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arXiv:2001.11237 · Physical Review A

Witnessing negative conditional entropy

Mahathi Vempati, Nirman Ganguly, Indranil Chakrabarty, Arun Kumar Pati

Published Jul 19, 2021• 22 citations
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Abstract

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.

DOI
10.1103/physreva.104.012417

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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.
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.
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.
Quantum information theory explores numerous properties that surpass classical paradigms, offering novel applications and benefits. Among these properties, negative conditional von Neumann entropy (CVNE) is particularly significant in entangled quantum systems, serving as an indicator of potential advantages in various information-theoretic tasks, despite its indirect observability. In this paper, we investigate the relationship between CVNE and the violation of Bell inequalities. Our goal is to establish upper bounds on CVNE through semidefinite programming applied to entangled qubits and qutrits, utilizing selected Bell operators. Our findings reveal that a semi-device-independent certification of negative CVNE is achievable and could be practically beneficial. We further explore two types of robustness: robustness against detection efficiency loopholes, measured by relative violation, and robustness against white noise and imperfections in state preparation, measured by critical visibility. Additionally, we analyze parametrized families of Bell inequalities to identify optimal parameters for different robustness criteria. This study demonstrates that different Bell inequalities exhibit varying degrees of robustness depending on the desired properties, such as the type of noise resistance or the target level of negative CVNE. By bridging the gap between Bell inequalities and CVNE, our research enhances understanding of the quantum properties of entangled systems and offers insights for practical quantum information processing tasks.
In this work, we prove that while all measures of mixedness can be used to witness entanglement, no measure of mixedness is more sensitive than the negativity of the partial transpose. However, computing either the negativity or differences between von Neumann entropies to witness entanglement requires complete knowledge of the joint density matrix (and is therefore not practical at high dimension). In light of this, we examine joint vs marginal purities as a witness of entanglement, (which can be obtained directly through interference measurements) and find that comparing purities is actually more sensitive at witnessing entanglement than using von Neumann entropies while also providing tight upper and lower bounds to it in the high-entanglement limit.
Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying Tr [ W σ sep ] ⩾ 0 for all separable states σ sep ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, u W ⩾ Tr [ W σ sep ] ⩾ 0. An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that W + M = u W I ⊗ I . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. W + M = K is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.