Remote Minimum-Error Discrimination of <Emphasis Type="Italic">N</Emphasis> Nonorthogonal Symmetric Qudit States

We present a scheme for implementing a remote minimum-error discrimination (MD) among N linearly independent nonorthogonal symmetric qudit states. The probability of correct guesses is in agreement with the optimal probability for local MD among the N nonorthogonal states. The procedure we use is a remote probability operator measure (POM). We show that this remote POM can be performed as a remote von Neumann measurement by remote basis transformation. We construct a quantum network for realizing the remote MD using local operations, classical communications and shared entanglement (LOCCSE), and thus provide a feasible physical means to realize the remote MD.     

Minimum-error state discrimination and quantitative wave particle duality

We establish the link between minimum-error discrimination of two pure states and wave particle duality in two-path interferometers. In particular, the upper bound of the probability of success discrimination is derived directly from the corresponding duality relation. It is already known that quantum state discrimination can produce wave particle duality relations, here we show the converse is also true: Wave particle duality can be used to obtain information about state discrimination.     

A non-symmetric transition probability in quantum mechanics

The purpose of this paper is to show the possibility of introducing a non-symmetric transition probability in quantum mechanics, starting from the axiomatic basis of the theory known as the “quantum logic approach”. The quantum-mechanical postulates are later reformulated in the language of the transition probability and the operations (filtering procedures) transforming pure states of a physical system into themselves, ans some advantages of this formulation are shown consisting mainly in resolving the old troubles connected with quantum logics (e.g., the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law).As a consequence of the axioms assumed, the representation theorem for the logic of proposition is deduced.     

On quantum operations as quantum states

We formalize Jamiolkowski’s correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in Choi’s proof of the operator sum representation of Completely Positive-preserving linear maps; we go further and show that all of the important theorems concerning quantum operations can be derived directly from those concerning quantum states. As we do so the discussion first provides an elegant and original review of the main features of quantum operations. Next (in the second half of the paper) we find more results stemming from our formulation of the correspondence. Thus, we provide a factorizability condition for quantum operations, and give two novel Schmidt-type decompositions of bipartite pure states. By translating the composition law of quantum operations, we define a group structure upon the set of totally entangled states. The question whether the correspondence is merely mathematical or can be given a physical interpretation is addressed throughout the text: we provide formulae which suggest quantum states inherently define a quantum operation between two of their subsystems, and which turn out to have applications in quantum cryptography.     

A Framework for Bounding Nonlocality of State Discrimination

We consider the class of protocols that can be implemented by local quantum operations and classical communication (LOCC) between two parties. In particular, we focus on the task of discriminating a known set of quantum states by LOCC. Building on the work in the paper Quantum nonlocality without entanglement (Bennett et al., Phys Rev A 59:1070–1091, 1999), we provide a framework for bounding the amount of nonlocality in a given set of bipartite quantum states in terms of a lower bound on the probability of error in any LOCC discrimination protocol. We apply our framework to an orthonormal product basis known as the domino states and obtain an alternative and simplified proof that quantifies its nonlocality. We generalize this result for similar bases in larger dimensions, as well as the “rotated” domino states, resolving a long-standing open question (Bennett et al., Phys Rev A 59:1070–1091, 1999).     

Discriminating States: the quantum Chernoff bound

We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.     

No-signaling principle can determine optimal quantum state discrimination

We provide a general framework of utilizing the no-signaling principle in derivation of the guessing probability in the minimum-error quantum state discrimination. We show that, remarkably, the guessing probability can be determined by the no-signaling principle. This is shown by proving that, in the semidefinite programing for the discrimination, the optimality condition corresponds to the constraint that quantum theory cannot be used for a superluminal communication. Finally, a general bound to the guessing probability is presented in a closed form.     

Controlled teleportation of an arbitrary n-qudit state using nonmaximally entangled GHZ states

In this paper, we explicitly present a general scheme for controlled quantum teleportation of an arbitrary multi-qudit state with unit fidelity and non-unit successful probability using d-dimensional nonmaximally entangled GHZ states as the quantum channel and generalized d-dimensional Bell states as the measurement basis. The expression of successful probability for controlled teleportation is present depending on the degree of entanglement matching between the quantum channel and the generalized Bell states. And the formulae for the selection of operations performed by the receiver are given according to the results measured by the sender and the controller.      

