Device-independent tests of quantum channels
aa r X i v : . [ qu a n t - ph ] M a r Device-independent tests of quantum channels
Michele Dall’Arno, ∗ Sarah Brandsen, † and Francesco Buscemi ‡ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan (Dated: March 17, 2017)We develop a device-independent framework for testing quantum channels. That is, we falsify ahypothesis about a quantum channel based only on an observed set of input-output correlations.Formally, the problem consists of characterizing the set of input-output correlations compatible withany arbitrary given quantum channel. For binary (i.e., two input symbols, two output symbols)correlations, we show that extremal correlations are always achieved by orthogonal encodings andmeasurements, irrespective of whether or not the channel preserves commutativity. We furtherprovide a full, closed-form characterization of the sets of binary correlations in the case of: i) anydihedrally-covariant qubit channel (such as any Pauli and amplitude-damping channels), and ii)any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as anyerasure, depolarizing, universal cloning, and universal transposition channels).
I. INTRODUCTION
Any physical experiment is based upon the observationof correlations among events at various points in spaceand time, along with some assumptions about the under-lying physics. Naturally, in order to be operational anysuch assumption must have been tested as a hypothesisin a previous experiment. Ultimately, to break an other-wise circular argument, experiments involving no furtherassumptions are required – that is, device-independenttests.Formally, a hypothesis consists of a circuit [9], whichis usually assumed to have a global causal structure (fol-lowing special relativity), and its components, which areusually assumed to be governed by classical or quantumtheories and thus representable by channels.Denoting a hypothesis (circuit) by X , the set of cor-relations compatible with X is denoted by S ( X ). Then,hypothesis X is falsified, along with any other hypothesis Y such that S ( Y ) ⊆ S ( X ), as soon as the observed cor-relation does not belong to S ( X ) (This inclusion relationinduces an ordering among channels which is reminis-cent of that introduced by Shannon [1] among classicalchannels). Therefore, from the theoretical viewpoint, theproblem of falsifying a hypothesis X can be recast [2] asthat of characterising the set S ( X ) of compatible corre-lations.Since (discrete, memoryless) classical channels are bydefinition input-output correlations (conditional proba-bilities), the characterisation of S ( X ) is trivial in classicaltheory as it is a polytope easily related to the correlationdefining the channel. On the contrary, the problem is farfrom trivial in quantum theory: due to the existence ofsuperpositions of states and effects, the set S ( X ) can bestrictly convex. ∗ [email protected] † [email protected] ‡ [email protected] In this work we address the problem of device-independent tests of quantum channels, in particular thecharacterization of the set S nm ( X ) of m -inputs/ n -outputscorrelations p j | i obtainable through an arbitrary givenchannel X , upon the input of an arbitrary preparation { ρ i } m − i =0 and the measurement of an arbitrary POVM { π j } n − j =0 , that is p j | i := Tr[ X ( ρ i ) π j ] = i '!& ρ i X *-+, π j j . (1)The analogous problems of device-independent tests ofquantum states and measurements have been recentlyaddressed in Ref. [15] and Ref. [16], respectively.An alternative formulation for the problem consideredhere can be given in terms of a “game” involving twoparties: an experimenter, claiming to be able to preparequantum states, feed them through some quantum chan-nel X , and then perform measurements on the output,and a skeptical theoretician, willing to trust observedcorrelations only. If the experimenter produces some correlations lying outsides of S nm ( X ), then the theoreti-cian must conclude that the actual channel X ′ is notworse than X at producing correlations, but this is notsufficient to support the experimenter’s claim. Indeed,in order to convince the theoretician, the experimentermust produce the entire set S nm ( X ): in fact, it is suffi-cient to produce a set of correlations whose convex hull contains S nm ( X ). Then, the theoretician must concludethat whatever channel the experimenter actually has isat least as good as X at producing correlations, and