Computing the Distance Between Quantum Channels: Usefulness of the Fano Representation Giuliano Benenti, Giuliano Strini

Computing the distance between quantum channels: usefulness of the Fano representation Giuliano Benenti, Giuliano Strini

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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, UniversitàdegliStudidell’Insubria,viaValleggio11,22100Como,Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy 3Dipartimento di Fisica, UniversitàdegliStudidiMilano,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 matrix) 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 maximum 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]. B. The K-algorithm Hence, the maximization in the diamond norm is over the 6realparametersθ1,θ2,θ3,andφ1,φ2,φ3.Ofcourse, We consider a special and unnormalized state in the the number of parameters can be reduced for specific extended Hilbert space K⊗H: channels when there are symmetries. Let E and E denote the two single-qubit superoper- 1 2 |αi = X |jKi|jHi, (16) ators we would like to distinguish. The output states 0 0 j ξ1 ≡ (IK ⊗E1)ξ and ξ2 ≡ (IK ⊗E2)ξ can be conveniently computed using the Fano representation. Any two-qubit where dim(K)=dim(H)and{|jKi}, {|jHi} are or- state can be written in the Fano form as follows [17–19]: thonormal bases for K, H.Thestate|αi is, up to a normalization factor, a maximally entangled state in K⊗H. 1 ξ = X R σ ⊗ σ , (11) We define an operator σ on K⊗H: 4 αβ α β α,β=x,y,z,I σ =(IK ⊗E)(|αihα|), (17) where σx, σy,andσz are the Pauli matrices, σI ≡ 11, and where E = E1 −E2 is the difference of two quantum oper- Rαβ =Tr[(σα ⊗ σβ)ξ]. (12) ations but is not a quantum operation itself. That is, E is linear but not trace preserving and completely positive. Note that the normalization condition Tr(ξ)=1implies Using the singular value decomposition of σ,wecan RII =1.Moreover,thecoefficientsRαβ are real due to write the hermicity of ξ.Eqs.(11)and(12)allowustogofrom M the standard representation for the density matrix (in the (i) (i) {|0i≡|00i, |1i≡|01i, |2i≡|10i, |3i≡|11i} basis) to the σ = X |u ihv |, (18) Fano representation, and vice versa. It is convenient to i=1 write the coefficients R as a column vector, αβ where M is the rank of σ and |u(i)i, |v(i)i are vectors in K⊗H E R =[R ,R ,R ,R ,R ,R , .Theoperatorσ completely specifies and can xx xy xz xI yx yy be exploited to express E as T Ryz,RyI,Rzx,Rzy,Rzz,RzI,RIx,RIy,RIz,RII] . M (13) E(X)=X A(i)XB(i)†. (19) I ⊗E I ⊗E Then the quantum operations K 1 and K 2 map, in i=1 0 (2) 0 the Fano representation, R into R1 = M1 R and R2 = (2) (2) (2) The operators A(i) and B(i) can be derived by gener- M2 R,respectively,whereM1 and M2 are affine transformation matrices. Such matrices have a simple alizing the construction of the operator-sum representa- block structure: tion for quantum operations (see, for instance, Sec. 8.2.4 in Ref. [2]). Given a generic state |ψi = Pj ψj |jHi in H, M(2) = I(1) ⊗M(1), (14) ˜ ? i i we define a corresponding state in K: |ψi = Pj ψj |jKi. Next, we define (1) (1) with I and Mi 4 × 4affinetransformationmatri- (i) ˜ (i) ces corresponding to the quantum operations IK and A = hψ|u i. (20) (1) Ei (of course, I is the identity matrix). Matrices (1) | (i)i M directly determines the transformation, induced by For instance, in the single-qubit case u = i (i) (i) (i) (i) T Ei,ofthesingle-qubitBloch-spherecoordinates(x,y,x). [u1 ,u2 ,u3 ,u4 ] and we obtain Given the Fano representation for a single qubit, ρ = 1 T 0 ≡M(1) " u(i) u(i) # 2 Pα rασα,thenr =[x,y,z,1] and ri i r.We (i) 1 3 0 A = (i) (i) . (21) point out that, while one could compute ξi from the u2 u4 Kraus representation of the superoperator IK ⊗Ei,the advantage of the Fano representation is that the matrix Similarly, we define elements of Mi are directly related to the transformation of the expectation values of the system’s polarization B(i) = hψ˜|v(i)i. (22) measurements [1, 14, 15]. 0 Finally, we compute the trace distance between ξ1 and Finally, it can be checked that with the above defined op- 0 (i) (i) E ξ2 as erators A , B we can express by means of Eq. (19). We can now give explicit expressions for ΨA and ΨB 0 0 in Eq. (5): ||ξ1 − ξ2||1 = X |λk|, (15) k (i) (j)? [ΨA(X)]ij = X A XmnA , (23) 0 0 αm αn where λ1,...,λ4 are the eigenvalues of ξ1 − ξ2. α,m,n 4

