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arXiv:1004.4110 · Journal of Physics B Atomic Molecular and Optical Physics

Computing the distance between quantum channels: usefulness of the Fano representation

Giuliano Benenti, G. Strini

Published Oct 22, 2010• 39 citations
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Abstract

The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions. In this paper, we show that the diamond norm can be conveniently and in a physically transparent way computed by means of a Monte-Carlo algorithm based on the Fano representation of quantum states and quantum operations. The effectiveness of this algorithm is illustrated for several single-qubit quantum channels.

DOI
10.1088/0953-4075/43/21/215508

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The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions. In this paper, we show that the diamond norm can be conveniently and in a physically transparent way computed by means of a Monte-Carlo algorithm based on the Fano representation of quantum states and quantum operations. The effectiveness of this algorithm is illustrated for several single-qubit quantum channels.
The diamond norm is a norm defined over the space of quantum transformations. This norm has a natural operational interpretation: it measures how well one can distinguish between two transformations by applying them to a state of arbitrarily large dimension. This interpretation makes this norm useful in the study of quantum interactive proof systems. In this note we exhibit an efficient algorithm for computing this norm using convex programming. Independently of us, Watrous recently showed a different algorithm to compute this norm. An immediate corollary of this algorithm is a slight simplification of the argument of Kitaev and Watrous that QIP \subseteq \EXP.
In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantifyaverage-casestatistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an$\delta −approximate4−design,with-approximate 4-design, with−approximate4−design,with\delta =o(d^{-8}),thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimension, then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that for each dimension,thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimensiondthisratioisatmostthis ratio is at mostthisratioisatmostd^{\frac {1}{2}}, d, d^{\frac {3}{2}}$for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.
The diamond distance between two quantum channels determines the optimal probability of distinguishing them from a single use. We give a quantum algorithm that estimates the diamond distance between two unknown ddd-dimensional unitary channels to within additive error ε\varepsilonε, using O(d/ε)O(\sqrt{d}/\varepsilon)O(d​/ε) queries to the unitary channels and their inverses. The query complexity is optimal, as it matches the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).
We prove that learning an unknown quantum channel with input dimension dAd_AdA​, output dimension dBd_BdB​, and Choi rank rrr to diamond distance ε\varepsilonε requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when dA=rdBd_A= rd_BdA​=rdB​, and Ω ⁣(dAdBrε2log⁡(dBr/ε))Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)Ω(ε2log(dB​r/ε)dA​dB​r​) channel queries when dA≤rdB/2d_A\le rd_B/2dA​≤rdB​/2. These lower bounds improve upon the best previous Ω(dAdBr)Ω(d_A d_B r)Ω(dA​dB​r) bound by introducing explicit, near-optimal ε\varepsilonε-dependence. Moreover, when dA≤rdB/2d_A\le rd_B/2dA​≤rdB​/2, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.
The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Gutoski and Watrous [In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS’05), Lecture Notes in Computer Science, Vol. 3404 (Springer, 2005), pp. 605–616. The generalized result states that for any two convex sets S0, S1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of S0 ∈ S0 from any choice of S1 ∈ S1 with bias proportional to the minimal distance between the sets S0 and S1 as measured by this norm. A similar discrimination result for channels then follows as a special case.