On the extreme points of quantum channels

Friedland, Shmuel; Loewy, Raphael

doi:10.1016/j.laa.2016.02.001

Cited by 10 publications

(19 citation statements)

References 16 publications

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“…It is also shown recently, from a semi-algebraic geometry approach, that the set of extreme channels dominates the set of generalized extreme channels [13]. For clarity, we denote an extreme channel as E e , a generalized extreme channel as E g , and a quasi-extreme channel as E q .…”

Section: Convex Set Of Quantum Channelsmentioning

confidence: 99%

“…It is well known that an M -dimensional while rank-k hermitian matrix contains k(2M −k) parameters (also see Ref. [13]), then a rank-d generalized extreme qudit channel contains…”

Section: Extreme Qutrit-to-qubit and Qubit-to-qutrit Channelsmentioning

confidence: 99%

“…This means any convex combination of channels still leads to a valid quantum channel. The convex set of quantum channels and also other quantum objects such as states and observables have been a major focus of mathematical characterization lately [9][10][11][12][13], and exploring the convexity can benefit tasks involving quantum channels, such as quantum channel simulation [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Convex decomposition of dimension-altering quantum channels

Wang

2016

Int. J. Quantum Inform.

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Quantum channels, which are completely positive and trace preserving mappings, can alter the dimension of a system; e.g., a quantum channel from a qubit to a qutrit. We study the convex set properties of dimension-altering quantum channels, and particularly the channel decomposition problem in terms of convex sum of extreme channels. We provide various quantum circuit representations of extreme and generalized extreme channels, which can be employed in an optimization to approximately decompose an arbitrary channel. Numerical simulations of low-dimensional channels are performed to demonstrate our channel decomposition scheme.

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Section: Convex Set Of Quantum Channelsmentioning

confidence: 99%

“…It is well known that an M -dimensional while rank-k hermitian matrix contains k(2M −k) parameters (also see Ref. [13]), then a rank-d generalized extreme qudit channel contains…”

Section: Extreme Qutrit-to-qubit and Qubit-to-qutrit Channelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convex decomposition of dimension-altering quantum channels

Wang

2016

Int. J. Quantum Inform.

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“…In particular, a tight bound on the number of generalized extreme channels, i.e., channels which lie in the closure of the set of all extreme channels, required for such a convex decomposition is not known [5]. The set of extreme channels has been described by Friedland and Loewy [6] using the framework of semi-algebraic geometry. In contrast to his work, we consider the set of extreme channels in the framework of differential geometry.…”

Section: Introductionmentioning

confidence: 99%

Smooth manifold structure for extreme channels

Iten

Colbeck

2018

Journal of Mathematical Physics

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A quantum channel from a system A of dimension dA to a system B of dimension dB is a completely positive trace-preserving map from complex dA × dA to dB × dB matrices, and the set of all such maps with Kraus rank r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the rdB ×dA dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system A to system B of a fixed Kraus rank r is a smooth submanifold of dimension 2rdAdB − d 2 A − r 2 of the set of all Choi-states of rank r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from A to B arbitrarily well.

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“…This is equivalent to the assumption that µ is trace preserving, i.e. [3]. A more general definiton can be stated as in [4,7].…”

Section: Representation Of Quantum Channelsmentioning

confidence: 99%

Infinite dimensional generalizations of Choi’s Theorem

Friedland

2019

Special Matrices

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In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi's characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A map µ is called a quantum channel, if it is trace preserving, and µ is called a quantum subchannel if it decreases the trace of a positive operator. We give simple neccesary and sufficient condtions for µ to be a quantum subchannel. We show that µ is a quantum subchannel if and only if it has Hellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

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On the extreme points of quantum channels

Cited by 10 publications

References 16 publications

Convex decomposition of dimension-altering quantum channels

Convex decomposition of dimension-altering quantum channels

Smooth manifold structure for extreme channels

Infinite dimensional generalizations of Choi’s Theorem

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