A Generalization of Quantum Stein’s Lemma

Brandão, Fernando G. S. L.; Plenio, Martin B.

doi:10.1007/s00220-010-1005-z

Cited by 104 publications

(191 citation statements)

References 62 publications

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“…Furthermore, it was also shown in Proposition II.1 of Ref. [30] that in this case the relative entropy of a resource can be expressed as in Eq. (4).…”

Section: Sðρ∥σþ;mentioning

confidence: 88%

Reversible Framework for Quantum Resource Theories

2015

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In recent years it has been recognized that properties of physical systems such as entanglement, athermality, and asymmetry, can be viewed as resources for important tasks in quantum information, thermodynamics, and other areas of physics. This recognition was followed by the development of specific quantum resource theories (QRTs), such as entanglement theory, determining how quantum states that cannot be prepared under certain restrictions may be manipulated and used to circumvent the restrictions. Here we discuss the general structure of QRTs, and show that under a few assumptions (such as convexity of the set of free states), a QRT is asymptotically reversible if its set of allowed operations is maximal, that is, if the allowed operations are the set of all operations that do not generate (asymptotically) a resource. In this case, the asymptotic conversion rate is given in terms of the regularized relative entropy of a resource which is the unique measure or quantifier of the resource in the asymptotic limit of many copies of the state. This measure also equals the smoothed version of the logarithmic robustness of the resource. Classical and quantum information theories can be viewed as examples of theories of interconversions among different resources [1]. These resources are classified as being quantum or classical, dynamic or static, noisy or noiseless, and therefore enable a plethora of quantum information processing tasks [2,3]. For example, quantum teleportation can be viewed as a resource interconversion task in which one entangled bit (a quantum static noiseless resource) is transformed by local operations and classical communication (LOCC) into a single use of a quantum channel (a quantum dynamic noiseless resource) [4]. Just as the restriction of LOCC leads to the theory of entanglement [5], in general, every restriction on quantum operations defines a resource theory, determining how quantum states that cannot be prepared under the restriction may be manipulated and used to circumvent the restriction.The scope of quantum resource theories (QRTs) goes far beyond quantum information science. In recent years a lot of work has been done formulating QRTs in different areas of physics, such as the resource theory of athermality in quantum thermodynamics [6][7][8][9][10][11], the resource theory of asymmetry [12,13] (which led to generalizations of important theorems in physics such as Noether's theorem [13]), the resource theory of non-Gaussianity in quantum optics [14,15], the resource theory of stabilizer computation in quantum computing [16], noncontextuality in the foundations of quantum physics [17], and more recently it was suggested that non-Markovian evolution can be formulated as a resource theory [18]. In addition, tools and ideas from quantum resource theories have been applied in many-body physics (see, e.g., Ref. [19] and references therein), and even for a universal formulation of the uncertainty principle [20]. Furthermore, very recently an abstract formulation using concepts from categor...

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“…Furthermore, it was also shown in Proposition II.1 of Ref. [30] that in this case the relative entropy of a resource can be expressed as in Eq. (4).…”

Section: Sðρ∥σþ;mentioning

confidence: 88%

Reversible Framework for Quantum Resource Theories

2015

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“…Therefore, under asymptotically non-entangling operations the amount of entanglement of any multipartite state is completely determined by how distinguishable the latter is from a state that only contains classical correlations. Furthermore, we showed in Corollary III.3 of [12] that…”

Section: Corollary Iii4 For Two Multipartite Statesmentioning

confidence: 84%

“…As shown in Ref. [12] and discussed in section IV, this measure is related to the optimal rate of discrimination from many copies of an entangled state to a separable states. Therefore, under asymptotically non-entangling operations the amount of entanglement of any multipartite state is completely determined by how distinguishable the latter is from a state that only contains classical correlations.…”

Section: Corollary Iii4 For Two Multipartite Statesmentioning

confidence: 90%

A Reversible Theory of Entanglement and its Relation to the Second Law

Brandão

Plenio

2010

Commun. Math. Phys.

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171

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We consider the manipulation of multipartite entangled states in the limit of many copies under quantum operations that asymptotically cannot generate entanglement. As announced in [Brandão and Plenio, Nature Physics 4, 8 (2008)], and in stark contrast to the manipulation of entanglement under local operations and classical communication, the entanglement shared by two or more parties can be reversibly interconverted in this setting. The unique entanglement measure is identified as the regularized relative entropy of entanglement, which is shown to be equal to a regularized and smoothed version of the logarithmic robustness of entanglement.Here we give a rigorous proof of this result, which is fundamentally based on a certain recent extension of quantum Stein's Lemma proved in [Brandão and Plenio, Commun. Math. 295, 791 (2010)], giving the best measurement strategy for discriminating several copies of an entangled state from an arbitrary sequence of non-entangled states, with an optimal distinguishability rate equal to the regularized relative entropy of entanglement. We moreover analyse the connection of our approach to axiomatic formulations of the second law of thermodynamics.

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“…These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

mentioning

confidence: 99%

“…Luckily, it is possible to circumvent this obstruction. For example, Renner's exponential de Finetti theorem employs the "almost de Finetti states" and has an error bound that decreases exponentially in n − k [9], being very useful in dealing with cryptography or information theory problems [9,13,14].…”

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confidence: 99%

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Smith²

2015

Phys. Rev. Lett.

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We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.

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A Generalization of Quantum Stein’s Lemma

Cited by 104 publications

References 62 publications

Reversible Framework for Quantum Resource Theories

Reversible Framework for Quantum Resource Theories

A Reversible Theory of Entanglement and its Relation to the Second Law

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

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