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arXiv:1703.01059 · Physical Review A

Non-negativity of conditional von Neumann entropy and global unitary operations

Subhasree Patro, Indranil Chakrabarty, Nirman Ganguly

Published Dec 1, 2017• 28 citations
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Abstract

Conditional von Neumann entropy is an intriguing concept in quantum information theory. In the present work, we examine the effect of global unitary operations on the conditional entropy of the system. We start with a set containing states with a non-negative conditional entropy and find that some states preserve the non-negativity under unitary operations on the composite system. We call this class of states the absolute conditional von Neumann entropy non-negative (ACVENN) class. We characterize such states for $2\ensuremath{\bigotimes}2$--dimensional systems. From a different perspective the characterization accentuates the detection of states whose conditional entropy becomes negative after the global unitary action. Interestingly, we show that this ACVENN class of states forms a set which is convex and compact. This feature enables the existence of Hermitian witness operators. With these we can distinguish the unknown states which will have a negative conditional entropy after the global unitary operation. We also show that this has immediate application to superdense coding and state merging, as the negativity of the conditional entropy plays a key role in both these information processing tasks. Some illustrations followed by analysis are also provided to probe the connection of such states with absolutely separable states and absolutely local states.

DOI
10.1103/physreva.96.062102

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Conditional von Neumann entropy is an intriguing concept in quantum information theory. In the present work, we examine the effect of global unitary operations on the conditional entropy of the system. We start with a set containing states with a non-negative conditional entropy and find that some states preserve the non-negativity under unitary operations on the composite system. We call this class of states the absolute conditional von Neumann entropy non-negative (ACVENN) class. We characterize such states for 2\ensuremath⨂22\ensuremath{\bigotimes}22\ensuremath⨂2--dimensional systems. From a different perspective the characterization accentuates the detection of states whose conditional entropy becomes negative after the global unitary action. Interestingly, we show that this ACVENN class of states forms a set which is convex and compact. This feature enables the existence of Hermitian witness operators. With these we can distinguish the unknown states which will have a negative conditional entropy after the global unitary operation. We also show that this has immediate application to superdense coding and state merging, as the negativity of the conditional entropy plays a key role in both these information processing tasks. Some illustrations followed by analysis are also provided to probe the connection of such states with absolutely separable states and absolutely local states.
Quantum states that possess negative conditional von Neumann entropy provide quantum advantage in several information-theoretic protocols including superdense coding, state merging, distributed private randomness distillation, and one-way entanglement distillation. While entanglement is an important resource, only a subset of entangled states have negative conditional von Neumann entropy. In this work, we characterize the class of density matrices having non-negative conditional von Neumann entropy as convex and compact. This allows us to prove the existence of a Hermitian operator (a witness) for the detection of states having negative conditional entropy for bipartite systems in arbitrary dimensions. We show two constructions of such witnesses. For one of the constructions, the expectation value of the witness in a state is an upper bound to the conditional entropy of the state. We pose the problem of obtaining a tight upper bound to the set of conditional entropies of states in which an operator gives the same expectation value. We solve this convex optimization problem numerically for a two qubit case and find that this enhances the usefulness of our witnesses. We also find that for a particular witness, the estimated tight upper bound matches the value of conditional entropy for Werner states. We explicate the utility of our work in the detection of useful states in several protocols.
Quantum states that possess negative conditional von Neumann entropy provide quantum advantage in several protocols including superdense coding, state merging, distributed private randomness distillation and one-way entanglement distillation. While entanglement is an important resource, only a subset of entangled states have negative conditional von Neumann entropy. Despite this utility, a proper resource theory for conditional von Neumann entropy has not been developed, unlike that of entanglement. We pave the way for such a resource theory by characterizing the class of density matrices having non-negative conditional von Neumann entropy as convex and compact. This allows us to prove the existence of a Hermitian operator (a witness) for the detection of states having negative conditional entropy for bipartite systems in arbitrary dimensions. We show constructions of this witness and explicate its utility in the detection of useful states in the above-mentioned protocols. We provide a local decomposition of the witness and probe its implications in the context of the uncertainty principle.
Negative quantum conditional entropy states are key ingredients for information theoretic tasks such as superdense coding, state merging and one-way entanglement distillation. In this work, we ask: how does one detect if a channel is useful in preparing negative conditional entropy states? We answer this question by introducing the class of A-unital channels, which we show are the largest class of conditional entropy non-decreasing channels. We also prove that A-unital channels are precisely the completely free operations for the class of states with non-negative conditional entropy. Furthermore, we study the relationship between A-unital channels and other classes of channels pertinent to the resource theory of entanglement. We then prove similar results for ACVENN: a previously defined, relevant class of states and also relate the maximum and minimum conditional entropy of a state with its von Neumann entropy. The definition of A-unital channels naturally lends itself to a procedure for determining membership of channels in this class. Thus, our work is valuable for the detection of resourceful channels in the context of conditional entropy.
Chapter 7 discusses quantum conditional entropy, extending the concept of conditional majorization and introducing the notion of negative quantum conditional entropy. The chapter starts with the basic definition of conditional entropy, exploring its key properties like monotonicity and additivity. It further delves into the concepts of conditional min- and max-entropies, emphasizing their roles in quantifying uncertainty in quantum states and their operational significance in quantum information theory. The text presents conditional entropy as a measure sensitive to the effects of entanglement, showing that negative conditional entropy is a distinctive feature of quantum systems, contrasting with the classical domain where entropy values are nonnegative. This negativity is particularly pronounced in the context of maximally entangled states and is connected to the fundamental differences between classical and quantum information processing. Moreover, the chapter includes theorems and exercises to solidify understanding, like the invariance of conditional entropy under local isometric channels and its reduction to entropy for product states. It concludes by underscoring the inevitability of negative conditional entropy in quantum systems, a topic of both theoretical and practical importance in the quantum domain.
The action of global unitary operations can have intriguing effects on the non-local manifestations of a quantum state. States which do not violate Bell-CHSH inequality, can violate it when transformed by a global unitary. This phenomena assumes significance when one exercises to augment non-local resources(in terms of violation of Bell-CHSH inequality) from these seemingly useless states(i.e., in terms of non-local tasks). However, equally intriguing is the existence of states from which no non-local resource can be generated with a global unitary. The present work confirms the existence of such a set, pertaining to two qubit states. The set exhibits counter intuitive features by containing within it some entangled states which remain Bell-CHSH local on the action of any unitary. Furthermore through an analytic characterization of the set we lay down a generic prescription through which one can operationally identify states which can generate non-local resources under the action of a unitary on the composite system. It has also been shown that our prescription remains valid for any linear Bell inequality.
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.