theexperimenter’s claim is accepted.It is hence clear that the problem of device-independent tests of quantum channels induces a pre-ordering relation among quantum channels: X (cid:23) Y ifand only if S nm ( X ) ⊇ S nm ( Y ). (The order also dependsupon m and n , but for compactness we drop the indexeswhenever they are clear from the context). In order tocharacterize such preorder, for any given channel X , weneed to i) provide the experimenter with all the statesand measurements generating the extremal correlationsof S nm ( X ), and ii) provide the theoretician with a fullclosed-form characterization of the set S nm ( X ) of com-patible correlations.As a preliminary result, we find that the sets S nm ( X )coincide for any d -dimensional unitary and dephasingchannels, for any d , m , and n (this is an immedi-ate consequence of a remarkable result by Frenkel andWeiner [17].) Upon considering only the binary case m = n = 2, our first result is to show that any corre-lation on the boundary of S ( X ) is achieved by a pairof commuting pure states – irrespective of whether X is a commutativity-preserving channel. Then, we de-rive the complete closed-form characterization of S ( X )for: i) any given dihedrally-covariant qubit channel,including any Pauli and amplitude-damping channels;and ii) any given universally-covariant commutativity-preserving channel, including any erasure, depolarizing,universal 1 → X as the d -dimensional identity chan-nel I d , one recovers device-independent dimension testsanalogous to those discussed in Refs. [20–23], in whichcase the aforementioned ordering induced by the inclu-sion S nm ( I d ) ⊆ S nm ( I d ) ⇔ d ≤ d is of course to-tal. However, the completeness of our characterizationof S ( X ) implies that our framework detects all corre-lations incompatible with the given hypothesis, unlikeRefs. [20–24] where the set of correlations is tested onlyalong an arbitrarily chosen direction.Let us provide a preview of some consequences of ourresults: • Any Pauli channel P ~λ : ρ → λ ρ + P k =1 λ k σ k ρσ † k is compatible with p if and only if | p | − p | | − | p | − p | | ≤ max k ∈ [1 , | λ + λ k ) − | ; • any amplitude-damping channel A λ : ρ → A ρA † + A ρA † with A = | ih | + √ λ | ih | and A = √ − λ | ih | is compatible with p if and only if (cid:0) √ p | p | − √ p | p | (cid:1) ≤ λ ; • any d -dimensional erasure channel E d : ρ → λρ ⊕ (1 − λ ) Tr[ ρ ] φ for some pure state φ is compatiblewith p if and only if | p | − p | | ≤ λ ; • any d -dimensional depolarizing channel D λd : ρ → λρ + (1 − λ ) Tr[ ρ ] /d is compatible with p if andonly if | p | − p | | ≤ λ, | p | − p | | − | p | − p | | ≤ dλ − λ + dλ ; • the d -dimensional universal optimal 1 → C d is compatible with p if andonly if | p | − p | | ≤ dd + 1 ; • any d -dimensional universal optimal transposi-tion [19] channel T d is compatible with p if andonly if | p | − p | | ≤ d + 1 , | p | − p | | − | p | − p | | ≤ . This paper is structured as follows. We will intro-duce our framework and discuss the case of unitary andtrace class channels in Section II. For the binary case, in-troduced in Section III, we will solve the problem forany qubit dihedrally-covariant channel in Section IV,and for any arbitrary-dimensional universally-covariantcommutativity-preserving channel in Section V. In Sec-tion VI we will provide a natural geometrical interpreta-tion of our results, and in Section VII we will summarizeour results and present further outlooks.
II. GENERAL RESULTS
We will make use of standard definitions and results inquantum information theory [25]. Since S nm ( X ) is convexfor any n and m , the hyperplane separation theorem [26,27] states that p / ∈ S nm ( X ) if and only if there exists an m × n real matrix w such that p T · w − W ( X , w ) > , (2)where p T · w := P i,j p j | i w i,j , and W ( X , w ) := max q ∈ S nm ( X ) w T · q, (3)We call w a channel witness and W ( X , w ) its thresholdvalue for channel X .Although Eq. (2) generally allows one to detect some conditional probability distributions p not belonging to S nm ( X ) for any arbitrarily fixed witness w , here our aimis to detect any such p . Direct application of Eq. (2)is impractical, as one would need to consider all of theinfinitely many witnesses w . Notice however that Eq. (2)can be rewritten through negation by stating that p ∈ S nm ( X ) if and only if for any m × n witness w one has p T · w − W ( X , w ) ≤ , We then have our first preliminary result.
Lemma 1.
