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arXiv:quant-ph/0303088 · Physical Review Letters

Locking Classical Correlations in Quantum States

David Peter Divincenzo, Michał Horodecki, Debbie Leung, John A. Smolin, Barbara M. Terhal

Published Feb 12, 2004• 236 citations
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Abstract

We show that there exist bipartite quantum states which contain a large locked classical correlation that is unlocked by a disproportionately small amount of classical communication. In particular, there are (2n+1)-qubit states for which a one-bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. This phenomenon is impossible classically. However, states exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.

DOI
10.1103/physrevlett.92.067902

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We show that there exist bipartite quantum states which contain a large locked classical correlation that is unlocked by a disproportionately small amount of classical communication. In particular, there are (2n+1)-qubit states for which a one-bit message doubles the optimal classical mutual information between measurement results on the subsystems, from n/2 bits to n bits. This phenomenon is impossible classically. However, states exhibiting this behavior need not be entangled. We study the range of states exhibiting this phenomenon and bound its magnitude.
We consider the classical correlations that two observers can extract by measurements on a bipartite quantum state and we discuss how they are related to the quantum mutual information of the state. We show with several examples how complementarity gives rise to a gap between the quantum and the classical correlations and we relate our quantitative finding to the so-called classical correlation locked in a quantum state. We derive upper bounds for the sum of classical correlation obtained by measurements in different mutually unbiased bases and we show that the complementarity gap is also present in the deterministic quantum computation with one quantum bit.
It is known that the maximum classical mutual information, which can be achieved between measurements on pairs of quantum systems, can drastically underestimate the quantum mutual information between them. In this article, we quantify this distinction between classical and quantum information by demonstrating that after removing a logarithmic-sized quantum system from one half of a pair of perfectly correlated bitstrings, even the most sensitive pair of measurements might yield only outcomes essentially independent of each other. This effect is a form of information locking but the definition we use is strictly stronger than those used previously. Moreover, we find that this property is generic, in the sense that it occurs when removing a random subsystem. As such, the effect might be relevant to statistical mechanics or black hole physics. While previous works had always assumed a uniform message, we assume only a min-entropy bound and also explore the effect of entanglement. We find that classical information is strongly locked almost until it can be completely decoded. Finally, we exhibit a quantum key distribution protocol that is 'secure' in the sense of accessible information but in which leakage of even a logarithmic number of bits compromises the secrecy of all others.
The intuition that the amount of classical correlations between two systems be bounded by their size does not hold true in general for quantum states. In the setting of information locking, measurements on a pair of quantum systems that appear to be completely uncorrelated can become maximally correlated with a small increment in the size of one of the systems. A new information locking scheme based on generic unitary channels is presented and a strengthened definition of locking based on a measure of indistinguishability is used. The new definition demonstrates that classical information can be kept arbitrarily low until it can be completely decoded. Unlike previous locking results, non-uniform input messages are allowed and shared entanglement between the pair of quantum systems is considered. Whereas past locking results relied on schemes with an explicit "key" register, this requirement is eliminated in favor of an arbitrary quantum subsystem. Furthermore, past results considered only projective measurements at the receiver. Here locking effects can be shown even in the case where the receiver is armed with the most general type of measurement. The locking effect is found to be generic and finds applications in entropic security and models for black hole evaporation.
The mutual information has been successfully used as a measure of correlation strength between quantum systems, especially for bipartite systems. Here we examine the use of this measure for multipartite quantum systems. For system of qubits we find that the difference between `classical' and `quantum' regimes of correlation strength amounts to just 1 bit of information, at most. We show that the information content of a correlation can be expanded into correlations between pairwise components and demonstrate that in terms of this information-based measure of correlation the GHZ states are the only states that simultaneously optimize these pairwise correlations for systems of qubits.
There exist bipartite quantum states that cost entanglement to create, yet from which no singlet can be distilled. This extreme irreversibility is known as bound entanglement. A quarter-century ago, Gisin and Wolf asked whether classical information theory admits the same phenomenon. Are there correlations, shared by two parties and an eavesdropper, that cost secret bits to create, yet from which none can be distilled? This question is part of a broader program exploring the relation between quantum and classical information theory. We answer it in the affirmative, providing a simple example, a distribution of two bits and a trit. Bound entanglement thus has a classical counterpart, bound information. An even more direct correspondence was originally conjectured, namely that measuring purifications of bound-entangled states yields distributions with bound information. The very states that motivated the conjecture, however, yield no bound information when measured in the standard basis, whereas suitable measurements of purifications of separable states do. The analogy between bound entanglement and bound information thus lies in the accounting of resources, not in a correspondence between individual quantum states and the classical probability distributions induced by measuring them.
In this paper, two conjectures which were proposed in Luo et al (2010 Phys. Rev. A 82 052122 ) on the correlations in a bipartite state ρ AB are studied. If the mutual information I (ρ AB ) between two quantum systems A and B before any measurement is considered as the total amount of correlations in the state ρ AB , then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations C (ρ AB ) in the state ρ AB are defined by the maximizing mutual information between two quantum systems A and B after von Neumann measurements on system B . We show that it is upper bounded by the von Neumann entropies of both subsystems A and B , which answered the conjecture on the classical correlation. If the quantum correlations Q (ρ AB ) in the state ρ AB are defined by Q (ρ AB ) = I (ρ AB ) − C (ρ AB ), we also show that it is upper bounded by the von Neumann entropy of subsystem B . It is also found that Q (ρ AB ) is upper bounded by the von Neumann entropy of subsystem A for a class of states.