Showing papers in "Physical Review A in 2000"

Three qubits can be entangled in two inequivalent ways

Abstract: Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the Greenberger-Horne-Zeilinger state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher-dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.

Ultra-bright source of polarization-entangled photons

Abstract: Utilizing the process of spontaneous parametric down-conversion in a novel crystal geometry, a source of polarization-entangled photon pairs has been provided that is more than ten times brighter, per unit of pump power, than previous sources, with another factor of 30 to 75 expected to be readily achievable. A high level of entanglement between photons emitted over a relatively large collection angle, and over a 10-nm bandwidth, is a characteristic of the invention. As a demonstration of the source capabilities, a 242-σ violation of Bell's inequalities was attained in fewer than three minutes, and near-perfect photon correlations were achieved when the collection efficiency was reduced. In addition, both the degree of entanglement, and the purity of the state are readily tunable. The polarization entangled photon source can be utilized as a light source for the practice of quantum cryptography.

Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures

Abstract: We apply the full power of modern electronic band-structure engineering and epitaxial heterostructures to design a transistor that can sense and control a single-donor electron spin. Spin-resonance transistors may form the technological basis for quantum information processing. One- and two-qubit operations are performed by applying a gate bias. The bias electric field pulls the electron wave function away from the dopant ion into layers of different alloy composition. Owing to the variation of the g factor (Si: g1.998,Ge:g1.563), this displacement changes the spin Zeeman energy, allowing single-qubit operations. By displacing the electron even further, the overlap with neighboring qubits is affected, which allows two-qubit operations. Certain silicon-germanium alloys allow a qubit spacing as large as 200 nm, which is well within the capabilities of current lithographic techniques. We discuss manufacturing limitations and issues regarding scaling up to a large size computer.

Entanglement and quantum computation with ions in thermal motion

Abstract: With bichromatic fields, it is possible to deterministically produce entangled states of trapped ions. In this paper we present a unified analysis of this process for both weak and strong fields, for slow and fast gates. Simple expressions for the fidelity of creating maximally entangled states of two or an arbitrary number of ions under nonideal conditions are derived and discussed.

Security against individual attacks for realistic quantum key distribution

Abstract: I prove the security of quantum key distribution against individual attacks for realistic signals sources, including weak coherent pulses and down-conversion sources. The proof applies to the Bennett-Brassard 1984 protocol with the standard detection scheme (no strong reference pulse). I obtain a formula for the secure bit rate per time slot of an experimental setup, which can be used to optimize the performance of existing schemes for the considered scenario.

Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity

Abstract: We study the amount of classical communication needed for distributed quantum-information processing. In particular, we introduce the concept of ``remote preparation'' of a quantum state. Given an ensemble of states, Alice's task is to help Bob in a distant laboratory to prepare a state of her choice. We find several examples of an ensemble with an entropy S where the remote preparation can be done with a communication cost lower than the amount $(2S)$ required by standard teleportation. We conjecture that, for an arbitrary N-dimensional pure state, its remote preparation requires $2{\mathrm{log}}_{2}N$ bits of classical communication, as in standard teleportation.

Optimization of entanglement witnesses

Abstract: An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.

Minimum classical bit for remote preparation and measurement of a qubit

Abstract: We show that a qubit chosen from equatorial or polar great circles on a Bloch sphere can be remotely prepared with one cbit from Alice to Bob if they share one ebit of entanglement. Also we show that any single-particle measurement on an arbitrary qubit can be remotely simulated with one ebit of shared entanglement and communication of one cbit.

Heating of trapped ions from the quantum ground state

Abstract: We have investigated motional heating of laser-cooled ${}^{9}{\mathrm{Be}}^{+}$ ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes. The results show that heating is due to electric-field noise from the trap electrodes that exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating.

Theory of quantum secret sharing

Abstract: I present a variety of results on the theory of quantum secret sharing. I show that any mixed state quantum secret sharing scheme can be derived by discarding a share from a pure state scheme, and that the size of each share in a quantum secret sharing scheme must be at least as large as the size of the secret. I show that the only constraints on the existence of quantum secret sharing schemes with general access structures are monotonicity (if a set is authorized, so are larger sets) and the no-cloning theorem. I also discuss some aspects of sharing classical secrets using quantum states. In this situation, the size of each share can sometimes be half the size of the classical secret.

