Computing the distance between quantum channels: usefulness of the Fano representation Giuliano Benenti, Giuliano Strini To cite this version: Giuliano Benenti, Giuliano Strini. Computing the distance between quantum channels: usefulness of the Fano representation. Journal of Physics B: Atomic, Molecular and Optical Physics, IOP Publish- ing, 2010, 43 (21), pp.215508. 10.1088/0953-4075/43/21/215508. hal-00569863 HAL Id: hal-00569863 https://hal.archives-ouvertes.fr/hal-00569863 Submitted on 25 Feb 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 24 June 2010 Computing the distance between quantum channels: Usefulness of the Fano representation Giuliano Benenti1,2,∗ and Giuliano Strini3, † 1CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Universit`adegliStudidell’Insubria,viaValleggio11,22100Como,Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy 3Dipartimento di Fisica, Universit`adegliStudidiMilano,viaCeloria16,20133Milano,Italy (Dated: June 24, 2010) 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. PACS numbers: 03.65.Yz, 03.67.-a I. INTRODUCTION cated semidefinite programming or convex optimization. On the other hand, analytical solutions are limited to special classes of channels [7, 8]. In this paper, we pro- Quantum information processes in a noisy environment pose a simple and easily parallelizable Monte-Carlo al- are conveniently described in terms of quantum chan- gorithm based on the Fano representation of quantum nels,thatis,linear,tracepreserving,completelypositive states and quantum operations. We show that our al- maps on the set of quantum states [1, 2]. The problem gorithm provides reliable results for the case, most sig- of discriminating quantum channels is of great interest. nificant for present-day implementations of quantum in- For instance, knowing the correct noise model might pro- formation processing, of single-qubit quantum channels. vide useful information to devise efficient error-correcting Furthermore, in the Fano representation quantum opera- strategies, both in the fields of quantum communication tions are described by affine maps whose matrix elements and quantum computation. have precise physical meaning: They are directly related It is therefore natural to consider distances between to the evolution of the expectation values of the system’s quantum channels, that is to say, we would like to quan- polarization measurements [1, 14, 15]. tify how similarly two channels E1 and E2 act on quantum The paper is organized as follows. After reviewing in states, or in other words to determine if there are input Sec. II basic definitions of the distance between quantum states ρ on which the two channels produce output states channels, we discuss in Sec. III two numerical Monte- 0 0 ρ1 = E1(ρ)andρ2 = E2(ρ)thataredistinguishable.The Carlo strategies for computing the diamond norm. The 0 0 0 0 trace norm of ρ1−ρ2 represents how well ρ1 and ρ2 can be first one is based on the Kitaev’s characterization of the distinguished by a measurement [3]: the more orthogonal diamond norm. The second one is based on the Fano rep- two quantum states are, the easier it is to discriminate resentation of quantum states and quantum operations. them. The trace distance of two quantum channels is The two methods are compared in Sec. IV for a few phys- 0 0 then obtained after maximizing the trace norm of ρ1 −ρ2 ically significant single-qubit quantum channels. Finally, over the input state ρ.However,thetracenormisnot our conclusions are drawn in Sec. V. agoodmeasureofthedistancebetweenquantumchan- nels. Indeed, in general the presence in the input state of entanglement with an ancillary system can help discrim- II. THE DIAMOND NORM inating quantum channels [4–8]. This fact is captured by the diamond norm [9, 10]: the trace distance between the overall output states (including the ancillary system) is A. Basic definitions optimized over all possible input states, including those entangled. We consider the following problem: given two quantum The computation of the diamond norm is not known channels E1 and E2,andasinglechanneluse,chosenuni- to be straightforward and only a limited number of al- formly at random from {E1, E2},wewishtomaximizethe gorithms have been proposed [11–13], based