Quantum Entanglement and Reduced Density Matrices

We investigate entanglement and separability criteria of multipartite (n-partite) state by examining ranks of its reduced density matrices. Firstly, we construct the general formula to determine the criterion. A rank of origin density matrix always equals one, meanwhile ranks of reduced matrices have various ranks. Next, separability and entanglement criterion of multipartite is determined by calculating ranks of reduced density matrices. In this article we diversify multipartite state criteria into completely entangled state, completely separable state, and compound state, i.e. sub-entangled state and sub-entangledseparable state. Furthermore, we also shorten the calculation proposed by the previous research to determine separability of multipartite state and expand the methods to be able to differ multipartite state based on criteria above.     

Classical and Quantum Coherent States

p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics 41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

A Classical Analog of Random Quantum States

We examine the statistical properties of a pure quantum state randomly chosen with respect to the uniform measure in a Hilbert space. Namely, we consider the distribution of outcomes of a fixed measurement performed on the random quantum state. We show that such distribution is completely analogous to the distribution of measurement outcomes of an a priori unknown classical random system. In particular, Shannon entropies of both distributions coincide. We study this correspondence between quantum and classical random systems and clarify its origin.     

On the Classical Capacity of General Quantum Gaussian Measurement

In this paper, we consider the classical capacity problem for Gaussian measurement channels. We establish Gaussianity of the average state of the optimal ensemble in the general case and discuss the Hypothesis of Gaussian Maximizers concerning the structure of the ensemble. Then, we consider the case of one mode in detail, including the dual problem of accessible information of a Gaussian ensemble. Our findings are relevant to practical situations in quantum communications where the receiver is Gaussian (say, a general-dyne detection) and concatenation of the Gaussian channel and the receiver can be considered as one Gaussian measurement channel. Our efforts in this and preceding papers are then aimed at establishing full Gaussianity of the optimal ensemble (usually taken as an assumption) in such schemes.     

Quantum Fisher Information for Density Matrices with Arbitrary Ranks

We provide a new expression of the quantum Fisher information （QFI） for a general system. Utilizing this expression, the QFI for a non-full rank density matrix is only determined by its support. This expression can bring convenience for an infinite-dimensional density matrix with a finite support. Besides, a matrix representation of the QFI is also given.     

New Inequality for Density Matrices of Single Qudit States

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the “qubit portrait” bound for the distance between the qutrit states are considered in explicit form.     

Two-Tape Finite Automata with Quantum and Classical States

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and two-way two-tape deterministic finite automata (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called two-way two-tape finite automata with quantum and classical states (2TQCFA). First, we give efficient 2TFA algorithms for identifying languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as L_square={aⁿbⁿ² | n ? N}L_{mathit{square}}={a^{n}b^{n^{2}}mid nin mathbf{N}}. Third, we show that {aⁿb^nk | n ? N}{a^{n}b^{n^{k}}mid nin mathbf{N}} can be recognized by (k+1)-tape deterministic finite automata ((k+1)TFA). Finally, we introduce k-tape automata with quantum and classical states (kTQCFA) and prove that {aⁿb^nk | n ? N}{a^{n}b^{n^{k}}mid nin mathbf{N}} can be recognized by kTQCFA.     

Detecting Entanglement of States by Entries of Their Density Matrices

For any bipartite systems, a universal entanglement witness of rank-4 for pure states is obtained and a class of finite rank entanglement witnesses is constructed. In addition, a method of detecting entanglement of a state only by entries of its density matrix with respect to some product basis is obtained.     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

Classical Limits of Euclidean Gibbs States for Quantum Lattice Models

Models of quantum and classical particles on a lattice ^d are considered. The classical model is obtained from the corresponding quantum model when the reduced mass of the particle m = / #x210F;² tends to infinity. For these models, the convergence of the Euclidean Gibbs states, when m + , is described in terms of the weak convergence of local Gibbs specifications, determined by conditional Gibbs measures. In fact, it is shown that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with pair interactions possessing the translation invariance, has also been shown.     

Entropy and Entanglement Bounds for Reduced Density Matrices of Fermionic States

Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.     

Lipschitz-Continuity of the Integrated Density of States for Gaussian Random Potentials

The integrated density of states of a Schrödinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate.     

Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems

We consider ground states in relatively bounded quantum perturbations of classical lattice models. We prove general results about such perturbations (existence of the spectral gap, exponential decay of truncated correlations, analyticity of the ground state), and also prove that in particular the AKLT model belongs to this class if viewed on a large enough length scale. This immediately implies a general perturbation theory about this model. On leave from Institute for Information Transmission Problems, Moscow The author is an Irish Research Council for Science, Engineering and Technology Postdoctoral Fellow     

Secret Key Distillation for Continuous Variable Quantum Key Distribution against Gaussian Classical Eve

The continuous variable quantum key distribution is expected to provide high secret key rate without single photon source and detector, while the lack of the effective key distillation method makes it unpractical under the high loss condition. Here we present a single-bit-reverse-reconciliation protocol against Oaussian classical Eve, which can distill the secret key through practical imperfect error correction with high efficiency. The simulation results show that this protocol can distill secret keys even when the transmission fibre is longer than 150 km, which may make the continuous variable scheme to outvie the single photon one.