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FROM JOINT CONVEXITY OF QUANTUM RELATIVE ENTROPY TO A CONCAVITY THEOREM OF LIEB
"... Abstract. This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen an ..."
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Abstract. This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb. 1.
Structure of sufficient quantum coarsegrainings
"... B(K) be a coarsegraining and D1, D2 be density matrices on H. In this paper the consequences of the existence of a coarsegraining β: B(K) → B(H) satisfying βT(Ds) = Ds are given. (This means that T is sufficient for D1 and D2.) It is shown that Ds = ∑ r p=1 λs(p)S H s (p)R H (p) (s = 1, 2) shou ..."
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B(K) be a coarsegraining and D1, D2 be density matrices on H. In this paper the consequences of the existence of a coarsegraining β: B(K) → B(H) satisfying βT(Ds) = Ds are given. (This means that T is sufficient for D1 and D2.) It is shown that Ds = ∑ r p=1 λs(p)S H s (p)R H (p) (s = 1, 2) should hold with pairwise orthogonal summands and with commuting factors and with some probability distributions λs(p) for 1 ≤ p ≤ r (s = 1, 2). This decomposition allows to deduce the exact condition for equality in the strong subaddivity of the von Neumann entropy.
On the power of twoparty quantum cryptography
, 2009
"... We study quantum protocols among two distrustful parties. Under the sole assumption of correctness—guaranteeing that honest players obtain their correct outcomes—we show that every protocol implementing a nontrivial primitive necessarily leaks information to a dishonest player. This extends known i ..."
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We study quantum protocols among two distrustful parties. Under the sole assumption of correctness—guaranteeing that honest players obtain their correct outcomes—we show that every protocol implementing a nontrivial primitive necessarily leaks information to a dishonest player. This extends known impossibility results to all nontrivial primitives. We provide a framework for quantifying this leakage and argue that leakage is a good measure for the privacy provided to the players by a given protocol. Our framework also covers the case where the two players are helped by a trusted third party. We show that despite the help of a trusted third party, the players cannot amplify the cryptographic power of any primitive. All our results hold even against quantum honestbutcurious adversaries who honestly follow the protocol but purify their actions and apply a different measurement at the end of the protocol. As concrete examples, we establish lower bounds on the leakage of standard universal twoparty primitives such as oblivious transfer.
Remarks on Kim’s strong subadditivity matrix inequality: extensions and equality conditions
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Entanglement Distillation  A Discourse on Bound Entanglement in Quantum Information Theory
, 2006
"... In recent years entanglement has been recognised as a useful resource in quantum information and computation. This applies primarily to pure state entanglement which is, due to interaction with the environment, rarely available. Decoherence provides the main motivation for the study of entanglement ..."
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In recent years entanglement has been recognised as a useful resource in quantum information and computation. This applies primarily to pure state entanglement which is, due to interaction with the environment, rarely available. Decoherence provides the main motivation for the study of entanglement distillation. A remarkable effect in the context of distillation is the existence of bound entangled states, states from which no pure state entanglement can be distilled. The concept of entanglement distillation also relates to a canonical way of theoretically quantifying mixed state entanglement. This thesis is, apart from a review chapter on distillation, mainly a theoretical study of bound entanglement and the two major open problems in their classification. The first of these is the classification of PPT bound entanglement (separability problem). After having reviewed known tools we study in detail the multipartite permutation criteria, for which we present new results in their classification. We solve an open problem on the existence of certain PPT states. The Schmidt number of a quantum state is a largely unvalued concept, we analyse it in detail and introduce the Schmidt robustness. The notion of Schmidt number is exploited in the study of the second
Some bipartite states do not arise from channels,” quantph/0303141
"... It is wellknown that action of a quantum channel on a state can be represented, using an auxiliary space, as the partial trace of an associated bipartite state. Recently, it was observed that for the bipartite state associated with the optimal average input of the channel, the entanglement of forma ..."
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It is wellknown that action of a quantum channel on a state can be represented, using an auxiliary space, as the partial trace of an associated bipartite state. Recently, it was observed that for the bipartite state associated with the optimal average input of the channel, the entanglement of formation is simply the entropy of the reduced density matrix minus the Holevo capacity. It is natural to ask if every bipartite state can be associated with some channel in this way. We show that the answer is negative. PACS number 03.67; MSC classification 82P68. 1
Stronger subadditivity of entropy
 Phys. Rev. A
, 2005
"... The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[] = −Tr ( ln ) of a density matrix 123 on the product of three Hilbert spaces satisfies S[123] − S[23] ≤ S[12] − S[2]. We strengthen this to S[123] −S[12] ≤ ∑ α nα ( ..."
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The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[] = −Tr ( ln ) of a density matrix 123 on the product of three Hilbert spaces satisfies S[123] − S[23] ≤ S[12] − S[2]. We strengthen this to S[123] −S[12] ≤ ∑ α nα ( S[α 23] −S[α2]), where the nα are weights and the α 23 are partitions of 23. Correspondingly, there is a strengthening of the theorem that the map A ↦ → Tr exp[L + lnA] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl entropy and entropy inequalities for quantum gases. 1
Concurrence and foliations induced by some 1qubit channels
, 2003
"... We start with a short introduction to the roof concept. An elementary discussion of phasedamping channels shows the role of antilinear operators in representing their concurrences. A general expression for some concurrences is derived. We apply it to 1qubit channels of length two, getting the ind ..."
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We start with a short introduction to the roof concept. An elementary discussion of phasedamping channels shows the role of antilinear operators in representing their concurrences. A general expression for some concurrences is derived. We apply it to 1qubit channels of length two, getting the induced foliations of the state space, the optimal decompositions, and the entropy of a state with respect to these channels. For amplitudedamping channels one obtains an expression for the Holevo capacity allowing for easy numerical calculations. 1