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arXiv:2306.04077 · Journal of Physics A Mathematical and Theoretical

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

David W. Kribs, Jeremy Levick, Rajesh Pereira, Mizanur Rahaman

Published Feb 23, 2024• 2 citations
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Abstract

Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

DOI
10.1088/1751-8121/ad2cb0

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Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.
Abstract In this chapter we visit the ‘church of the larger Hilbert space’: we take a closer look at the relationship of pure and mixed quantum states via purification, and discuss a similar connection between completely positive and trace-preserving (CPTP) maps and unitary dynamics on a larger Hilbert space. We show that such CPTP maps, which represent quantum channels, are equivalent to operations admitting a Kraus decomposition, as well as to unitary dynamics on a larger Hilbert space with factoring initial condition, the famous Stinespring dilation of quantum operations. In this context we introduce the Choi-Jamiołkowski isomorphism, which expresses a duality between maps between two Hilbert spaces and states on tensor products of these Hilbert spaces, and we review several important examples of quantum channels such as the dephasing channel, amplitude-damping channel, and depolarizing channel. Finally, we examine operations that are not possible in quantum theory, focusing on the no-cloning theorem.
It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. This result is novel even for the subclass of Schur maps and stands in sharp contrast to the resolution of the asymptotic quantum Birkhoff conjecture by Haagerup and Musat, who demonstrated that tensor powers of some unital quantum channels maintain a persistent positive distance from the set of mixed unitary channels. Remarkably, our results show that this gap vanishes in finite time when considering ordinary powers within a semigroup. Building on this, we define the mixed unitary index of a unital quantum channel as the minimum time (or power) beyond which all subsequent maps become mixed unitary. We demonstrate that for any fixed dimension d≥3d \geq 3d≥3, there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some t>0t > 0t>0, it remains non-mixed unitary for all ttt sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all t≥0t \geq 0t≥0.
Completely positive trace-preserving maps SSS, also known as quantum channels, arise in quantum physics as a description of how the density operator ρρρ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space H\mathscr{H}H that ρρρ acts on is finite-dimensional, then every SSS must have a fixed point, i.e., a density operator ρ0ρ_0ρ0​ with S(ρ0)=ρ0S(ρ_0)=ρ_0S(ρ0​)=ρ0​. In infinite dimension, SSS need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that SSS leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.
Let (M,τ)(\mathcal{M},τ)(M,τ) and (N,τ′)(\mathcal{N},τ^{\prime})(N,τ′) be tracial von-Neumann algebras and let φ:M→Nφ:\mathcal{M}\to\mathcal{N}φ:M→N be a strictly completely positive, trace preserving map. Given a positive, invertible B∈MB\in\mathcal{M}B∈M with τ(B)=1τ(B)=1τ(B)=1, a state on M\mathcal{M}M given by a positive A∈L1(M,τ)A\in L^1(\mathcal{M}, τ)A∈L1(M,τ) is said to be recoverable if R(φ(A))=A\mathcal{R}(φ(A))=AR(φ(A))=A where R\mathcal{R}R is the Petz recovery map corresponding to BBB and φφφ. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of R∘φ\mathcal{R}\circφR∘φ. We show that there exists a completely positive, trace preserving map ψ:M→Mψ:\mathcal{M}\to\mathcal{M}ψ:M→M such that ψ(A)ψ(A)ψ(A) is recoverable for all AAA and (R∘φ)n→ψ(\mathcal{R}\circφ)^n\toψ(R∘φ)n→ψ in norm as operators on Lp(M,τ)L^p(\mathcal{M},τ)Lp(M,τ) for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in L1L^1L1. Finally, we prove an interesting decomposition theorem for normal states on M\mathcal{M}M.
We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.