Quantum cryptography with squeezed states

Abstract: A quantum key distribution scheme based on the use of squeezed states is presented. The states are squeezed in one of two field quadrature components, and the value of the squeezed component is used to encode a character from an alphabet. The uncertainty relation between quadrature components prevents an eavesdropper from determining both with enough precision to determine the character being sent. Losses degrade the performance of this scheme, but it is possible to use phase sensitive amplifiers to boost the signal and partially compensate for their effect.

Quantum Cryptography using larger alphabets

Abstract: Like all of quantum information theory, quantum cryptography is traditionally based on two-level quantum systems. In this paper, a protocol for quantum key distribution based on higher-dimensional systems is presented. An experimental realization using an interferometric setup is also proposed. Analyzing this protocol from the practical side, one finds an increased key creation rate while keeping the initial laser pulse rate constant. Analyzing it for the case of intercept/resend eavesdropping strategy, an increased error rate is found compared to two-dimensional systems, hence an advantage for the legitimate users to detect an eavesdropper.

Dense coding for continuous variables

Abstract: A scheme to achieve dense quantum coding for the quadrature amplitudes of the electromagnetic field is presented. The protocol utilizes shared entanglement provided by nondegenerate parametric down-conversion in the limit of large gain to attain high efficiency. For a constraint in the mean number of photons \(\bar n\) associated with modulation in the signal channel, the channel capacity for dense coding is found to be \(1 + \bar n + {\bar n^2}\), which always beats coherent-state communication and surpasses squeezed-state communication for \(\bar n > 1\). For \(\bar n > > 1\), the dense coding capacity approaches twice that of either scheme.

Fidelity and Concurrence of conjugated states

Abstract: We prove some properties of fidelity (transition probability) and concurrence, the latter defined by a straightforward extension of Wootters' notation. Choose a conjugation and consider the dependence of fidelity or of concurrence on conjugated pairs of density operator. These functions turn out to be concave or convex roofs. Optimal decompositions are constructed. Some applications to two and tripartite systems illustrate the general theorems.

Exact and asymptotic measures of multipartite pure-state entanglement

Abstract: Hoping to simplify the classification of pure entangled states of multi $(m)\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ quantum systems, we study exactly and asymptotically (in $n)$ reversible transformations among $n\mathrm{th}$ tensor powers of such states (i.e., $n$ copies of the state shared among the same $m$ parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter---their partial entropy---which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states having an $m\ensuremath{-}\mathrm{w}\mathrm{a}\mathrm{y}$ Schmidt decomposition are similarly parametrizable, with the partial entropy across any nontrivial partition representing the number of standard quantum superposition or ``cat'' states $|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉$ asymptotically interconvertible to the state in question. For general $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the $m=4$ cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the $m=4$ cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties $m$ we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for $mg2$ we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible $[o(n)]$ amount of quantum communication.

Coherent quantum feedback

Abstract: In the conventional picture of quantum feedback control, sensors perform measurements on the system, a classical controller processes the results of the measurements, and actuators supply semiclassical potentials to alter the behavior of the quantum system. In this picture, the sensors tend to destroy coherence in the process of making measurements, and although the controller can use the actuators to act coherently on the quantum system, it is processing and feeding back classical information. This paper proposes an alternative method for quantum feedback control, in which the sensors, controller, and actuators are quantum systems that interact coherently with the system to be controlled. In this picture, the controller gets, processes, and feeds back quantum information. Controllers that operate using such quantum feedback loops can perform tasks such as entanglement transfer that are not possible using classical feedback. Necessary and sufficient conditions are presented for Hamiltonian quantum systems to be controllable and observable using both classical and quantum feedback.

