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arXiv:2111.14645 · Physical Review A

Catalytic transformations in coherence theory

Priyabrata Char, Dipayan Chakraborty, Amit Bhar, Indrani Chattopadhyay, Debasis Sarkar

Published Jan 3, 2023• 13 citations
arXivHTMLPDF

Abstract

In 2016, Winter and Yang [A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] provided an operational meaning to relative entropy of coherence and coherence of formation by introducing coherence distillation and dilation protocols in an asymptotic setup. Here we have provided relative entropy of coherence [via incoherent operations (IO)] and coherence of formation (via IO) and quantum incoherent relative entropy [via local quantum incoherent operations with classical communications (LQICC)] as a measure using the concept of a catalyst. We have proved an existential correspondence between asymptotic and catalytic state transformation using IO, local incoherent operations with classical communications, and LQICC. We have also discussed two very important protocols, assisted distillation and quantum incoherent state merging using a catalyst. Monotone properties of relative entropy of coherence, coherence of formation, and quantum incoherent relative entropy under the catalytic transformation are also discussed here.

DOI
10.1103/physreva.107.012404

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In 2016, Winter and Yang [A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] provided an operational meaning to relative entropy of coherence and coherence of formation by introducing coherence distillation and dilation protocols in an asymptotic setup. Here we have provided relative entropy of coherence [via incoherent operations (IO)] and coherence of formation (via IO) and quantum incoherent relative entropy [via local quantum incoherent operations with classical communications (LQICC)] as a measure using the concept of a catalyst. We have proved an existential correspondence between asymptotic and catalytic state transformation using IO, local incoherent operations with classical communications, and LQICC. We have also discussed two very important protocols, assisted distillation and quantum incoherent state merging using a catalyst. Monotone properties of relative entropy of coherence, coherence of formation, and quantum incoherent relative entropy under the catalytic transformation are also discussed here.
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Maximal Rényi Relative Entropy for α>2α>2α>2
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Abstract It is well known that the majorization condition is the necessary and sufficient for the deterministic transformations of pure-coherent states under incoherent operation (IO). In this paper, we present the protocol 1 (product-form) which provides a method for the pure-coherent transformation under IO and overcomes the restriction of the number of Kraus operators of Protocol I (Torun and Yildiz 2018 Phys. Rev. A 97 052331). Then, we show that the corresponding uncorrelated-catalytic protocol (discarding the catalytic state) is not incoherent-preserving. We construct the protocol 2 such that the uncorrelated-catalytic transformation with four-dimensional catalyst state can be implemented probabilistically under dephasing-covariant IO. Especially, the protocol 2 provides the target state from the catalyst-assisted transformation and its equivalent states with respect to Pauli operators <?CDATA X,ZX,ZX,Z?> X , Z .
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.
Quantum catalysts can overcome otherwise impossible quantum state transformations without being consumed, and allowing them to become correlated with the output makes this assistance substantially more powerful. This raises a fundamental question for entanglement theory: which limitations on state manipulation remain when such correlated catalysts are freely available? We answer this question in the positive-partial-transpose (PPT) resource theory, which allows a substantially broader class of operations than local operations and classical communication (LOCC). We identify general conditions under which regularized relative-entropy measures become strongly superadditive, and use them to construct monotones that constrain correlated catalytic PPT transformations without any knowledge of the catalyst. In particular, we prove that the regularized PPT relative entropy is fully additive and strongly superadditive, resolving an open problem in entanglement theory. Most importantly, these constraints show that even arbitrary correlated catalysts cannot restore asymptotic reversibility: for an explicit state, the optimal entanglement distillation rate remains strictly smaller than the entanglement cost. Thus, substantial catalytic assistance does not remove some of the fundamental limitations of mixed-state entanglement manipulation.
Abstract In this chapter the quantification of entanglement is discussed. Beginning with the pure-state entanglement measure called the entropy of entanglement, we discuss the paradigm of local operations and classical communication (LOCC) and its relation to majorization via Nielsen’s majorization theorem. We then turn to the asymptotic setting and discuss the formation and distillation of entanglement and the related entanglement measures: entanglement cost and distillable entanglement, and we examine the notions of distillability and bound entanglement. This brings us to a more general discussion of entanglement measures and monotones, and their desired properties, during which we present the entanglement of formation and concurrence, squashed entanglement, as well as the tangle and its relation to monogamy of entanglement, but also quantities like the relative entropy of entanglement, the Hilbert-Schmidt measure, and the (logarithmic) negativity. Finally we turn to the construction of entanglement witnesses and their geometric interpretation via the Bertlmann-Narnhofer-Thirring theorem