arXiv:0909.3744

Extremal Channels for a genaral Quantum system

Kuldeep Dixit, E. C. G. Sudarshan

Published Sep 21, 2009• 0 citations

Abstract

Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level system, the extremal channel can be characterized in terms of $N^2$-$N$ real parameters coupled with rotations. We give a representation for $N$= 2, 3, 4.

DOI: 10.48550/arxiv.0909.3744

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arXiv:0909.3744

Extremal Channels for a genaral Quantum system

Kuldeep Dixit, E. C. G. Sudarshan

Published Sep 21, 2009• 0 citations

arXiv HTML PDF

Abstract

N

-level system, the extremal channel can be characterized in terms of

N^2

N

real parameters coupled with rotations. We give a representation for

N

= 2, 3, 4.

DOI: 10.48550/arxiv.0909.3744

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N

-level system, the extremal channel can be characterized in terms of

N^2

N

real parameters coupled with rotations. We give a representation for

N

= 2, 3, 4.

\|\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.

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 $.

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.