on compli- probability of correctly identifying the quantum channel. It seems natural to reformulate the optimization problem into the problem of finding an input state (density ma- trix) ρ in the Hilbert space H such that the error prob- ∗Electronic address: [email protected] ability in the discrimination of the output states E1(ρ) † Electronic address: [email protected] and E2(ρ)isminimal.Inthiscase,theminimalerror 2 probability reads Note that the space H is traced out in the definition of ΨA,ΨB,ratherthanthespaceR.Finally,itturnsout 1 ||E −E || p0 = − 1 2 1 , that [9, 10] E 2 4 (1) ||E|| = Fmax(ΨA, ΨB), (6) ||E||1 ≡ max ||E(ρ)||1, ρ where F (Ψ , Ψ )isthemaximumoutputfidelityof √ max A B † ΨA and ΨB,definedas where ||X||1 ≡ Tr X X denotes the trace norm. The superoperator trace distance ||E1 −E2||1 is, how- Fmax(ΨA, ΨB)=maxF [ΨA(ρ1), ΨB(ρ2)], (7) ever, not a good definition of the distance between two ρ1,ρ2 quantum operations. The point is that in general it is possible to exploit quantum entanglement to increase the where ρ1,ρ2 are density matrices in H,andthefidelity distinguishability of two quantum channels. In this case, F is defined as an ancillary Hilbert space K is introduced, the input state q 1/2 1/2 ξ is a density matrix in K⊗H, and the quantum opera- F (ΨA, ΨB)=Tr ΨA ΨBΨA . (8) tions are trivially extended to K.Thatistosay,theout- put states to discriminate are (IK ⊗E1)ξ and (IK ⊗E2)ξ, Note that ΨA,ΨB are not density matrices: the condi- where IK is the identity map acting on K.Theminimal tions Tr(ΨA)=1,Tr(ΨB)=1arenotsatisfied. error probability reads 1 ||E −E || p = − 1 2 , III. COMPUTING THE DIAMOND NORM E 2 4 (2) We numerically compute the distance (induced by the ||E|| ≡ max ||(IK ⊗E)ξ||1, ξ diamond norm) between two quantum channels E1 and E2 using two Monte-Carlo algorithms. The first one is based where ||E|| denotes the diamond norm of E.Itisclear on the direct computation of ||E1 −E2||,withtheoutput from definition (2) that states (IK ⊗E1)ξ and (IK ⊗E2)ξ in Eq. (2) computed from ξ taking advantage of the Fano representation of ||E|| = ||IK ⊗E||1 ≥||E||1 (3) quantum states and quantum operations. The second Monte-Carlo algorithm uses the Kitaev’s representation 0 and therefore pE ≤ pE,sothatitcanbeconvenientto of the diamond norm to compute the maximum output use an ancillary system to better discriminate two quan- fidelity Fmax of Eq. (7). In the following, we will refer tum operations after a single channel use. The two quan- to the two above Monte-Carlo algorithms as F-algorithm tum channels E1 and E2 become perfectly distinguishable and K-algorithm, respectively. For the sake of simplicity (pE =0)whentheirdiamonddistance||E1 −E2|| =2. we will confine ourselves to one-qubit quantum channels, It turns out that the diamond norm does not depend even though the two algorithms can be easily formulated on K,provideddim(K) ≥ dim(H)[9].Duetothecon- for two- or many-qubit channels. vexity of the trace norm, it can be shown that the maxi- mum in both Eqs. (1) and (2) is achieved for pure input states [16]. A. The F-algorithm In this section we describe the F-algorithm, which we B. Kitaev’s characterization of the diamond norm will use to directly compute the diamond norm (2), with the maximum taken over a large number of randomly Kitaev provided a different equivalent characterization chosen input states ξ.Aconvexityargumentshows of the diamond norm, see, e.g., [9, 10]. Any superoper- that it is sufficient to optimize over pure input states ator (not necessarily completely positive) E : L(H) → ξ = |ΨihΨ| [16]. For one-qubit channels, it is enough to L(H), with L(H)spaceoflinearoperatorsmappingH to add a single ancillary qubit when computing the diamond itself, can be expressed as norm [9]. Therefore, we can write E † (X)=TrR(AXB ), (4) |Ψi = C00|00i + C01|01i + C10|10i + C11|11i, (9) where X ∈ L(H), A and B linear operators from H to with R⊗H, with R auxiliary Hilbert space and dim(R) ≤ [dim(H)]. It is then possible to define completely positive C00 =cosθ1 cos θ2, iφ1 superoperators ΨA, ΨB : L(H) → L(R): C01 =cosθ1 sin θ2e , iφ (10) C10 =sinθ1 cos θ3e 2 , † † iφ ΨA(X)=TrH(AXA ), ΨB(X)=TrH(BXB ). (5) C11 =sinθ1 sin θ3e 3 , 3 ∈ π ∈ where the angles θi 0, 2 and the phase φi [0, 2π].