A channel X : L ( H ) → L ( K ) is compatiblewith conditional probability distribution p if and only if max w (cid:2) p T · w − W ( X , w ) (cid:3) ≤ . (4)Let us start by considering an arbitrary d -dimensionalunitary channel U d : ρ → U ρU † , for some unitary U ∈ L ( H ) with dim H = d . If d ≥ m , the maximiza-tion in Eq. (3) is trivial, since the input labels i ∈ [1 , m ]can all be encoded on orthogonal states, so that any m × n conditional probability distribution q can in fact be ob-tained. However, if d < m , the evaluation of the witnessthreshold W ( U d , w ) for any witness w is far from obvious.The solution immediately follows from a recent, remark-able result by Frenkel and Weiner [17]. It turns out that W ( U d , w ) is attained on extremal conditional probabilitydistributions q compatible with the exchange of a classi-cal d -level system, namely, those q where q j | i = 0 or 1 forany i and j , and such that q j | i = 0 for at most d differentvalues of j . Frenkel and Weiner’s result hence guaranteesthat the threshold W ( U d , w ) can be provided in closedform since, for any m and n , the number of such ex-tremal classical conditional probabilities is finite, i.e., theset S nm ( U d ) is a polytope . Any probability p lying outside S nm ( U d ) can thus be detected by testing the violation ofEq. (4) for a finite number of witnesses w , correspondingto the faces of the polytope. Moreover, the set S nm ( U d )of distributions compatible with any d -dimensional uni-tary channel U d coincides with the set S nm ( F λd ) of dis-tributions compatible with any d -dimensional dephasingchannel F λd : ρ → λρ + (1 − λ ) P k h k | ρ | k i | k ih k | .At the opposite end of the unitary channels, there sit trace-class channels T : ρ → σ for some arbitrary butfixed state σ . In this case, no information about i (theinput label) can be communicated. Of course, the set S nm ( T ) of correlations achievable through any trace-classchannel T does not depend on the particular choice of σ : a trace-class channel simply means that no commu-nication is available. For any trace-class channel T andany witness w , it immediately follows that the threshold W ( T , w ) is achieved by conditional probabilities q suchthat q j | i = 1 for a single value of j , and therefore is givenby W ( T , w ) = max j P i w i,j . As a consequence, the set S nm ( T ) is a polytope with n vertices, and any probabil-ity p lying outside S nm ( T ) can be detected by testing theviolation of Eq. (4) for a finite number of witnesses w . III. BINARY CONDITIONAL PROBABILITYDISTRIBUTION
In the remainder of this work we will consider the casewhere p is a binary input-output conditional probabilitydistributions (i.e. m = n = 2).First, we show that it suffices to consider diagonalor anti-diagonal witnesses with positive entries summingup to one. Indeed, for any witness w , the witness w ′ := α ( w + β ), where α > β is such that β i,j is independent of j , leaves Eq. (4) invariant for any con-ditional probability distribution p and channel X , since w ′ · p = α ( p T · w + P i β i, ).By taking β i,j = − min k w i,k for any i and j , the wit-ness w ′ is diagonal, anti-diagonal, or has a single non-null column. We first consider the latter case. Clearly, themaximum in Eq. (3) is attained when p is a vertex ofthe polytope S ( T ) of probabilities compatible with anytrace-type channel T , and therefore Eq. (4) is alwaysverified. Then we consider the case of diagonal and anti-diagonal witnesses. By taking α − = P i | w i, − w i, | onerecovers the normalization condition P i,j w i,j = 1, thusproving the statement.Therefore, upon denoting with w ± ( ω ) the diagonal andanti-diagonal witnesses given by w + ( ω ) := (cid:18) ω − ω (cid:19) , w − ( ω ) := (cid:18) ω − ω (cid:19) , where ω ∈ [ − , Lemma 2.