(i) (j)? 1 [ΨB(X)]ij = X BαmXmnBαn , (24) α,m,n ≤ ≤ × with 1 i, j M.Therefore,ΨA and ΨB are M M 0.5 matrices. In the single-qubit case, M ≤ 4andtocompute the diamond norm through Eq. (7) we need to calculate eigenvalues and eigenvectors of matrices of size M.A ρ ρ=ρ z 0 simpler but less eﬃcient decomposition of E,ΨA,ΨB is 2 1 provided in Appendix A. For single-qubit channels the optimization (7) is over

6realparameters,3forρ1 and 3 for ρ2 (for instance, -0.5 the Bloch-sphere coordinates of ρ1 and ρ2). As discussed in Sec. III B, the same number of parameters are needed in the F-method. However, the F-method has -1 computational advantages in that only the eigenvalues of -1 -0.5 0 0.5 1 0 0 x the 4 × 4matrixξ1 − ξ2 are required, while in the K- method we need to evaluate both eigenvalues and eigenvectors of matrices in general of the same size (M =4). FIG. 1: Schematic drawing of the trace (and diamond) distance distance between the bit-flip and the phase-flip chan- Moreover, the singular-values decomposition of matrix nels. σ is needed. Besides computational advantages, the F- method is physically more transparent, as it is based on affine maps, whose matrix elements have physical mean- with 0 ≤ c1,c2 ≤ 1. We then readily obtain from Eq. (26) ing, being directly related to the transformation of the that expectation values of the system’s polarization measurements [1, 14, 15]. ||E|| = ||E1 −E2|| =max{1 − c1, 1 − c2}. (28) For this example, the diamond norm coincides with the IV. EXAMPLES FOR SINGLE-QUBIT trace norm, ||E|| = ||E||1,andthereforehasasimple QUANTUM CHANNELS geometrical interpretation: For a single qubit the trace distance between two single-qubit states is equal to the In this section, we illustrate the working of the F- and Euclidean distance between them on the Bloch ball [2]. E E K-methods for the case, most significant for present-day Superoperators 1 and 2 map the Bloch sphere into an implementations, of single-qubit quantum channels. ellipsoid with x (for the bit-flip channel) and z (for the phase-flip channel) as symmetry axis. If we call (x, y, z), 0 0 0 0 0 0 (x1,y1,z1), and (x2,y2,z2), the initial Bloch-sphere co- A. Pauli channels ordinates and the new coordinates after application of quantum operations E1 and E2,respectively,weobtain We start by considering the case of Pauli channels, 0 0 0 x1 = x, y1 = c1y, z1 = c1z, (29) Ei(ρ)= X (qα)iσαρσα, X(qα)i =1,i= 1, 2, 0 0 0 x2 = c2x, y2 = c2y, z2 = z. α=x,y,z,I α (25) The geometrical meaning of the trace norm for the for which the diamond norm can be evaluated analyti- present example is clear from Fig. 1: the length of the 0 0 0 cally [7]: line segment ρρ2 = ρ1ρ2 is the trace (and the diamond) distance ||E1 −E2||1 = ||E1 −E2|| (note that in this figure ||E|| = ||E1 −E2|| = X |(qα)1 − (qα)2|, (26) 1 − c2 > 1 − c1). α As a further example, we discuss a special instance of the channels considered in Ref. [7]: this value of the diamond norm being achieved for maximally entangled input states. The Pauli-channel case will 1 1 1 E (ρ)= ρ + σ ρσ + σ ρσ , serve as a testing ground for the F- and K-algorithms and 1 2 4 x x 4 y y help us develop a physical and geometrical intuition. (30) Let us focus on a couple of significant examples. We E2(ρ)=σzρσz. first consider the bit-flip and the phase-flip channels: In this case, 1+c1 1 − c1 E1(ρ)= ρ + σxρσx, 1 1 2 2 0 0 0 x1 = x, y1 = y, z1 =0, (27) 2 2 (31) 1+c2 1 − c2 E2(ρ)= ρ + σzρσz, 0 0 0 2 2 x2 = −x, y2 = −y, z2 = z. 5

