Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

Haagerup, Uffe; Musat, Magdalena

doi:10.1007/s00220-011-1216-y

Cited by 126 publications

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“…noisy operations [46][47][48], generated by preparing ancillas in the microcanonical state, turning on a unitary dynamics, and discarding the ancillas;3. unital channels [49,50], defined as the quantum processes that preserve the microcanonical state.These three sets are strictly different: the set of random unitary channels is strictly contained in the set of noisy operations [51], and the latter is strictly contained in the set of unital channels [52]. In spite of this, the three sets are equivalent in terms of state convertibility [48].…”

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confidence: 99%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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Microcanonical thermodynamics studies the operations that can be performed on systems with welldefined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: (i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, (ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and (iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.1. random unitary channels [43][44][45], arising from unitary dynamics with randomly fluctuating parameters;2. noisy operations [46][47][48], generated by preparing ancillas in the microcanonical state, turning on a unitary dynamics, and discarding the ancillas;3. unital channels [49,50], defined as the quantum processes that preserve the microcanonical state.These three sets are strictly different: the set of random unitary channels is strictly contained in the set of noisy operations [51], and the latter is strictly contained in the set of unital channels [52]. In spite of this, the three sets are equivalent in terms of state convertibility [48]. This means that all the natural candidates for the sets of free operations induce the same notion of resource. This resource is generally called purity, and plays a fundamental role in many quantum protocols [53].In this paper we extend the paradigm of microcanonical thermodynamics from quantum theory to arbitrary physical theories [54][55][56][57][58][59][60][61]. We propose two minimal requirements a probabilistic theory must satisfy in order to support a microcanonical description, and, when these requirements are satisfied, we provide a general operational definition of random reversible, noisy, and unital operations. We then focus on a special class of theories, called sharp theories with purification, which are appealing for the foundations of thermodynamics [62], and have also been studied for their computational power [63, 64] and interference properties [65]. In sharp theories with purification, we show that the three sets of operations satisfy the same inclusion relations as in quantum theory, with...

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confidence: 99%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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“…of the Rota Dilation constructed with [Ric, and the proof of [HaM,Theorem 5.3] is also hyperfinite. Using the above ideas, one can prove the following theorem.…”

Section: Using (32) Proposition 22 and Lemma 32 We Obtain Thatmentioning

confidence: 99%

“…Now, we are ready to give the proof of Theorem 1.5. Proof of Theorem 1.5 : By [Ric,pages 4371] and the proof of [HaM,Theorem 5.3], for any t 0, the operator T t = (T t 2 ) 2 admits a Rota dilation (see [HaM,Definition 5.1] and [JMX,page 124] for a precise definition)…”

Section: Then the Banach Spacementioning

confidence: 99%

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Analytic semigroups on vector valued noncommutative L^p-spaces

Arhancet

2013

Studia Math.

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We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or a R-analytic semigroup (Tt ⊗ IdE) t 0 on the vector valued noncommutative L p -space L p (M, E). Moreover, we give applications to the H ∞ (Σ θ ) functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu. IntroductionIt is shown in [JMX] that whenever (T t ) t 0 is a noncommutative diffusion semigroup on a von Neumann algebra M equipped with a faithful normal state such that each T t satisfies the Rota dilation property, then the negative generator of itsis the open sector of angle 2θ around the positive real axis (0, +∞). Our first principal result is an extension of this theorem to the vector valued case. We use a different approach using R-analyticity instead of square functions.In order to describe our result, we need several definitions. (1) T is completely positivewhere (σ φ t ) t∈R and (σ ψ t ) t∈R denote the automorphism groups of the states φ and ψ respectively.In particular, when (M, φ) = (N, ψ), we say that T is a φ-Markov map.A linear map T : M → N satisfying conditions (1) − (3) above is normal. If, moreover, condition (4) is satisfied, then it is known that there exists a unique completely positive, unital mapThe next definition is a variation of the one of [AnD] (see also [Ric] and [HaM] Neumann algebra P with QWEP equipped with a faithful normal state χ , and * -monomorphisms J 0 : N → P and J 1 : M → P such that J 0 is (φ, χ)-Markov and J 1 is (ψ, χ)-Markov, satisfying, moreover, T = J * 0 • J 1 . We say that T is hyper-factorizable if the same property is true with a hyperfinite von Neumann algebra P .

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“…Here, we focus on density matrices in coupled quantum system M n (C)⊗ M n (C) whose restrictions are the same or I /n. A set of such density matrices is considered in many literature [7,[11][12][13][14]16]. A density matrix whose restrictions are I /n corresponds to a unital completely positive trace-preserving map.…”

Section: Example 6 Consider Mubs In M 6 (C)mentioning

confidence: 99%

Examples of conditional SIC-POVMs

Ohno

Petz

2015

Quantum Inf Process

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The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.

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Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

Cited by 126 publications

References 32 publications

Microcanonical thermodynamics in general physical theories

Microcanonical thermodynamics in general physical theories

Analytic semigroups on vector valued noncommutative L^p-spaces

Examples of conditional SIC-POVMs

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Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

Cited by 126 publications

References 32 publications

Microcanonical thermodynamics in general physical theories

Microcanonical thermodynamics in general physical theories

Analytic semigroups on vector valued noncommutative Lp-spaces

Examples of conditional SIC-POVMs

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Analytic semigroups on vector valued noncommutative L^p-spaces