The maximum in Eq. 4 is attained for adiagonal or anti-diagonal witness, namely max w ( p T · w − W ( X , w ))= max ω ∈ [ − , ( p T · w ± ( ω ) − W (cid:0) X , w ± ( ω ) (cid:1) ) . Any extremal distribution q in Eq. (3) can be repre-sented by states ρ and ρ and a POVM { π , π } suchthat q j | i = Tr[ X ( ρ i ) π j ]. Since w ± ( ω ) is diagonal or anti-diagonal, Eq. (3) represents the maximum probability ofsuccess in the discrimination of states { ρ , ρ } with priorprobabilities given by the non-null entries of w , in thepresence of noise X , namely W ( X , w ± ( ω ))= 12 max ρ ,ρ { π ,π } [(1 + ω ) Tr[ X ( ρ ) π ] + (1 − ω ) Tr[ X ( ρ ) π ]] . It is a well-known fact [28] that the solution of the opti-mization problem over POVMs is given as a function ofthe Helstrom matrix defined as H ω ( ρ , ρ ) := 1 + ω ρ − − ω ρ , as follows W ( X , w ± ( ω )) = 12 max ρ ,ρ [1 + ||X ( H ω ( ρ , ρ )) || ] , (5)where ||·|| denotes the operator 1-norm.It is easy to see that without loss of generality onecan take ρ and ρ such that [ ρ , ρ ] = 0. Indeed, let {| k i} be a basis of eigenvectors of the Helstrom matrix H ω ( ρ , ρ ). The complete dephasing channel F d on thebasis {| k i} is such that H ω ( ρ , ρ ) = F d ( H ω ( ρ , ρ )) = H ω ( σ , σ ) , where σ i := F d ( ρ i ) and therefore [ σ , σ ] = 0. By apply-ing channel X we have the following identity X ( H ω ( ρ , ρ )) = X ( H ω ( σ , σ ))Therefore, the encoding { σ i } performs as well as the en-coding { ρ i } , and thus without loss of generality we cantake the supremum in Eq. (5) over commuting encodingsonly.Moreover, one can see that without loss of generalityone can take σ i to be orthogonal pure states. Indeed,let σ i = P k µ k | i | k ih k | be a spectral decomposition of σ i .Due to the convexity of the trace norm we have ||X ( H ω ( σ , σ )) || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k,l µ k | µ l | X ( H ω ( | k ih k | , | l ih l | )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X k,l µ k | µ l | ||X ( H ω ( | k ih k | , | l ih l | )) || ≤ max k,l ||X ( H ω ( | k ih k | , | l ih l | )) || . Then we have the following preliminary result.
Lemma 3.
The maximum in Eq. (3) is given by an or-thonormal pure encoding, namely W (cid:0) X , w ± ( ω ) (cid:1) := max | φ i , | φ ih φ | φ i =0
12 [1 + ||X ( H ω ( φ , φ )) || ] , and by an orthogonal POVM such that π is the projectoron the positive part of H ω ( φ , φ ) and π = − π . Here, for any pure state | φ i we denote with φ := | φ ih φ | the corresponding projector. IV. DIHEDRALLY COVARIANT QUBITCHANNEL
Let us start with the case where X : L ( H ) → L ( K ) isa qubit channel, i.e. dim H = dim K = 2. Since Paulimatrices span the space of qubit Hermitian operators,any qubit state ρ can be parametrized in terms of Paulimatrices, i.e. ρ = 12 ( ~σ T · ~x ) , | ~x | ≤ , (6)where ~σ = ( σ x , σ y , σ z ) T and ~x are the vectors of Paulimatrices and their real coefficients, respectively. Analo-gously, any qubit channel X can be parametrized in termsof Pauli matrices, i.e. X ( ρ ) = 12 (cid:16) ~σ T · ( A~x + ~b ) (cid:17) , where A i,j = Tr [ σ i X ( σ j )] and b i = Tr [ σ i X ( X ( H ω ( φ , φ )) assumesa very simple form given by X ( H ω ( φ , φ )) = 12 (cid:20) ω (cid:16) A~x + ω~b (cid:17) T · ~σ (cid:21) , whose eigenvalues are (cid:16) ω ± (cid:12)(cid:12)(cid:12) A~x + ω~b (cid:12)(cid:12)(cid:12) (cid:17) . Thus, thewitness threshold W ( X , w ± ( ω )) in Eq. (3) can be readilycomputed by means of Lemma 3 as W (cid:0) X , w ± ( ω ) (cid:1) = 12 | ω | , max ~x | ~x | ≤ (cid:12)(cid:12)(cid:12) A~x + ω~b (cid:12)(cid:12)(cid:12) . Notice that this expression is the maximum betweentwo strategies. The first one is given by the trivial POVMand thus corresponds to trivial guessing. The secondone can be further simplified by means of the followingsubstitutions. Let A = V DU be a polar decompositionof matrix A with U and V unitaries and D diagonal andpositive-semidefinite with eigenvalues ~d (accordingly ~c := − V † ~b ). By unitary invariance of the 2-norm one hasmax ~x | ~x | ≤ (cid:12)(cid:12)(cid:12) A~x + ω~b (cid:12)(cid:12)(cid:12) = max ~x | ~x | ≤ | D~x − ω~c | . By defining ~y := D~x one hasmax ~x | ~x | ≤ | D~x − ω~c | = max ~y,~z | D − ~y + ( − D − D ) ~z | ≤ | ~y − ω~c | , where ( · ) − denotes the Moore-Penrose pseudoin-verse. By explicit computation it follows that[ D − ] T (cid:0) − D − D (cid:1) = 0, and therefore vectors D − ~y and (cid:0) − D − D (cid:1) ~z are orthogonal. Then for any op-timal ( ~y, ~z ) one has that ( ~y,
