Showing papers in "Journal of Mathematical Physics in 2002"
TL;DR: In this paper, the authors introduce the notion of pseudo-hermiticity and show that every non-Hermitian Hamiltonian with a real spectrum is pseudo-HERMIAN.
Abstract: We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
1,320 citations
TL;DR: In this article, the authors studied the topological quantum error-correcting surface codes (surface codes) introduced by Kitaev, where qubits are arranged in a two-dimensional array on a surface of nontrivial topology.
Abstract: We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z(2) lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.
1,176 citations
TL;DR: In this article, it was shown that a diagonalizable Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilINear operator, and the eigenvalues of H are real or come in complex conjugate pairs.
Abstract: We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
793 citations
TL;DR: In this article, the authors give necessary and sufficient conditions for the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.
Abstract: We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.
785 citations
TL;DR: In this article, tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles were constructed.
Abstract: This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β=1,2,4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
656 citations
TL;DR: In this paper, a polynomial measure of global entanglement was defined, which applies to any number of spin-12 particles, and evaluated for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces.
Abstract: We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-12 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.
447 citations
TL;DR: The quantum de Finetti representation theorem as discussed by the authors is a quantum analog of the classical de Finettis representation theorem on exchangeable probability assignments, where probabilities are taken to be degrees of belief instead of objective states of nature.
Abstract: We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than...
416 citations
TL;DR: In this article, the authors give the general solution of a two-loop integral, the so-called C-topology, in terms of multiple nested sums and discuss some important properties of nested sums, in particular they satisfy a Hopf algebra.
Abstract: Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer. Examples such as expansions of generalized hypergeometric functions or Appell functions are discussed. As a further application, we give the general solution of a two-loop integral, the so-called C-topology, in terms of multiple nested sums. In addition, we discuss some important properties of nested sums, in particular we show that they satisfy a Hopf algebra.
382 citations
TL;DR: In this article, the authors consider the problem of reversing quantum dynamics, with the goal of preserving an initial state's quantum entanglement or classical correlation with a reference system, and exhibit an approximate reversal operation, adapted to the initial density operator and the "noise" dynamics to be reversed.
Abstract: We consider the problem of reversing quantum dynamics, with the goal of preserving an initial state’s quantum entanglement or classical correlation with a reference system. We exhibit an approximate reversal operation, adapted to the initial density operator and the “noise” dynamics to be reversed. We show that its error in preserving either quantum or classical information is no more than twice that of the optimal reversal operation. Applications to quantum algorithms and information transmission are discussed.
295 citations
TL;DR: In this paper, the authors introduced the concept of entanglement of purification, which is a measure of both quantum as well as classical correlations in a quantum state, and showed that the entangement cost of creating a state rho asymptotically from maximally entangled states, with negligible communication.
Abstract: We introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the (regularized) entanglement of purification is equal to the entanglement cost of creating a state rho asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in H(2)circle times H(2).
273 citations
TL;DR: For the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo-Schumacher-Westmoreland capacity are additive as discussed by the authors.
Abstract: We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo–Schumacher–Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for classical capacity) that can occur.
TL;DR: In this article, Epstein et al. presented self-contained proofs of the strong subadditivity inequality for von Neumann's quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps.
Abstract: This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s the...
TL;DR: In this article, it was shown that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels.
Abstract: A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p>4.79. The same example (with p=∞) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.
TL;DR: In this paper, the authors apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two-and three-level quantum systems, where the aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences.
Abstract: We apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two- and three-level quantum systems. The aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences. Sub-Riemannian geometry and singular-Riemannian geometry provide a natural framework for this minimization, where the optimal control is expressed in terms of geodesics. We first show that in two-level systems the well-known technique of “π-pulse transfer” in the rotating wave approximation emerges naturally from this minimization. In three-level systems driven by two resonant fields, we also find the counterpart of the “π-pulse transfer.” This geometrical picture also allows one to analyze the population transfer by adiabatic passage.
TL;DR: In this paper, the controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras.
Abstract: The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases
TL;DR: In this article, it was shown that the Holevo bound is the classical information capacity of unital qubit channels, and an explicit formula for this capacity was given for product channels.
Abstract: Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary. As a byproduct this proves that the Holevo bound is the classical information capacity of such qubit channels, and provides an explicit formula for this classical capacity. Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions. The proof relies on a new bound for the p-norm of an output state from the half-noisy phase-damping channel.
TL;DR: In this paper, a direct method for establishing integrable couplings of TD hierarchy is proposed, which is obtained by constructing a suitable transformation of Lax pairs and a new Lie algebra.
Abstract: A direct method for establishing integrable couplings is proposed in this paper. As an example illustration, integrable couplings of TD hierarchy are obtained by constructing a suitable transformation of Lax pairs and a new Lie algebra.
TL;DR: In this paper, the coexistence of first and third-order integrals of motion in two-dimensional classical and quantum mechanics is studied, and all potentials that admit such integrals and all their integrals are found explicitly.