Multivalued logic gates for quantum computation

Abstract: We develop a multivalued logic for quantum computing for use in multi-level quantum systems, and discuss the practical advantages of this approach for scaling up a quantum computer. Generalizing the methods of binary quantum logic, we establish that arbitrary unitary operations on any number of d-level systems $(dg2)$ can be decomposed into logic gates that operate on only two systems at a time. We show that such multivalued logic gates are experimentally feasible in the context of the linear ion trap scheme for quantum computing. By using d levels in each ion in this scheme, we reduce the number of ions needed for a computation by a factor of ${\mathrm{log}}_{2}d.$

Quantum feedback control and classical control theory

Abstract: We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer-based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential.

Schmidt number for density matrices

Abstract: We introduce the notion of a Schmidt number of a bipartite density matrix. We show that k-positive maps witness the Schmidt number, in the same way that positive maps witness entanglement. We determine the Schmidt number of the family of states that is made from mixing the completely mixed state and a maximally entangled state. We show that the Schmidt number does not necessarily increase when taking tensor copies of a density matrix $\ensuremath{\rho};$ we give an example of a density matrix for which the Schmidt numbers of $\ensuremath{\rho}$ and $\ensuremath{\rho}\ensuremath{\bigotimes}\ensuremath{\rho}$ are both $2.$

Entangling power of quantum evolutions

Abstract: We analyze the entangling capabilities of unitary transformations U acting on a bipartite ${(d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2})$-dimensional quantum system. To this aim we introduce an entangling power measure $e(U)$ given by the mean linear entropy produced acting with U on a given distribution of pure product states. This measure admits a natural interpretation in terms of quantum operations. For a uniform distribution explicit analytical results are obtained using group-theoretic arguments. The behavior of the features of $e(U)$ as the subsystem dimensions ${d}_{1}$ and ${d}_{2}$ are varied is studied both analytically and numerically. The two-qubit case ${d}_{1}{=d}_{2}=2$ is argued to be peculiar.

Improvement on teleportation of continuous variables by photon subtraction via conditional measurement

Abstract: We show that the recently proposed scheme of teleportation of continuous variables [S.L. Braunstein and H.J. Kimble, Phys. Rev. Lett. 80, 869 (1998)] can be improved by a conditional measurement of the entangled state shared by the sender and the recipient. The conditional measurement subtracts photons from the original entangled two-mode squeezed vacuum, by transmitting each mode through a low-reflectivity beam splitter and performing a joint photon-number measurement on the reflected beams. In this way the degree of entanglement of the shared state is increased and so is the fidelity of the teleported state.

Methodology for quantum logic gate construction

Abstract: We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation [Gottesman and Chuang, Nature (London) 402, 390 (1999)] and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the $\ensuremath{\pi}/8$ and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed.

Precision measurement of the Casimir force using gold surfaces

Abstract: A precision measurement of the Casimir force using metallic gold surfaces is reported. The force is measured between a large gold-coated sphere and flat plate using an atomic force microscope. The use of gold surfaces removes some theoretical uncertainties in the interpretation of the measurement. The forces are also measured at smaller surface separations. The complete dielectric spectrum of the metal is used in the comparison of theory to the experiment. The average statistical precision remains at the same 1% of the forces measured at the closest separation. These results should lead to the development of stronger constraints on hypothetical forces.

Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication

Abstract: We present experimental evidence for the synchronization of two semiconductor lasers exhibiting chaotic emission on subnanosecond time scales. The transmitter system consists of a semiconductor laser with weak to moderate coherent optical feedback and therefore exhibits chaotic oscillations. The receiver system is realized by a solitary semiconductor laser in which a fraction of the transmitter signal is coherently injected. We find that for a considerably large parameter range, synchronized receiver output can be achieved. We discuss the physical mechanism and demonstrate that the receiver acts as a chaos pass filter, which reproduces the chaotic fluctuations of the transmitter laser, but suppresses additionally encoded signals. Signal extraction at frequencies of up to 1 GHz has been achieved. Thus we provide a simple and robust optical chaos synchronization system that is promising for the realization of communication by sending signals with chaotic carriers.