0) is also optimal, since (cid:12)(cid:12) D − ~y + (cid:0) − D − D (cid:1) ~z (cid:12)(cid:12) ≥ (cid:12)(cid:12) D − ~y (cid:12)(cid:12) . Therefore we have W (cid:0) X , w ± ( ω ) (cid:1) = 12 [1 + max ( ω, ∆( ω ))] , (7)where ∆( ω ) := max ~y | D − ~y | ≤ | ~y − ω~c | . (8)The maximum in Eq. (8) is a quadratically constrainedquadratic optimization problem, which is known to beNP-hard in general. However, ∆( ω ) has a simple ge-ometrical interpretation: it is the maximum Euclideandistance of vector ω~c and ellipsoid (cid:12)(cid:12) D − ~y (cid:12)(cid:12) ≤
1. Thisinterpretation suggests symmetries under which the op-timization problem becomes feasible. In particular, wetake vector ~c to be parallel to one of the axis of the ellip-soid (cid:12)(cid:12) D − ~y (cid:12)(cid:12) ≤
1, namely c = c = 0 (up to irrelevantpermutations of the computational basis).This configuration corresponds to a D -covariant chan-nel X , where D is the dihedral group of the symme-tries of a line segment, consisting of two reflections anda π -rotation. This configuration is depicted in Fig. 1.In particular, a qubit channel X is D -covariant if andonly if there exist unitary representations U k ∈ R × and V k ∈ R × of D such that AU k ~x + ~b = V k ( A~x + ~b ) . (9) ( a ) ( b ) ~c ( c ) ~c Figure 1. Bloch-sphere representation of: [ (a) , (b) ] dihe-drally covariant channels X mapping the sphere into an ellip-soid (a) centered in the Bloch sphere (e.g. any Pauli channel P ~λ ), or (b) translated by a vector ~c which is parallel to one ofthe axis of the ellipsoid (e.g. any amplitude damping channel A λ ); (c) non-dihedrally covariant channel X , as the ellipsoidis translated by a vector ~c which is not parallel to any of theaxis of the ellipsoid. Up to unitaries, the most general unitary representationof D in R × is given by W = σ z ⊕ , W = − σ z ⊕ , W = − ⊕ , where W and W are reflections and W is a π -rotation.We take U k := U † W k U and V k := V W k V † . Then byexplicit computation we have AU k ~x + ~b = V k A~x + ~b, where we used the fact that [ D, W k ] = 0 for any k . There-fore, D covariance expressed by Eq. (9) is equivalent tothe requirement W k ~c = ~c , namely c = c = 0.Under the assumption of D -covariance, we take with-out loss of generality d ≥ d and c ≥
0. If also c = 0,we further take without loss of generality d ≥ d . Then,as formally proved in the Appendix, the maximum Eu-clidean distance ∆( ω ) in Eq. (8) can be explicitly com-puted, leading to the following result. Lemma 4.
The witness threshold W ( X , w ± ( ω )) of anyqubit D -covariant channel X is given by Eq. (7) where ∆( ω ) = d s c ω d − d , if | ω | < d − d d c ,d + c | ω | , otherwise.The optimal encoding is given by Eq. (6) with ~x = D − ~y and ~y = (cid:16) , ± d q − c d ω ( d − d ) , c d ωd − d (cid:17) T if | ω | ≤ d − d d c (0 , , ± d ) T otherwise . Using Lemma 4 and Lemma 1, Eq. (4) becomes themaximum over ω of the minimum of two functions. Themaximum is attained either in the maxima 0, ± ω , or ± − , ω := ( d − d )( p | − p | ) c q c d − ( d − d )( p | − p | ) , (the limit should be considered if c = 0), or in theirintersection ± ω given by ω := s d ( d − d ) d − d − d c , if ( d − d ) > d c ,d − c , otherwise.We can then state our first main result, formally provedin the Appendix, namely a complete and closed-formcharacterization of the set S ( X ) of conditional probabil-ity distributions compatible with any qubit D -covariantchannel X . Theorem 1.
