Whether the coherent channel-discrimination bias $p_s^{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)-\tfrac{1}{2}$ is a metric on the space of quantum channels

In coherent quantum channel discrimination (Wilde, ISIT 2020), a verifier applies one of two channels in a coherent superposition via controlled isometric extensions, and a prover attempts to distill a maximally entangled Bell state; the optimal success probability $p_s^{\rm coh}$ is a superchannel-monotone, semidefinite-program-computable distinguishability measure. The source posed the open question of whether the bias $d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^1):=p_s^{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)-\tfrac{1}{2}$ is a metric on quantum channels, in contrast to incoherent discrimination, where the analogous bias equals one quarter of the diamond distance and hence is a metric. The source's own propositions already give non-negativity and symmetry and the sufficiency direction of identity of indiscernibles; the surviving open core is the triangle inequality, together with the unproved necessity direction ($d_{\rm coh}=0$ only for identical channels).

Background

A quantum channel is a completely positive, trace-preserving linear map $\mathcal{N}_{A\to B}$ taking density operators on an input system $A$ to density operators on an output system $B$; every channel admits an isometric extension, i.e. it can be realized as $\mathcal{N}(\rho)=\mathrm{Tr}_E[U\rho U^\dagger]$ for an isometry $U$ into system $B$ plus an environment $E$. In standard ("incoherent") binary channel discrimination with equal prior probabilities, a preparer chooses $\mathcal{N}^0$ or $\mathcal{N}^1$, and a tester who may entangle the input with a reference system and measure the output guesses which channel was applied. The optimal success probability $p_s^{\rm inc}$ satisfies $p_s^{\rm inc}(\mathcal{N}^0,\mathcal{N}^1)=\tfrac{1}{2}+\tfrac{1}{4}\|\mathcal{N}^0-\mathcal{N}^1\|_\diamond$, where $\|\cdot\|_\diamond$ is the diamond norm (the completely bounded norm of the channel difference, obtained by maximizing the trace norm of the output difference over all entangled input states). Consequently the incoherent bias $p_s^{\rm inc}-\tfrac{1}{2}$ is a metric on channels. This is one of the two standard operational meanings of the diamond norm, and the reason it is the default channel distance in, e.g., fault-tolerance analysis.

Wilde (2020) introduced a coherent version of this task, formulated as a quantum interactive proof game between a verifier and a prover. The verifier holds a control qubit $R_1$ in the state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$; conditioned on the control value $i\in\{0,1\}$, the verifier applies the isometric extension of $\mathcal{N}^i$ to the system $A$ sent by the prover, so that the two channels are called in superposition. The prover prepares an arbitrary pure state $\psi$ on a reference and the input, may attach ancillas, and applies an arbitrary unitary to the output, reference, and ancillary systems. The verifier accepts if the control qubit and a prover-held flag system project onto a maximally entangled (Bell) state $|\Phi\rangle$, i.e. a pure joint state of two equal-dimension systems of the form $d^{-1/2}\sum_{k=1}^{d}|k\rangle|k\rangle$; the optimal acceptance probability over all prover strategies is $p_s^{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)$, the coherent channel discrimination success probability. The paper proves that $p_s^{\rm coh}$ is independent of the chosen isometric extensions, does not increase under any quantum superchannel (a physical supermap taking channels to channels, realizable as pre- and post-processing), obeys $\tfrac{1}{2}\le p_s^{\rm coh}\le 1$, and is computable by a semidefinite program, i.e. a convex optimization problem over positive semidefinite matrices that can be solved to arbitrary numerical precision in time polynomial in the dimension. A variational expression (Proposition 1) writes $p_s^{\rm coh}=\big[\tfrac{1}{2}\sup_{\{P^i\}:\,\sum_i (P^i)^\dagger P^i=I}\big\|\sum_{i\in\{0,1\}}(\mathcal{N}^i)^\dagger(P^i)\big\|_\infty\big]^2$, where the $P^i$ are operators on output plus ancilla systems and $(\mathcal{N}^i)^\dagger$ denotes the adjoint map; the paper also bounds $p_s^{\rm coh}\le p_s^{{\rm inc},{\rm unc}}\le p_s^{\rm inc}\le\sqrt{p_s^{\rm coh}}$, where $p_s^{{\rm inc},{\rm unc}}$ is the success probability of an intermediate incoherent variant with uncomputing. Unlike the incoherent case, however, no representation of the coherent bias as a norm of the channel difference is known, so the geometric structure of $d_{\rm coh}:=p_s^{\rm coh}-\tfrac{1}{2}$ is undetermined: the concluding section of the source poses, as an open question, whether $d_{\rm coh}$ is a metric on the space of quantum channels, i.e. whether it satisfies non-negativity, symmetry, vanishing only for identical channels, and the triangle inequality. Non-negativity and symmetry follow from the source's own results (Proposition 2 and the Proposition 1 expression), and $d_{\rm coh}=0$ whenever the channels are identical; what is neither proved nor refuted anywhere is the converse vanishing statement and, the substantive core, the triangle inequality $d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^2)\le d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)+d_{\rm coh}(\mathcal{N}^1,\mathcal{N}^2)$.

