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arXiv:2301.05488

From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative Construction for Infinite Dimensions

Frederik vom Ende

Published Jan 13, 2023• 0 citations
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Abstract

We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map $Φ$ there exists an auxiliary Hilbert space $\mathcal K$ in a pure state $|ψ\rangle\langleψ|$ as well as a unitary operator $U$ on system plus environment such that $Φ$ equals $\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|ψ\rangle\langleψ|)U^*)$. The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of $Φ$" together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.

DOI
10.48550/arxiv.2301.05488

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We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map ΦΦΦ there exists an auxiliary Hilbert space K\mathcal KK in a pure state ∣ψ⟩⟨ψ∣|ψ\rangle\langleψ|∣ψ⟩⟨ψ∣ as well as a unitary operator UUU on system plus environment such that ΦΦΦ equals tr⁡K(U((⋅)⊗∣ψ⟩⟨ψ∣)U∗)\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|ψ\rangle\langleψ|)U^*)trK​(U((⋅)⊗∣ψ⟩⟨ψ∣)U∗). The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of ΦΦΦ" together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.
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.
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.
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We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.
Given any pair of quantum channels Φ1,Φ2\Phi_1,\Phi_2Φ1​,Φ2​ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries V1,V2V_1,V_2V1​,V2​, we prove that there exists a unitary UUU on the environment such that $\|V_1-(\mathbbm1\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}$. Moreover, we provide a simple example which shows that the factor 2\sqrt22​ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.