Any given binary conditional probabilitydistribution p is compatible with any given qubit D -covariant channel X if and only if max ω ∈ Ω ( p T · w ± ( ω ) − W (cid:0) X , w ± ( ω ) (cid:1) ) ≤ , (10) where Ω := { , ± ω , ± ω , ± } ∩ [ − , . As applications of Theorem 1, let us explicitly charac-terize the sets of binary conditional probability distribu-tions compatible with two relevant examples of qubit D -covariant channels: the Pauli and amplitude-dampingchannels.Any Pauli channel can be written as P ~λ : ρ → λ ρ + P k =1 λ k σ k ρσ † k , where ~σ = ( σ x , σ y , σ z ) are the Pauli ma-trices. One has that c = 0 and d = max k ∈ [1 , | λ + λ k ) − | ≥ d , thus ω = ∞ and ω = d and the maximum inEq. (10) is attained for ω = ± ω . Thus, upon applyingTheorem 1, one has the following result. Corollary 1.
Any given binary conditional probabilitydistribution p is compatible with the Pauli channel P ~λ ifand only if | p | − p | | − | p | − p | | ≤ max k ∈ [1 , | λ + λ k ) − | . Any amplitude-damping channel can be written as A λ ( ρ ) = P k =0 A k ρA † k , where A = | ih | + √ λ | ih | and A = √ − λ | ih | . As shown in the Appendix, one hasthat c = 1 − λ and d = λ , d = d = √ λ , and thus themaximum in Eq. (10) is attained for ω = ± ω or ω = ± Corollary 2.
Any given binary conditional probabilitydistribution p is compatible with the amplitude-dampingchannel A λ if and only if (cid:0) √ p | p | − √ p | p | (cid:1) ≤ λ. V. UNIVERSALLY-COVARIANTCOMMUTATIVITY-PRESERVING CHANNELS
Let us now move to the arbitrary dimensional case.We trade generality regarding the dimension for general-ity regarding the symmetry of the channel, and assume universal covariance . A channel X : L ( H ) → L ( K ) isuniversally covariant if and only if there exist unitaryrepresentations U g ∈ L ( H ) and V g ∈ L ( K ) of the spe-cial unitary group SU ( d ) with d := dim H , such that forevery state ρ ∈ L ( H ) one has X ( U g ρU † g ) = V g X ( ρ ) V † g . (11)From universal covariance it immediately follows that any orthonormal pure encoding attains the witnessthreshold W ( X , w ± ( ω )) in Eq. (5). Indeed, for any or-thonormal pure states { φ i } let U be the unitary such that φ i = U | i ih i | U † . Then one has ||X ( H ω ( φ , φ )) || = (cid:12)(cid:12)(cid:12)(cid:12) X (cid:0) H ω (cid:0) U | ih | U † , U | ih | U † (cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) V X ( H ω ( | ih | , | ih | )) V † (cid:12)(cid:12)(cid:12)(cid:12) = ||X ( H ω ( | ih | , | ih | )) || , where the second equality follows from Eq. (11), and thethird from the invariance of trace distance under unitarytransformations. Then we have the following result. Lemma 5.
The witness threshold W ( X , w ± ( ω )) of anyuniversally covariant channel X is given by W (cid:0) X , w ± ( ω ) (cid:1) = 12 [1 + ||X ( H ω ( | ih | , | ih | )) || ] . (12) The optimal encoding is given by any pair of orthonormalpure states.
