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arXiv:1907.07707 · Journal of Physics A Mathematical and Theoretical

Generalized Holevo theorem and distinguishability notions

Diego G. Bussandri, Pedro W Lamberti

Published Nov 28, 2019• 4 citations
arXivHTMLPDF

Abstract

Abstract We present a generalization of the Holevo theorem through distance measures between quantum states, showing that each of these leads to an alternative Holevo theorem. This result involves two quantities: the distance-based Holevo quantity and the generalized accessible information I d . Additionally, we consider three distinguishability notions relevant to quantum cryptography, applying the resulting new inequalities to qubits ensembles. For some well-known distinguishability notion, we show that for any ensemble of two qubits the generalized quantities— I d and —are equal. On the other hand, by using a paradigmatic example, we show that the quantities corresponding to the Bures distance captures not only the non-commutativity of the ensemble but also its purity.

DOI
10.1088/1751-8121/ab5cfa

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Abstract We present a generalization of the Holevo theorem through distance measures between quantum states, showing that each of these leads to an alternative Holevo theorem. This result involves two quantities: the distance-based Holevo quantity and the generalized accessible information I d . Additionally, we consider three distinguishability notions relevant to quantum cryptography, applying the resulting new inequalities to qubits ensembles. For some well-known distinguishability notion, we show that for any ensemble of two qubits the generalized quantities— I d and —are equal. On the other hand, by using a paradigmatic example, we show that the quantities corresponding to the Bures distance captures not only the non-commutativity of the ensemble but also its purity.
A basic property of distinguishability is that it is non-increasing under further quantum operations. Following this, we generalize two measures of distinguishability of pure states--fidelity and von Neumann entropy, to mixed states as self-consistent measures. Then we extend these two measures to quantum operations. The information-theoretic point of the generalized Holevo quantity of an ensemble of quantum operations is constructed. Preferably it is an additive measure. The exact formula for SU(2) ensemble is presented. With the aid of the formula, we show Jozsa-Schlienz paradox that states as a whole are less distinguishable while all pairwise are more distinguishable in an ensemble of quantum states, also occurs in an ensemble of quantum operations, even in the minimal dimensional case SU(2) ensemble.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.
The accessible information acc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed chi(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the acc(E) and chi(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information acc(E) of an ensemble E = {(p, rho_0), (1-p, rho_1)}, with 0 <= p <= 1, as functions of p and chi(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states rho_0,rho_1. As an intermediate step, therefore, we give new relations between the two quantities acc(E), chi(E) and the fidelity B(rho_0,rho_1).
The notion of divergence information of an ensemble of probability distributions was introduced by Jain, Radhakrishnan and Sen in Jain et al. (2002; 2009) in the context of the ‘substate theorem’. Since then, divergence has been recognised as a more natural measure of information in several situations in both quantum and classical communication. We construct ensembles of probability distributions for which divergence information may be significantly smaller than the more standard Holevo information. As a result, we establish that bounds previously shown for Holevo information are weaker than similar ones shown for divergence information.
Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be described by distinct probability distributions. In this work, we quantify the inaccessible information of a model in terms of the maximum distinguishability of probability distributions associated to any pair of ensembles with identical density operators, as quantified by the total variation distance of the distributions. We obtain a family of lower bounds on this maximum distinguishability in terms of experimentally measurable quantities. In the case of an ideal qubit this leads to a lower bound of, approximately, 0.07. These bounds can also be interpreted as a new class of robust preparation non-contextuality inequalities. Our non-contextuality inequalities are phrased in terms of generalizations of max-relative entropy and trace distance for general operational theories, which could be of independent interest. Under sufficiently strong noise any quantum system becomes preparation non-contextual, i.e., can be described by models with zero inaccessible information. Using our non-contextuality inequalities, we show that this can happen only if the noise channel has the average gate fidelity less than or equal to 1/D(1+1/2+...+1/D), where D is the dimension of the Hilbert space.