We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.
We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.
Given an n×n distance matrix, we construct an infinite family of distances dp (where p≥2) on the complex projective space P(Cn) modelling the space of pure states of an n-level quantum system. The construction can be seen as providing a natural way to isometrically embed any given finite metric space into the space of pure quantum states 'spanned' upon it. In order to show that the maps dp are indeed distance functions -- in particular, that they satisfy the triangle inequality -- we employ methods of analysis, multilinear algebra and convex geometry, obtaining a nontrivial auxiliary convexity result in the process. In addition, a way of extending the distances dp to mixed states is proposed for a broad class of distance matrices.
Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.
Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
A quantum ensemble ${({p}_{x},{\ensuremath{\rho}}_{x})}$ is a set of quantum states each occurring randomly with a given probability. Quantum ensembles are necessary to describe situations with incomplete a priori information, such as the output of a stochastic quantum channel (generalized measurement), and play a central role in quantum communication. In this paper, we propose measures of distance and fidelity between two quantum ensembles. We consider two approaches: the first one is based on the ability to mimic one ensemble given the other one as a resource and is closely related to the Monge-Kantorovich optimal transportation problem, while the second one uses the idea of extended-Hilbert-space (EHS) representations which introduce auxiliary pointer (or flag) states. Both types of measures enjoy a number of desirable properties. The Kantorovich measures, albeit monotonic under deterministic quantum operations, are not monotonic under generalized measurements. In contrast, the EHS measures are. This property can be regarded as a generalization of the monotonicity under deterministic maps of the trace distance and the fidelity between states. The EHS measures are equivalent to convex optimization problems and are bounded by the Kantorovich measures which are equivalent to linear programs. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides an interpretation of the fidelity between mixed states as the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures. These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.
Niels Bohr supposedly said that if quantum mechanics did not make you dizzy then you did not understand it. I think that the same can be said about statistical inference. Robert D. Cousins In this chapter we quantify how easy it may be too distinguish probability distributions from each other (a discussion that was started in Chapter 2). The issue is a very practical one and arises whenever one is called upon to make a decision based on imperfect data. There is no unique answer because everything depends on the data – the l 1 -distance appears if there has been just one sampling of the distributions, the relative entropy governs the approach to the ‘true’ distribution as the number of samplings goes to infinity, and so on. The quantum case is even subtler. A quantum state always stands ready to produce a large variety of classical probability distributions, depending on the choice of measurement procedure. It is no longer possible to distinguish pure states from each other with certainty, unless they are orthogonal. The basic idea behind the quantum distinguishability measures is the same as that which allowed us, in Section 5.3, to relate the Fubini–Study metric to the Fisher–Rao metric. We will optimize over all possible measurements. Classical distinguishability measures If a distance function has an operational significance as a measure of statistical distinguishability, then we expect it to be monotone (and decreasing) under general stochastic maps. Coarse graining means that information is being discarded, and this cannot increase distinguishability.
The counter-intuitive features of Quantum Mechanics make it possible to solve problems and perform tasks that are beyond the abilities of classical computers and classical communication devices. The area of quantum information processing studies how representing information by quantum states can help achieving such improvements.In this research, we use basic notions of quantum information (mainly entanglement, Bloch sphere, and geometrical distances between quantum states) for analyzing relations of quantum states to each other and quantum cryptographic protocols.Entanglement is an important feature of quantum states. Intuitively (and, partly, inaccurately), entanglement represents quantum (non-classical) correlations between several different quantum systems. Entanglement is one of the most important quantum phenomena, and it has many uses in quantum information, quantum communication, and quantum computing.Some quantum states can be geometrically represented by the Bloch sphere: the unit sphere in the three-dimensional Euclidean space. The "standard" quantum states, to which the laws of Quantum Mechanics directly apply, are called pure states. Other states are the mixed states: probability distributions ("mixtures") of several pure states. The points on the Bloch sphere are the pure states, and those inside the Bloch sphere are the mixed states. This geometrical representation is useful and intuitive for many purposes.We provide a geometrical analysis of entanglement for all the quantum mixed states of rank 2 (all the mixtures of exactly two pure states): for any such state (in any dimension), we define a generalized Bloch sphere by using the two pure states, and we analyze this state and its neighbor states inside this Bloch sphere. We look at the set of non-entangled states ("separable states") in the Bloch sphere and characterize it into exactly five possible classes. We give examples for each class and prove that there are no other classes. In addition, we suggest possible definitions of "entanglement measures" by using the "trace distance" from the nearest separable state.Many types of distances between quantum states can be defined. One of the most useful distances is the trace distance, which bounds the "distinguishability" between the states. The trace distance is very useful in quantum information and quantum cryptography, and it also has a simple geometrical interpretation: it is half of the Euclidean distance between the states in the Bloch sphere.Quantum key distribution (QKD) protocols make it possible for two participators to achieve the classically-impossible task of generating a secret random shared key even if their adversary is computationally unlimited. Several important QKD protocols, including the first protocol of Bennett and Brassard (BB84), have their unconditional security proved against adversaries performing the most general attacks in a theoretical (idealized) setting. We discuss a slightly different protocol, named "BB84-INFO-z", and prove it secure against a broad class of attacks (the collective attacks). Moreover, we make use of the "trace distance" for making our security proof more "composable" than similar security proofs for BB84: namely, for making a step towards proving that the secret key remains secret even when the two participators actually use it for cryptographic purposes.