0 1 10

-1 10

-2 10 0.5 -3 10 δ z -4 10

ρ ρ ρ -5 2 0 1 10

-6 10

-7 10 0 1 2 3 4 5 6 -0.5 10 10 10 10 10 10 10 Nr

FIG. 3: Error δ in evaluating the diamond distance between -1 channels (30) by means of the F-algorithm after Nr randomly -1 -0.5 0 0.5 1 chosen initial conditions. Three different runs are shown, to- x 2/3 gether with the 1/Nr (full line) and (1/Nr) (dashed line) dependences. FIG. 2: Schematic drawing of the trace distance distance between the two Pauli channels of Eq. (30). initial conditions requested√ to get a point within a trace distance√ smaller than δ from this curve is of the order ||E|| 3 of (1/ δ)np−1,thusleadingtoδ(N ) ∼ 1/N . The trace norm 1 = 2 ,asshowninFig.2,where r r 0 0 With regard to the K-algorithm, we have observed in ||E||1 is given by the length of the segment ρ ρ .On 1 2 the example of Eq. (30) the same 1/N convergence to the the other hand, the two channels E1 and E2 are perfectly r distinguishable, as we readily obtain from Eq. (26) that expected asymptotic value ||E|| =2.However,thecost ||E|| =2,thisvaluebeingachievedbymeansofmaxi- per initial condition in the K-algorithm is much larger mally entangled input states. Therefore, in this example than in the F-algorithm, in agreement with the general discussion of Sec. III B. Furthermore, the physical mean- ||E|| > ||E||1,thatis,entangledinputstatesimprovethe distinguishability of the two channels. ing of the F-method is much more transparent. For instance, for the Pauli channels (25) the affine transforma- Channels (30) are very convenient to illustrate the con- M(1) 0 I(1) ⊗M(1) vergence properties of the F- and K-algorithms. Let tion matrix i such that Ri =( i )R has a us first consider the F-algorithm. If the maximum of simple diagonal structure: ||ξ0 −ξ0 || ,withξ0 =(I ⊗E )ξ (i =1, 2) is taken over N 1 2 1 i K i r M(1) random initial conditions, then the obtained value differs i =diag[(cx)i, (cy)i, (cz)i, 1], (32) from the diamond norm ||E|| by an amount δ(N )which r where must converge to zero when Nr →∞.Togetconver- gence to ||E|| for channels (30) it is enough to optimize (cx)i =1− 2[(qy)i +(qz)i], over real initial conditions. Numerical results, shown in Fig. 3 for a few runs, are consistent with δ(Nr) ∼ 1/Nr. (cy)i =1− 2[(qz)i +(qx)i], (33) Aroughargumentcanbeusedtoexplainthe1/Nr dependence. Assuming a smooth quadratic dependence of (cz)i =1− 2[(qx)i +(qy)i]. 0 0 the distance D(ξ) ≡||ξ1 − ξ2||1 on the parameters for ∈ π (1) optimization [the angles θi in (10), with θ1 0, 2 , Matrix M simply accounts for the transformation, in- ∈ i θ2,θ3 [0, 2π)] for ξ around the value ξ0 optimizing D, duced by Ei,ofthepolarizationmeasurementsforthe then |D(ξ) − D(ξ0)|∼δ when the distance between ξ 0 √ system qubit: αi =(cα)iα,withα = x, y, z Bloch-sphere and ξ0 is of the order of δ.ThenumberNr of ran- coordinates. domly distributed initial conditions typically√ requested We have checked the computational advantages of the to get√ a point satisfying ||ξ − ξ0||1 < δ is of the order F-method also for all the other examples discussed in this n of (1/ δ) p ,wherenp is the number of parameters for paper. For this reason in what follows we shall focus on 2/n optimization. Therefore, δ(Nr) ∼ (1/Nr) p .Intheex- this method only. 2/3 ample of Fig. 3, np =3leadingtoδ(Nr) ∼ (1/Nr) . On the other hand, numerical data exhibit a 1/Nr dependence. This fact has a simple explanation: the maximum B. Nonunital channels is achieved for Bell states, which are invariant under ro- tations. Hence, the maximum distance is obtained not In this section, we consider nonunital channels, that is, on a single point but on a curve and the number Nr of channels that do not preserve the identity. Therefore, in 6