Hamiltonian Formulation of Statistical Ensembles and Mixed States of Quantum and Hybrid Systems

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.     

Deterministic Joint Remote Preparation of an Arbitrary Two-Qubit State Using the Cluster State

Recently, deterministic joint remote state preparation (JRSP) schemes have been proposed to achieve 100% success probability. In this paper, we propose a new version of deterministic JRSP scheme of an arbitrary two-qubit state by using the six-qubit cluster state as shared quantum resource. Compared with previous schemes, our scheme has high efficiency since less quantum resource is required, some additional unitary operations and measurements are unnecessary. We point out that the existing two types of deterministic JRSP schemes based on GHZ states and EPR pairs are equivalent.     

Unambiguous discrimination between linearly independent quantum states

The theory of generalised measurements is used to examine the problem of discriminating unambiguously between non-orthogonal pure quantum states. Measurements of this type never give erroneous results, although, in general, there will be a non-zero probability of a result being inconclusive. It is shown that only linearly independent states can be unambiguously discriminated. In addition to examining the general properties of such measurements, we discuss their application to entanglement concentration.     

Unambiguous State Discrimination with Intrinsic Coherence

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.     

Controlled Remote Implementation of an Arbitrary Single-Qubit Operation with Partially Entangled Quantum Channel

We present a scheme for controlled remote implementation of an arbitrary single-qubit operation by using partially entangled states as the quantum channel. The sender can remote implement an arbitrary single-qubit operation on the remote receiver’s quantum system via partially entangled states under the controller’s control. The success probability for controlled remote implementation of quantum operation can achieve 1 if the sender and the controller perform proper projective measurements on their entangled particles. Moreover, we also discuss the scheme for remote sharing the partially unknown operations via partially entangled quantum channel. It is shown that the quantum entanglement cost and classical communication can be reduced if the implemented operation belongs to the restrict sets.     

Two Forms Schemes of Deterministic Remote State Preparation for Four-Qubit Cluster-Type State

Two deterministic schemes are put forward to preparing an arbitrary four-qubit cluster-type state remotely by using two Bell states as quantum channel. The coefficients of the prepared states can be not only real, but also complex. To accomplish the schemes, we introduce some novel sets of ingenious measurement basis vectors. Especially, for complex coefficients case, we give two different forms schemes. The receiver will reconstruct the initial state by means of some appropriate unitary operations. The outstanding advantage of the present schemes is that the success probability in all the considered remote state preparation (RSP) can reach 1.

     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

A Reversible Theory of Entanglement and its Relation to the Second Law

We consider the manipulation of multipartite entangled states in the limit of many copies under quantum operations that asymptotically cannot generate entanglement. In stark contrast to the manipulation of entanglement under local operations and classical communication, the entanglement shared by two or more parties can be reversibly interconverted in this setting. The unique entanglement measure is identified as the regularized relative entropy of entanglement, which is shown to be equal to a regularized and smoothed version of the logarithmic robustness of entanglement. Here we give a rigorous proof of this result, which is fundamentally based on a certain recent extension of quantum Stein’s Lemma, giving the best measurement strategy for discriminating several copies of an entangled state from an arbitrary sequence of non-entangled states, with an optimal distinguishability rate equal to the regularized relative entropy of entanglement. We moreover analyse the connection of our approach to axiomatic formulations of the second law of thermodynamics.     

Phase-space representation of quantum systems in the relative-state formulation

A phase-space representation of quantum systems within the framework of the relative-state formulation is proposed. To this end, relative-position and relative-momentum states are introduced and their properties are investigated in detail. Phase-space functions that represent a quantum state vector are constructed in terms of the relative-positive and relative-momentum states, and the quantum dynamics is investigated by using the phase-space functions. Furthermore, probability distributions in phase space are considered by means of the relativestate formulation, and it is shown that the phase-space probability distribution is closely related to the operational probability distribution. The marginal distribution, characteristic function, and operational uncertainty relation are also discussed.