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Structural characterization and condition for measurement statistics preservation of a unital quantum operation

Lee, KY, Chau, HF, Fung, FCH

Published Jan 1, 2013• 1 citation
OpenAlex

Abstract

We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation $\phi$ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with $\phi$, we can completely characterize the kind of quantum states that are fixed by $\phi$. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of St{\o}rmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.

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Structural characterization and condition for measurement statistics preservation of a unital quantum operation

Kai-Yan Lee, Chi-Hang Fred Fung, H F Chau

We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation $ϕ$ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with $ϕ$, we can completely characterize the kind of quantum states that are fixed by $ϕ$. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of Størmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.

arXiv:1112.1137•Journal of Physics A Mathematical and Theoretical•May 7, 2013•0 citations

Quantum operations fixing a convex cone of density operators on ${{\mathcal T}}(H)$

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The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity of two quantum operations and determine what kind of measurement statistics are preserved by a unital quantum operation. Our work extends the results of Lee et al (2013 J. Phys. A: Math. Theor. 46 205305) to the infinite dimensional case and also corrects part (a) of theorem 6 in Lee et al (2013 J. Phys. A: Math. Theor. 46 205305).

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Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

Marcell Gaál

Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten $p$ -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a $C^{\ast }$ -algebra.

Canadian Mathematical Bulletin•Mar 22, 2018•3 citations
We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation ϕ\phiϕ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with ϕ\phiϕ, we can completely characterize the kind of quantum states that are fixed by ϕ\phiϕ. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of St{\o}rmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.
We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation ϕϕϕ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with ϕϕϕ, we can completely characterize the kind of quantum states that are fixed by ϕϕϕ. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of Størmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.
Quantum operations fixing a convex cone of density operators on T(H){{\mathcal T}}(H)T(H)
The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity of two quantum operations and determine what kind of measurement statistics are preserved by a unital quantum operation. Our work extends the results of Lee et al (2013 J. Phys. A: Math. Theor. 46 205305) to the infinite dimensional case and also corrects part (a) of theorem 6 in Lee et al (2013 J. Phys. A: Math. Theor. 46 205305).
Abstract The max-relative entropy and the conditional min-entropy of a quantum state plays a central role in one-shot and zero-error quantum information theory. One attractive feature of this quantity is that it can be expressed as an optimization over the cone of positive semidefinite operators. Recently, it was shown that when replacing this cone with the cone of separable operators, a new type of conditional min-entropy emerges that admits an operational interpretation in terms of communicating classical information over a quantum channel. In this work, we explore more deeply the idea of building information-theoretic quantities from different base cones and determine which results in quantum information theory rely upon the positive semidefinite cone and which can be generalized. In terms of asymptotic information processing, we find that the standard equipartition properties break down if a given cone fails to approximate the positive semidefinite cone sufficiently well. We also show that the near-equivalence of the smooth max and Hartley entropies breaks down in this setting. We present parallel results for the extended conditional min-entropy, which requires extending the notion of k-superpositive channels to superchannels. On the other hand, we show that for classical-quantum states the separable cone is sufficient to re-cover the asymptotic theory, thereby drawing a strong distinction between the fully and partial quantum settings. We also present operational uses of this framework. We show that the cone restricted min-entropy of a Choi operator captures a measure of entanglement-assisted noiseless classical communication using restricted measurements. We also introduce a novel min-entropy-like quantity that captures the conditions for when one quantum channel can be transformed into another using bistochastic pre-processing. Lastly, we relate this framework to general conic norms and their non-additivity. Throughout this work, we concretely study generalized entropies in resource theories that capture locality and resource theories of coherence/Abelian symmetries.
Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient conditions for this property expressed in terms of the Kraus representation of quantum channels and operations are presented. The characterization of a positive map "preserving continuity of the entropy" (in the sense that continuity of the entropy on a set of input states implies continuity of its output entropy on this set) is obtained and its applications to the class of quantum operations are considered. The special relation between continuity properties of the output entropies of complementary quantum operations is established and its corollaries are discussed. Continuity conditions for the output entropy of positive maps considered as a function of a pair (map, input state) are also obtained. Several applications to the quantum information theory are presented in the last part of the paper.
Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.