arXiv:1609.05890

On Faces of the set of Quantum Channels

Raphael Loewy

Published Sep 18, 2016• 0 citations

Abstract

A linear map $L$ from ${\mathbb C}^{n \times n}$ into ${\mathbb C}^{n \times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\cal L}_n$ of all such quantum channels is known to be a compact convex set. While the extreme points of ${\cal L}_n$ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix $Z(L)$ associated with the quantum channel $L$, we compute the maximum dimension of a proper face of ${\cal L}_n$, and in addition the possible dimensions of faces generated by $L$ when $rank \ Z(L)=2 $.

DOI: 10.48550/arxiv.1609.05890

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arXiv:1609.05890

On Faces of the set of Quantum Channels

Raphael Loewy

Published Sep 18, 2016• 0 citations

arXiv HTML PDF

Abstract

A linear map

L

from

{\mathbb C}^{n \times n}

into

{\mathbb C}^{n \times n}

is called a quantum channel if it is completely positive and trace preserving. The set

{\cal L}_n

of all such quantum channels is known to be a compact convex set. While the extreme points of

{\cal L}_n

can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix

Z(L)

associated with the quantum channel

L

, we compute the maximum dimension of a proper face of

{\cal L}_n

, and in addition the possible dimensions of faces generated by

L

when

rank \ Z(L)=2

DOI: 10.48550/arxiv.1609.05890

Related papers

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Raban Iten, Roger Colbeck

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Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space $X$ to another one $Y$ is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space $X$ to another one $Y$ is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.

arXiv:2305.05795May 9, 20230 citations

A linear map

L

from

{\mathbb C}^{n \times n}

into

{\mathbb C}^{n \times n}

is called a quantum channel if it is completely positive and trace preserving. The set

{\cal L}_n

of all such quantum channels is known to be a compact convex set. While the extreme points of

{\cal L}_n

can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix

Z(L)

associated with the quantum channel

L

, we compute the maximum dimension of a proper face of

{\cal L}_n

, and in addition the possible dimensions of faces generated by

L

when

rank \ Z(L)=2

A quantum channel from a system

A

of dimension

d_A

to a system

B

of dimension

d_B

is a completely positive trace-preserving map from complex

d_A\times d_A

d_B\times d_B

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

rd_B\times d_A

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

2rd_Ad_B-d_A^2-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

B

arbitrarily well.

\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2

for quantum channels

\mathcal{E}

and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where

\|\mathcal{E}\|_2

is the Hilbert-Schmidt norm of

\mathcal{E}

and

\widetilde{\mathcal{E}}

is a complementary channel of

\mathcal{E}.

Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained.

X

to another one

Y

is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.