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arXiv:quant-ph/9710001 · Physical Review A

Quantum extension of conditional probability

Nicolas J. Cerf, Christoph Adami

Published Aug 1, 1999• 135 citations
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Abstract

We analyze properties of the quantum conditional amplitude operator [Phys. Rev. Lett. 79, 5194 (1997)], which plays a role similar to that of the conditional probability in classical information theory. The spectrum of the conditional operator that characterizes a quantum bipartite system is shown to be invariant under local unitary transformations and reflects its inseparability. More specifically, it is proven that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. A related separability criterion based on the non-negativity of the von Neumann conditional entropy is also exhibited.

DOI
10.1103/physreva.60.893

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We analyze properties of the quantum conditional amplitude operator [Phys. Rev. Lett. 79, 5194 (1997)], which plays a role similar to that of the conditional probability in classical information theory. The spectrum of the conditional operator that characterizes a quantum bipartite system is shown to be invariant under local unitary transformations and reflects its inseparability. More specifically, it is proven that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. A related separability criterion based on the non-negativity of the von Neumann conditional entropy is also exhibited.
This paper analyzes the properties of the quantum conditional amplitude operator.This operator plays a role similar to that of the conditional probability in classical information theory.One argues that the spectrum of the conditional operator that characterizes a quantum bipartite system is invariant under local unitary transformations,and shows its inseparability.It is proven that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding unity.A related separability condition based on the non-negativity of the von Neumann conditional entropy is obtained.
Inspired by the idea of conditional probabilities, we introduce a variant of conditional density operators. But unlike the conditional probabilities which are bounded by 1, the conditional density operators may have eigenvalues exceeding 1 for entangled states. This has the consequence that although any bivariate classical probability distribution has a natural separable decomposition in terms of conditional probabilities, we do not have a quantum analogue of this separable decomposition in general. The ``nonclassical'' eigenvalues of conditional density operators are indications of entanglement. The resulting separability criterion turns out to be equivalent to the reduction criterion introduced by Horodecki [Phys. Rev. A 59, 4206 (1999)] and Cerf et al. [Phys. Rev. A 60, 898 (1999)]. This supplies an intuitive probabilistic interpretation for the reduction criterion. The conditional density operators are also used to define a form of quantum conditional entropy which provides an alternative mechanism to reveal quantum discord.
Quantum discord, as defined by Olliver and Zurek (2002 Phys. Rev. Lett. 88 017901) as the difference of two natural quantum extensions of the classical mutual information, plays an interesting role in characterizing quantumness of correlations. Inspired by this idea, we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy. Our approach is intrinsic, in contrast to the Olliver–Zurek method that involves extrinsic local measurements. For this purpose, we introduce two alternative variants of quantum conditional entropies via conditional density operators, which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability. The significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
Abstract Quantum coherence characterizes the non-classical feature of a single party system with respect to a local basis. Based on a recently introduced resource framework, coherence can be regarded as a resource and be systematically manipulated and quantified. Operationally, coherence quantifies the intrinsic randomness of the outcome of the projective measurement in the system’s computational basis. However, such a relation is only proven when randomness is characterized by the von Neumann entropy. In this work, we consider several recently proposed coherence measures and relate them to the general uncertainties of the projective measurement outcome conditioned on all the other systems. Our work thus provides a unified framework for redefining several coherence measures via general conditional entropies. Based on the relation, we numerically calculate the coherence measures via semi-definite programming. Furthermore, we discuss the operational meaning of the unified definition. Our result highlights the close relation between single partite coherence and bipartite quantum correlation.
We explore separability of bipartite divisions of mixed Gaussian states based on the positivity of the Abe-Rajagopal (AR) qqq-conditional entropy. The AR qqq-conditional entropic characterization provide more stringent restrictions on separability (in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$) than that obtained from the corresponding von Neumann conditional entropy (q=1q=1q=1 case)---similar to the situation in finite dimensional states. Effectiveness of this approach, in relation to the results obtained by partial transpose criterion, is explicitly analyzed in three illustrative examples of two-mode Gaussian states of physical significance.