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On the extreme points of quantum channels

Shmuel Friedland, Raphael Loewy

Published Jan 1, 2013• 13 citations
OpenAlex

Abstract

Let Lm,n denote the convex set of completely positive trace preserving operators from Cm×m to Cn×n, i.e. quantum channels. We give a necessary condition for L∈Lm,n to be an extreme point. We show that generically, this condition is also sufficient. We characterize completely the extreme points of L2,2 and L3,2, i.e. quantum channels from qubits to qubits and from qutrits to qubits.

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On Faces of the set of Quantum Channels

Raphael Loewy

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 $.

arXiv:1609.05890•Sep 18, 2016•0 citations
Let Lm,n denote the convex set of completely positive trace preserving operators from Cm×m to Cn×n, i.e. quantum channels. We give a necessary condition for L∈Lm,n to be an extreme point. We show that generically, this condition is also sufficient. We characterize completely the extreme points of L2,2 and L3,2, i.e. quantum channels from qubits to qubits and from qutrits to qubits.
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 XXX to another one YYY 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.
Abstract In this note we study completely positive, trace-preserving (CPT) maps M 4 ( C ) ↦ M 4 ( C ) , in particular the class of examples given by quantum Markov chains (QMCs) describing the evolution of one qubit on a 2-vertex graph. For this class of channels, and regarding its convex structure, we examine the problem of finding the extreme points both with respect to CPT maps and with respect to the set of QMCs. This is done by considering the class of maps whose image is contained in the set of block-diagonal matrices. Motivated by previous results in the literature, a practical criterion for extremality is obtained via the Choi–Jamiołkowski isomorphism and appropriate direct-sum subspaces. The study of extreme maps is natural from a structural point of view, as these play the role of indecomposable operations and provide fundamental information about the geometry of the set of quantum channels.
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.
In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.
A linear map LLL from Cn×n{\mathbb C}^{n \times n}Cn×n into Cn×n{\mathbb C}^{n \times n}Cn×n is called a quantum channel if it is completely positive and trace preserving. The set Ln{\cal L}_nLn​ of all such quantum channels is known to be a compact convex set. While the extreme points of Ln{\cal L}_nLn​ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix Z(L)Z(L)Z(L) associated with the quantum channel LLL, we compute the maximum dimension of a proper face of Ln{\cal L}_nLn​, and in addition the possible dimensions of faces generated by LLL when $rank \ Z(L)=2 $.