Entangled Rings

Abstract: Consider a ring of N qubits in a translationally invariant quantum state We ask to what extent each pair of nearest neighbors can be entangled Under certain assumptions about the form of the state, we find a formula for the maximum possible nearest-neighbor entanglement We then compare this maximum with the entanglement achieved by the ground state of an antiferromagnetic ring consisting of an even number of spin-1/2 particles We find that, though the antiferromagnetic ground state does not maximize the nearest-neighbor entanglement relative to all other states, it does so relative to other states having zero z-component of spin

Probabilistic teleportation and entanglement matching

Abstract: Teleportation may be interpreted as sending and extracting quantum information through quantum channels. In this report, it is shown that to get the maximal probability of exact teleportation through a partially entangled quantum channel, the sender (Alice) need only operate a measurement that satisfies an ``entanglement matching'' to this channel. An optimal strategy is also provided for the receiver (Bob) to extract the quantum information by adopting a general evolution.

Optimal local implementation of nonlocal quantum gates

Abstract: We investigate the minimal resources that are required in the local implementation of nonlocal quantum gates in a distributed quantum computer. Both classical communication requirements and entanglement consumption are investigated. We present general statements on the minimal resource requirements and present optimal procedures for a number of important gates, including controlled-NOT (CNOT) and Toffoli gates. We show that one bit of classical communication in each direction is both necessary and sufficient for the nonlocal implementation of the quantum CNOT, while in general two bits in each direction is required for the implementation of a general two-bit quantum gate. In particular, the state swapper requires this maximum classical communication overhead. Extensions of these ideas to multiparty gates are presented.

Stationary solutions of the one-dimensional nonlinear Schrodinger equation: II. Case of attractive nonlinearity

Abstract: In this second of two papers, we present all stationary so- lutions of the nonlinear Schrodinger equation with box or pe- riodic boundary conditions for the case of attractive nonlin- earity. The companion paper has treated the case of repulsive nonlinearity. Our solutions take the form of stationary trains of bright solitons. Under box boundary conditions the solu- tions are the bounded analog of bright solitons on the innite line, and are in one-to-one correspondence with particle-in-a- box solutions to the linear Schrodinger equation. Under peri- odic boundary conditions we nd several classes of solutions: constant amplitude solutions corresponding to boosts of the condensate; the nonlinear version of the well-known particle- on-a-ring solutions in linear quantum mechanics; nodeless, real solutions; and a novel class of intrinsically complex, node- less solutions. The set of such solutions on the ring are de- scribed by the Cn character tables from the theory of point groups. We make experimental predictions about the form of the ground state and modulational instability. We show that, though this is the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces new and surprising phenomena in the stationary one-dimensional non- linear Schrodinger equation. We also note that in various limits the spectrum of the nonlinear Schrodinger equation re- duces to that of the box, the Rydberg, and the harmonic oscillator, the latter being for repulsive nonlinearity, thus in- cluding the three most common and important cases of linear quantum mechanics.

Nonlinear Landau-Zener tunneling

Abstract: A self-interacting two-level system depending on an external parameter is investigated. The most striking feature exhibited in this system is the presence of a nonzero tunneling probability in the adiabatic limit for large enough interaction strength. Possible experimental observation of this breakdown of adiabaticity using a Bose-Einstein condensate in an optical potential is suggested.

Sensitive Magnetometry based on Nonlinear Magneto-Optical Rotation

Abstract: Application of nonlinear magneto-optical (Faraday) rotation to magnetometry is investigated Our experimental setup consists of a modulation polarimeter that measures rotation of the polarization plane of a laser beam resonant with transitions in Rb Rb vapor is contained in an evacuated cell with antirelaxation coating that enables atomic ground-state polarization to survive many thousand wall collisions This leads to ultranarrow features $(\ensuremath{\sim}{10}^{\ensuremath{-}6} \mathrm{G})$ in the magnetic-field dependence of optical rotation The potential sensitivity of this scheme to sub-$\ensuremath{\mu}\mathrm{G}$ magnetic fields as a function of atomic density, light intensity, and light frequency is investigated near the $D1$ and $D2$ lines of ${}^{85}\mathrm{Rb}$ It is shown that through an appropriate choice of parameters the shot-noise-limited sensitivity to small magnetic fields can reach $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12} \mathrm{G}/\sqrt{\mathrm{Hz}}$