1 1 ρ 2

0.5 ρ 0.5 2 z

0 z ρ 0 1 2 -0.5 -0.5 Π2 1 -1 -1 -1 -0.5 ρ=ρ0 0.5 1 -1 -0.5 0 0.5 1 1 x x 0 Π 4 FIG. 5: Bloch-sphere visualization of the trace norm ||E||1 in the limiting cases θ =0(left)andθ = θ = π (right). Π4 x x z 2

Π π E 2 ogous) and (ii) θx = θz = 2 .Intheﬁrstcase, 1 maps the Bloch sphere into an ellipsoid with z as symmetry E I FIG. 4: Trace norm ||E|| = ||E −E || ,whereE and E axis [14], while 2 = .Thetracenormisgivenbythe 1 1 2 1 1 2 0 0 0 are displacement channels along the +x-and+z-axis of the length of the line segment ρρ2 = ρ1ρ2 shown in Fig. 5 Bloch sphere. (left),

2 ||E||1 =2sin θz. (36) contrast to the Pauli channels considered in Sec. IV A, single-qubit nonunital channels displace the center of the It is interesting to remark that, in contrast to the Pauli Bloch sphere. Such channel are physically relevant in the channels, the optimal input state is not a maximally en- description, for instance, of energy dissipation in open tangled input state. In the limiting case (i) the channels quantum systems, the simplest case being the amplitude are strictly better discriminated by means of an appro- damping channel [1, 2]. priate separable input state, i.e., the south pole of the As a ﬁrst illustrative example, we compute the dis- Bloch sphere (see the left plot of Fig. 5)) rather than by tance between two channels E1 and E2 corresponding to Bell states. Indeed, given the maximally entangled input displacements of the Bloch sphere along the +x-and+z- state direction [14]. In the Fano representation the aﬃne trans- 1 formation matrices M(1) corresponding to maps E have ξ = |ΨihΨ|, |Ψi = √ (|00i + |11i), (37) i i 2 asimplestructure:

2 2 we obtain  Cx 00Sx  (1) 0 C 00 M = x , (34) 000Cz − 1 1  00C 0     x  0 0 1 0000 0001 ξ1 − ξ2 = 2 . (38) 2  00Sz 0   2  Cz − 10 0 −Sz Cz 000   0 0 0 C 00 The trace distance between ξ1 and ξ2 is then computed M(1) z 2 =  2 2  , (35) by means of Eq. (15):  00Cz Sz  0001 2 4 − 2 S2 Sz + pSz +4(1 Cz) where we have used the shorthand notation Cx ≡ cos θx 0 0 z ||ξ1 − ξ2||1 = + and Cz ≡ cos θz.Thetwochannelsdependparametri- 2 4 π (39) cally on θx,θz ∈ 0, .Thelimitingcasesθx =0and 2 2 − 4 − 2 θz =0correspondtoE1 = I and E2 = I,respectively. Sz pSz +4(1 Cz) π E + . Moreover, for θx = 2 quantum operation 1 maps the 0 0 0 4 Bloch ball onto the single point (x1 =1, y1 =0, z1 =0); π E || 0 − 0 || ||E|| for θz = 2 the mapping operated by 2 is onto the north As shown in Fig. 6, ξ1 ξ2 1 < 1 for any Cz > 0. 0 0 0 pole of the Bloch sphere, (x2 =0, y2 =0, z2 =1). In case (ii), for any initial state ρ,theBlochcoordi- 0 0 The numerically computed trace distance ||E1 −E2||1 nates of ρ1 and ρ2 are given by (1, 0, 0) and (0, 0, 1). The is shown in Fig. 4, as a function of the parameters θx trace norm is given√ by the distance between these two and θz.Wegatherednumericalevidencethatforsuch points, ||E||1 = 2[seeFig.5(right)].Inthiscase,given channels the diamond distance equals the trace distance. an input Bell state or any other√ two-qubit input state ξ, 0 0 It is interesting to examine the analytical solutions for the trace distance ||ξ1 − ξ2||1 = 2=||E||1.Thus,there two limiting cases: (i) θx =0(thecaseθz =0isanal- is no advantage in using an ancillary system. 7

2 e

c 1.5 n a t s i

d 1 2 e

c Π2 a 1 r 0.5 T 0 0 Π4 0 0 0.1 0.2 0.3 0.4 0.5 θz/π 3 8

34 0 FIG. 6: Comparison between ||E1 −E2||1 (upper curve) and || 0 − 0 || π ξ1 ξ2 1 (lower curve) for θx = 2 .

As a further example, we compute the distance between the depolarizing channel E and the nonunital 1 2 channel E2 corresponding to the displacement along the Π2 +z-axis of the Bloch sphere. The depolarizing channel 1 belongs to the class of Pauli channels (25), with qI =1−p and q = q = q = p .ThischannelcontractstheBloch 0 x y z 3 0 Π4 − 4 ≤ ≤ 3 sphere by a factor 1 3 p, with 0 p 4 . Fig. 7 shows the numerically computed ||E||1 and ||E|| 3 8 as well as their diﬀerence ||E|| −||E|| .Theuseofan 1 0 ancillary qubit improves the distinguishability of the two 34 channels. However, we remark again that maximally entangled input states can be detrimental. For instance, in the limiting case p =0,θ = π we obtain from z 2 √ ||E|| || 0 − 0 || 1+ 5 Eqs. (36) and (39) 1 =2> ξ1 ξ2 1 = 2 .A clear advantage is instead seen in another limiting case, 3 0.5 p = 4 , θz =0.Thefullydepolarizingchannelmaps each point of the Bloch ball onto its center, so that the Π2 trace distance is given by the radius of the Bloch sphere, ||E −E || 0 1 2 1 =1.Ontheotherhand,bymeansofEq.(26) Π ||E −E || 3 0 4 we obtain 1 2 = 2 . 38