Problem Statement

Determine whether the coherent channel-discrimination bias $d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^1):=p_s^{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)-\tfrac{1}{2}$ defines a metric on the space of quantum channels acting between finite-dimensional systems, where $p_s^{\rm coh}$ is the optimal Bell-state distillation success probability of the coherent discrimination game of Wilde (ISIT 2020), computable by the semidefinite program given there. Concretely, settle the two questions left open by the source and its own partial results: (i) does $d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)=0$ hold only when $\mathcal{N}^0=\mathcal{N}^1$ (that identical channels give vanishing bias is Proposition 2 of the source; the converse direction is unproved); and (ii) does the triangle inequality $d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^2)\le d_{\rm coh}(\mathcal{N}^0,\mathcal{N}^1)+d_{\rm coh}(\mathcal{N}^1,\mathcal{N}^2)$ hold for all channel triples (symmetry and non-negativity already follow from the source's Propositions 1-2, so they need no separate treatment). An admissible complete answer is either a proof that all metric axioms hold for arbitrary finite-dimensional channels, or an explicit finite-dimensional counterexample - a triple of channels violating the triangle inequality, or two distinct channels with vanishing bias - establishing that $d_{\rm coh}$ is not a metric.

References

1. Mark M. Wilde, Coherent Quantum Channel Discrimination, 2020 IEEE International Symposium on Information Theory (ISIT), pp. 1921-1926, 2020. DOI: 10.1109/ISIT44484.2020.9174425, arXiv:2001.02668, https://arxiv.org/abs/2001.02668
2. Soorya Rethinasamy, Rochisha Agarwal, Kunal Sharma, Mark M. Wilde, Estimating distinguishability measures on quantum computers, arXiv:2108.08406 (2021), published as Phys. Rev. A 108, 012409 (2023). DOI: 10.1103/PhysRevA.108.012409, https://arxiv.org/abs/2108.08406
3. Samad Khabbazi-Oskouei, Stefano Mancini, Milajiguli Rexiti, Quantum reading of quantum information, arXiv:2307.08821 (2023), published as J. Phys. A: Math. Theor. 56, 485302 (2023). DOI: 10.1088/1751-8121/ad075f, https://arxiv.org/abs/2307.08821
4. Qiang Lei, Liuheng Cao, Asutosh Kumar, Junde Wu, Dilation, Discrimination and Uhlmann's Theorem of Link Products of Quantum Channels, arXiv:2309.03052 (2023), published as Chinese Physics B 33, 030304 (2024). DOI: 10.1088/1674-1056/ad0ccb, https://arxiv.org/abs/2309.03052