Abstract: We consider here the coexistence of first- and third-order integrals of motion in two-dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e., the potentials are proportional to ℏ2, so their classical limit is free motion.
TL;DR: In this paper, it was shown that the Reeh-Schlieder property holds for states of quantum fields on real analytic curved space times if they satisfy an analytic microlocal spectrum condition.
Abstract: We show in this article that the Reeh–Schlieder property holds for states of quantum fields on real analytic curved space–times if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e., without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein–Gordon field are further investigated in the present work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground or KMS state of the Klein–Gordon field on a stationary real analytic space–time fulfills the analytic microlocal spectrum condition.
TL;DR: In this article, the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions, and the classical and quantum quadratic algebras associated with each of these potentials are determined.
Abstract: A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrodinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.
TL;DR: In this paper, the authors consider the case of electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting and derive the electric current whose expectation value gives the Hall effect in terms of an effective magnetic field.
Abstract: We consider electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting. Spectrum and eigenfunctions of the related Hamiltonian are obtained. We derive the electric current whose expectation value gives the Hall effect in terms of an effective magnetic field. We present a receipt to find the action which can be utilized in path integrals for noncommuting coordinates. In terms of this action we calculate the related Aharonov–Bohm phase and show that it also yields the same effective magnetic field. When magnetic field is strong enough this phase becomes independent of magnetic field. Measurement of it may give some hints on spatial noncommutativity. The noncommutativity parameter θ can be tuned such that electrons moving in noncommutative coordinates are interpreted as either leading to the fractional quantum Hall effect or composite fermions in the usual coordinates.
TL;DR: In this paper, it was shown that if an operator A is invariant under a quantum operation φ, it is called A a φ-fixed point, and sufficient conditions under which the answer is yes.
Abstract: Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.
TL;DR: In this paper, a modified proof of the recent result of C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal concerning entanglement-assisted classical capacity of a quantum channel is given.
Abstract: We give a modified proof of the recent result of C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal concerning entanglement-assisted classical capacity of a quantum channel and discuss the relation between entanglement-assisted and unassisted classical capacities.
TL;DR: Here it is proved that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/e, a result which matches the lower bound from counting volume up to constant factor.
Abstract: Quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and, furthermore, that the number of gates required for precision e is only polynomial in log 1/e. Here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/e, a result which matches the lower bound from counting volume up to constant factor.
TL;DR: In this paper, the Chernoff theorem is used to formulate and prove some rigorous results on representations for solutions of Schrodinger equations by the Hamiltonian Feynman path integrals (Feynman integrals over trajectories in the phase space).
Abstract: The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrodinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrodinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.
TL;DR: In this article, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced, by realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated.
Abstract: As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky–Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators, and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.
TL;DR: In this paper, it was shown that the pseudo-hermiticity property of diagonalizable pseudo-Hermitian operators is strictly connected with the existence of an antilinear symmetry.
Abstract: We inquire into some properties of diagonalizable pseudo-Hermitian operators, showing that their definition can be relaxed and that the pseudo-Hermiticity property is strictly connected with the existence of an antilinear symmetry. This result is then illustrated by considering the particular case of the complex Morse potential.
TL;DR: In this article, the authors give two characterization theorems for pseudo-Hermitian (possibly non-nondiagonalizable) Hamiltonians with a discrete spectrum that admit block-diagonalization with finite-dimensional diagonal blocks.
Abstract: We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.
TL;DR: In this article, a family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space, and two different sets of n commuting second-order operators are found, overlapping in the Hamiltonian alone.
Abstract: A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space. Two different sets of n commuting second-order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.
TL;DR: In this paper, it is shown that the ⋆-genvalue problem can be decomposed into separate harmonic oscillator equations for each dimension, and the angular momentum operator is derived and its eigenvalue problem is shown to be equivalent to the usual eigen value problem of the ε-genfunction related wave function, and two examples of symmetric non-commutative harmonic oscillators are analyzed.
Abstract: The noncommutative harmonic oscillator, with noncommutativity not only in position space but also in phase space, in arbitrary dimension is examined. It is shown that the ⋆-genvalue problem, which replaces the Schrodinger problem in this case, can be decomposed into separate harmonic oscillator equations for each dimension. The two-dimensional noncommutative harmonic oscillator (four noncommutative phase-space dimensions) is investigated in greater detail. The requirement of the existence of rotationally symmetric solutions leads to a two parameter harmonic oscillator which is completely solved in this case. The angular momentum operator is derived and its ⋆-genvalue problem is shown to be equivalent to the usual eigenvalue problem of the ⋆-genfunction related wave function. The ⋆-genvalues of the angular momentum are found to depend on the energy difference of the oscillations in the two dimensions. Furthermore two examples of a symmetric noncommutative harmonic oscillators are analyzed. The first is the noncommutative two-dimensional Landau problem with harmonic oscillator potential, which shows degeneracy in the energy levels for certain critical values of the noncommutativity parameters, and the second is the three-dimensional harmonic oscillator with noncommuting coordinates and momenta.