34 0 V. CONCLUSIONS

FIG. 7: From top to bottom: trace norm ||E||1 = ||E1 −E2||1, We have shown that the distance between two quan- diamond norm ||E||,andtheirdifference||E|| −||E||1; E1 is tum channels can be conveniently computed by means of the depolarizing channel, E2 the displacement along the +z- aMonte-CarloalgorithmbasedontheFanorepresenta- axis of the Bloch sphere. tion. The effectiveness of this algorithm is illustrated in the case, most relevant for present-day implementations of quantum information processing, of single-qubit quan- APPENDIX A: ALTERNATIVE tum channels. A main computational advantage of this DECOMPOSITION OF E algorithms is that it is easily parallelizable. Furthermore, being based on the Fano representation, is enlights the physical meaning of the involved quantum channels: the Given a superoperator E = E1 −E2,withE1, E2 quan- matrix elements of the affine map representing a quantum operations, we start from the Kraus representa- tum channel directly account for the evolution of the ex- tion [1, 2] of E1 and E2: pectation values of the system’s polarization measurements. More generally, we believe that the Fano rep- M1 M2 resentation provides a computationally convenient and † † E1(X)=X EkXEk, E2(X)=X FkXFk , (A1) physically trasparent representation of quantum noise. k=1 k=1 8 and define new operators: quired, but less efficient. Indeed, the maximum number of terms in (A3) is twice that of decomposition (19). (2k−1) 1 (2k) 1 This implies that, if Ψ ,Ψ are expressed in terms of A˜ = √ (Fk + Gk), A˜ = √ (Fk − Gk), A B 2 2 operators A˜(i), B˜(i) rather than A(i), B(i),toevaluate F (ΨA, ΨB)weneedtocomputeeigenvaluesandeigen- B˜(2k−1) = A˜(2k), B˜(2k) = A˜(2k−1), vectors of matrices of size 2M.Inthesingle-qubitcase, (A2) typically 2M =8. where k =1, ..., M ≡ max(M1,M2). Note that, if M1 > M2,wesetGk =0fork = M2 +1, ..., M1;vice-versa,if M1

2M E(X)=X A˜(i)XB˜(i)†. (A3) ACKNOWLEDGMENTS i=1

In contrast to (19), the present decomposition of E(X) We thank Massimiliano Sacchi for interesting com- is simpler, in that no singular value decomposition is re- ments on our work.

[1] G. Benenti, G. Casati, and G. Strini, Principles of Quan- [10] J. Watrous, Advanced Topics in Quantum Infor- tum Computation and Information,Vol.I:Basicconcepts mation Processing,chap.22,Lecturenotes(2004), (World Scientiﬁc, Singapore, 2004); Vol. II: Basic tools at http://www.cs.uwaterloo.ca/∼watrous/lecture- and special topics (World Scientiﬁc, Singapore, 2007). notes/701/. [2] M. A. Nielsen and I. L. Chuang, Quantum Computation [11] N. Johnston, D. W. Kribs, and V. I. Paulsen, Quantum and Quantum Information (Cambridge University Press, Inf. Comput. 9,16(2009). Cambridge, 2000). [12] J. Watrous, Theory of Computing 5,11(2009). [3] C. W. Helstrom, Quantum Detection and Estimation [13] A. Ben-Aroya and A. Ta-Shma, Quantum Inf. Comput. Theory (Academic, New York, 1976). 10,87(2010). [4] A. M. Childs, J. Preskill, and J. Renes, J. Mod. Opt. 47, [14] G. Benenti, S. Felloni, and G. Strini, Eur. Phys. J. D 38, 155 (2000). 389 (2006). [5] A. Ac´ın, Phys. Rev. Lett. 87,177901(2001). [15] G. Benenti and G. Strini, Phys. Rev. A 80,022318 [6] G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, Phys. (2009). Rev. Lett. 87,270404(2001). [16] B. Rosgen and J. Watrous, in Proceedings of the 20th [7] M. F. Sacchi, Phys. Rev. A 71,062340(2005);ibid. 72, Annual Conference on Computational Complexity, pages 014305 (2005). 344-354 (2005), preprint arXiv:cs/0407056. [8] G. M. D’Ariano, M. F. Sacchi, and J. Kahn, Phys. Rev. [17] U. Fano, Rev. Mod. Phys. 29,74(1957);ibid. 55,855 A 72,052302(2005). (1983). [9] A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical [18] F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47,838 and Quantum Computation,vol.47ofGraduateStudies (1981). in Mathematics (American Mathematical Society, Prov- [19] J. Schlienz and G. Mahler, Phys. Rev. A 52,4396(1995). idence, Rhode Island, 2002), Sec. 11.