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Quantum Information and Computation

Progress on the Kretschmann-Schlingemann-Werner Conjecture

Frederik vom Ende

Published Dec 1, 2023• 1 citation
Publisher

Abstract

Given any pair of quantum channels $\Phi_1,\Phi_2$ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries $V_1,V_2$, we prove that there exists a unitary $U$ 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 $\sqrt2$ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.

DOI
10.26421/qic23.15-16-5

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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 ∥V1−(\mathbbm1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄\|V_1-(\mathbbm1\otimes U)V_2\|_\infty\leq\sqrt{2\|\Phi_1-\Phi_2\|_\diamond}∥V1​−(\mathbbm1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​. 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.
Let Φ1,Φ2:Md(C)→Mn(C)Φ_1, Φ_2 : \mathbb{M}_d(\mathbb{C})\to \mathbb{M}_n(\mathbb{C})Φ1​,Φ2​:Md​(C)→Mn​(C) be two quantum channels with respective Stinespring isometries V1,V2:Cd→Cn⊗CmV_1, V_2 : \mathbb{C}^{d}\to \mathbb{C}^{n} \otimes \mathbb{C}^{m}V1​,V2​:Cd→Cn⊗Cm on any common dilation space Cm\mathbb{C}^{m}Cm. We prove that there exists a unitary UUU on Cm\mathbb{C}^{m}Cm such that ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄,\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond},∥V1​−(1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​, thus resolving vom Ende's modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.
Given any pair of quantum channels Φ1,Φ2Φ_1,Φ_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 ∥V1−(1⊗U)V2∥∞≤2∥Φ1−Φ2∥⋄\|V_1-({\bf1}\otimes U)V_2\|_\infty\leq\sqrt{2\|Φ_1-Φ_2\|_\diamond}∥V1​−(1⊗U)V2​∥∞​≤2∥Φ1​−Φ2​∥⋄​​. 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.
The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities 000 and 111. The remedy is the Arveson--Radon--Nikodym derivative TTT of one map with respect to the sum ΩΩΩ, whose transpose is the environment effect selecting that map in a Stinespring dilation of ΩΩΩ. In TTT, these quantities and all maximal fff-divergences of operator convex fff are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2∥T1−T2∥∞1/22\|T_1-T_2\|_\infty^{1/2}2∥T1​−T2​∥∞1/2​, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry V‾\overline{V}V from nnn uses of an unknown isometry channel V:Cd→CDV: \mathbb{C}^d\to\mathbb{C}^DV:Cd→CD. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity n=Θ(d[(D−d)/ε+1])n=Θ(d[(D-d)/ε+1])n=Θ(d[(D−d)/ε+1]) for achieving infidelity εεε. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity O(poly(D,1/ε))O(\mathrm{poly}(D,1/ε))O(poly(D,1/ε)). This task is extended to the multi-copy case V⊗n↦V‾⊗kV^{\otimes n}\mapsto \overline{V}^{\otimes k}V⊗n↦V⊗k. For fixed d<Dd<Dd<D and kkk, we show that the optimal fidelity for the multi-copy case is 1−kd(D−d)/n+o(n−1)1-kd(D-d)/n+o(n^{-1})1−kd(D−d)/n+o(n−1), and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-rrr quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank rrr is constant.
Consider quantum channels with input dimension d1d_1d1​, output dimension d2d_2d2​ and Kraus rank at most rrr. Any such channel must satisfy the constraint rd2≥d1rd_2\geq d_1rd2​≥d1​, and the parameter regime rd2=d1rd_2=d_1rd2​=d1​ is called the boundary regime. In this paper, we show an optimal query lower bound Ω(rd1d2/ε2)Ω(rd_1d_2/\varepsilon^2)Ω(rd1​d2​/ε2) for quantum channel tomography to within diamond norm error ε\varepsilonε in the away-from-boundary regime rd2≥2d1rd_2\geq 2d_1rd2​≥2d1​, matching the existing upper bound O(rd1d2/ε2)O(rd_1d_2/\varepsilon^2)O(rd1​d2​/ε2). In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions d1=d2=dd_1=d_2=dd1​=d2​=d with r≥2r\geq 2r≥2, in sharp contrast to the unitary case r=1r=1r=1 where Heisenberg scaling Θ(d2/ε)Θ(d^2/\varepsilon)Θ(d2/ε) is achievable.