arXiv subjects

Mark Girard

Publications and source records attributed to Mark Girard.

6 recordsLinked to original sources

arXiv2021

Certifying optimality for convex quantum channel optimization problems

We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.

quant-ph↗

arXiv2020

Jordan products of quantum channels and their compatibility

Given two quantum channels, we examine the task of determining whether they are compatible - meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). We show several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolaring channels are compatible.

quant-ph↗

arXiv2020

Convex cones in mapping spaces between matrix algebras

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The duals of such convex cones can be characterized in terms of ampliation maps, which can also be used to characterize many notions from quantum information theory---such as separability, entanglement-breaking maps, Schmidt numbers, as well as decomposable maps and $k$-positive maps in functional analysis. In fact, such characterizations hold if and only if the involved cone is a one-sided mapping cone. Through this analysis, we obtain mapping properties for compositions of cones from which we also obtain several equivalent statements of the PPT (positive partial transpose) square conjecture.

math.OA↗

arXiv2020

Twirling channels have minimal mixed-unitary rank

For a positive integer $d$ and a unitary representation $ρ:G\rightarrow\mathrm{U}(d)$ of a compact group $G$, the twirling channel for this representation is the linear mapping $Φ: M_d\rightarrow M_d$ defined as $Φ(X)=\int_{G}\mathrm{d}μ(g)\,ρ(g)Xρ(g^{-1})$ for every $X\in M_d$, where $μ$ is the Haar measure on $G$. Such channels are examples of mixed-unitary channels, as they are in the convex hull of the set of unitary channels of a fixed size. By Carathéodory's theorem, these channels can always be expressed as a finite linear combination of unitary channels. We consider the mixed-unitary rank twirling channels---which is the minimum number of distinct unitary conjugations required to express the channel as a convex combination of unitary channels---and show that the mixed-unitary rank of every twirling channel is always equal to its Choi rank, both of which are equal to the dimension of the von Neumann algebra generated by the representation. Moreover, we show how to explicitly construct minimal mixed-unitary decompositions for these types of channels and provide some examples.

quant-ph↗

arXiv2020

On the mixed-unitary rank of quantum channels

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality $N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the equality $N=2$ when $r=2$), and we exhibit the first known examples of mixed-unitary channels for which $N>r$. Specifically, we prove that there exist mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for infinitely many positive integers $d$, including every prime power $d$. We also examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.

quant-ph↗

arXiv2012

Gradient-based stopping rules for maximum-likelihood quantum-state tomography

When performing maximum-likelihood quantum-state tomography, one must find the quantum state that maximizes the likelihood of the state given observed measurements on identically prepared systems. The optimization is usually performed with iterative algorithms. This paper provides a gradient-based upper bound on the ratio of the true maximum likelihood and the likelihood of the state of the current iteration, regardless of the particular algorithm used. This bound is useful for formulating stopping rules for halting iterations of maximization algorithms. We discuss such stopping rules in the context of determining confidence regions from log-likelihood differences when the differences are approximately chi-squared distributed.

quant-ph↗