons and hence into an
electric current, called the photocurrent i
ˆ
. It is therefore
sensible to assume i
ˆ
⬀n
ˆ
=a
ˆ
†
a
ˆ
or i
ˆ
=qa
ˆ
†
a
ˆ
where q is a con-
stant 共Paul, 1995兲. In order to detect a quadrature of the
mode a
ˆ
, the mode must be combined with an intense
local oscillator at a 50:50 beam splitter. The local oscil-
lator is assumed to be in a coherent state with large
photon number, 兩
␣
LO
典. It is therefore reasonable to de-
scribe this oscillator by a classical complex amplitude
␣
LO
rather than by an annihilation operator a
ˆ
LO
. The
two output modes of the beam splitter, 共a
ˆ
LO
+a
ˆ
兲/
冑
2 and
共a
ˆ
LO
−a
ˆ
兲/
冑
2 共see Sec. II.D兲, may then be approximated
by
a
ˆ
1
= 共
␣
LO
+ a
ˆ
兲/
冑
2, a
ˆ
2
= 共
␣
LO
− a
ˆ
兲/
冑
2. 共19兲
This yields the photocurrents
i
ˆ
1
= qa
ˆ
1
†
a
ˆ
1
= q共
␣
LO
*
+ a
ˆ
†
兲共
␣
LO
+ a
ˆ
兲/2,
i
ˆ
2
= qa
ˆ
2
†
a
ˆ
2
= q共
␣
LO
*
− a
ˆ
†
兲共
␣
LO
− a
ˆ
兲/2. 共20兲
The actual quantity to be measured will be the differ-
ence photocurrent
␦
i
ˆ
⬅ i
ˆ
1
− i
ˆ
2
= q共
␣
LO
*
a
ˆ
+
␣
LO
a
ˆ
†
兲. 共21兲
By introducing the phase ⌰ of the local oscillator,
␣
LO
=兩
␣
LO
兩exp共i⌰兲, we recognize that the quadrature observ-
able x
ˆ
共⌰兲
from Eq. 共16兲 is measured 共without mode index
k兲. Now adjustment of the local oscillator’s phase ⌰
苸 关0,
␲
兴 enables us to detect any quadrature from the
whole continuum of quadratures x
ˆ
共⌰兲
. A possible way to
realize quantum tomography 共Leonhardt, 1997兲, i.e., the
reconstruction of the mode’s quantum state given by its
Wigner function, relies on this measurement method,
called 共balanced兲 homodyne detection. A broadband
rather than a single-mode description of homodyne de-
tection can be found in the work of Braunstein and
Crouch 共1991兲, who also investigate the influence of a
quantized local oscillator.
We have now seen that it is not too hard to measure
the quadratures of an electromagnetic mode. Unitary
transformations such as quadrature displacements
共phase-space displacements兲 can also be relatively easily
performed via the so-called feedforward technique, as
opposed to, for example, photon number displacements.
This simplicity and the high efficiency when measuring
and manipulating continuous quadratures are the main
reasons why continuous-variable schemes appear more
attractive than those based on discrete variables such as
the photon number.
In the following, we shall refer mainly to the conju-
gate pair of quadratures x
ˆ
k
and p
ˆ
k
共position and momen-
tum, i.e., ⌰= 0 and ⌰=
␲
/2兲. In terms of these quadra-
tures, the number operator becomes
n
ˆ
k
= a
ˆ
k
†
a
ˆ
k
= x
ˆ
k
2
+ p
ˆ
k
2
−
1
2
, 共22兲
using Eq. 共8兲.
Let us finally review some useful formulas for the
single-mode quadrature eigenstates,
x
ˆ
兩x典 = x兩x典, p
ˆ
兩p典 = p兩p典, 共23兲
where we have now dropped the mode index k. They are
orthogonal,
具x兩x
⬘
典 =
␦
共x − x
⬘
兲, 具p兩p
⬘
典 =
␦
共p − p
⬘
兲, 共24兲
and complete,
冕
−⬁
⬁
兩x典具x兩dx = 1,
冕
−⬁
⬁
兩p典具p兩dp = 1 . 共25兲
Just as for position and momentum eigenstates, the
quadrature eigenstates are mutually related to each
other by a Fourier transformation,
兩x典 =
1
冑
␲
冕
−⬁
⬁
e
−2ixp
兩p典dp, 共26兲
517
S. L. Braunstein and P. van Loock: Quantum information with continuous variables
Rev. Mod. Phys., Vol. 77, No. 2, April 2005

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Quantum information with continuous variables" ?

This article reviews the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field. 

Q2. What is the key ingredient in the generation of continuous-variable entanglement?

An important tool of many quantum communication protocols is entanglement, and the essential ingredient in the generation of continuous-variable entanglement is squeezed light. 

Q3. What is the optimal cloner for arbitrary coherent states?

For arbitrary coherent states, the optimal Gaussian cloner adds an excess noise ̄clon N ,M to the input state without changing its mean amplitude; the coherent-state copies are all in the same mixed state. 

Q4. What was the first thought to render secure key distribution impossible?

a line transmission below 50%, corresponding to line loss above 3 dB, was thought to render secure key distribution impossible. 

Q5. What is the main reason why continuous-variable schemes appear more attractive?

This simplicity and the high efficiency when measuring and manipulating continuous quadratures are the main reasons why continuous-variable schemes appear more attractive than those based on discrete variables such as the photon number. 

Q6. What is the quantum properties of light to be transferred to the atoms?

The quantum properties of light to be transferred to the atoms are those of a vacuum or a squeezed optical field, i.e., those of a pure Gaussian state of light. 

Q7. How did the concept of entanglement come to light?

Although important milestones in quantum information theory have been derived and expressed in terms of qubits or discrete variables, the notion of quantum entanglement itself came to light in a continuous-variable setting. 

Q8. What is the way to assess whether a continuousvariable teleportation scheme is?

Another possible way to assess whether a continuousvariable teleportation scheme is truly quantum is to check to what extent nonclassical properties such as squeezing or photon antibunching can be preserved in the teleported field Li, Li, et al., 2002 . 

Q9. What is the simplest way to measure the inferred variances of two modes?

The minimal inferred variances are also referred to as conditional variances Varcondx̂ and Varcondp̂ , a measure of the noise degrading the otherwise perfect correlations between the two modes. 

Q10. What formalism is more robust against a dissipative environment?

the nonlocality of a twomode squeezed state is more robust against a dissipative environment such as an absorbing optical fiber when it is based on the parity-spin formalism Filip and Mišta, 2002 rather than the phase-space formalism Jeong et al., 2000 . 

Q11. What is the fidelity of a teleportation with a single squeezed state?

Unless Alice and Bob have access to additional local squeezers,5 the maximum fidelity of coherent-state teleportation achievable with one single-mode squeezed state is Fav=1/ 2 in the limit of infinite squeezing van Loock and Braunstein, 2000a . 

Q12. What is the optimal cloning fidelity for turning N identical but arbitrary coherent states?

The optimal cloning fidelity for turning N identical but arbitrary coherent states into M identical approximate copies,Fclon,N,M coh st, = MN/ MN + M − N , 210was derived by Cerf and Iblisdir 2000a .