Equation (12) has a simple dependence on w in thecase when channel X is commutativity preserving, i.e.[ X ( ρ ) , X ( ρ )] = 0 whenever [ ρ , ρ ] = 0. Notice thatit suffices to check commutativity preservation for purestates, indeed a channel X is commutativity preservingif and only if [ X ( φ ) , X ( φ )] = 0 whenever h φ | φ i = 0.Necessity is trivial, and sufficiency follows by assuming[ ρ , ρ ] = 0, and considering a simultaneous spectral de-compositions of ρ = P k µ k φ k and ρ := P j ν j φ j . Thenone has [ X ( ρ ) , X ( ρ )] = X k,l µ k ν l [ X ( φ k ) , X ( φ l )]= 0 , where the last inequality follows from the fact that h φ l | φ k i = δ k,l . For a universally covariant channel X ,it immediately follows from Eq. (11) that it suffices tocheck commutativity preservation for an arbitrary pairof orthogonal pure states.In this case X ( | ih | ) and X ( | ih | ) admit a commonbasis of eigenvectors {| k i} , and thus a spectral decom-position of the Helstrom matrix X ( H ω ( | ih | , | ih | )) isgiven by X ( H ω ( | ih | , | ih | )) = X k ( α k ω + β k ) | k ih k | , where α k and β k are the half-sum and half-difference ofthe k -th eigenvectors of X ( | ih | ) and X ( | ih | ), respec-tively. Therefore Eq. (12) becomes W (cid:0) X , w ± ( ω ) (cid:1) = 12 X k | α k ω + β k | ! . Then, the optimization problem in Eq. (4) becomespiece-wise linear, thus the maximum is attained on theintersections of the piece-wise components given by γ k := β k /α k when such values belongs to the domain [ − , S ( X ) of conditional probability distributionscompatible with any arbitrary-dimensional universally-covariant commutativity-preserving channel X . Theorem 2.
Any given binary conditional probabilitydistribution p is compatible with any given arbitrary-dimensional universally-covariant commutativity-preserving channel X if and only if ( | p | − p | | ≤ P k | β k | , | p | − p | | ≤ ||X ( H γ k ( | ih | , | ih | )) || − γ k | p | − p | | , for any k such that γ k ∈ [ − , . As applications of Theorem 2, let us explicitly computethe binary conditional probability distributions compat-ible with any erasure, depolarizing, universal optimal1 → X ( | ih | ) , X ( | ih | )] = 0.Any erasure channel can be written as E λd : ρ → λρ ⊕ (1 − λ ) φ , where φ is some pure state. One cancompute that ~α = (cid:0) λ , λ , × d − , − λ (cid:1) and ~β = (cid:0) λ , − λ , × d − (cid:1) , thus upon applying Theorem 2 onehas the following Corollary. Corollary 3.
Any given binary conditional probabilitydistribution p is compatible with the erasure channel E λd if and only if | p | − p | | ≤ λ. Any depolarizing channel can be written as D λd : ρ → λρ + (1 − λ ) d . One can compute that ~α = (cid:0) λ + − λd × , − λd × d − (cid:1) and ~β = (cid:0) − λ , λ , × d − (cid:1) ,thus upon applying Theorem 2 one has the followingCorollary. Corollary 4.
Any given binary conditional probabilitydistribution p is compatible with the depolarizing channel D λd if and only if | p | − p | | ≤ λ, | p | − p | | − | p | − p | | ≤ dλ − λ + dλ . The universal optimal 1 → C λd : ρ → d +1 P S ( ρ ⊗ P S . By explicit com-putation one has C d ( | i ih i | ) = 12( d + 1) X k ( | k, i i + | i, k i )( h k, i | + h i, k | ) , and therefore [ C d ( | ih | ) , C d ( | ih | )] = 0, thus theuniversal optimal 1 → C d is a com-mutativity preserving channel. One can com-pute that ~α = (cid:16) d +1 × , d +1) × d − (cid:17) and ~β = (cid:16) − d +1 , d +1 , , − d +1) × d − , d +1) × d − (cid:17) ,thus upon applying Theorem 2 one has the followingCorollary. Corollary 5.
Any given binary conditional probabilitydistribution p is compatible with the universal optimal → cloning channel C d if and only if | p | − p | | ≤ dd + 1 . The universal transposition channel can be writ-ten as T d : ρ → d +1 (cid:0) ρ T + (cid:1) . One can com-pute that ~α = (cid:16) d +1) × , d +1 × d − (cid:17) and ~β = (cid:16) d +1) , − d +1) , × d − (cid:17) , thus upon applying Theo-rem 2 one has the following Corollary. Corollary 6.
Any given binary conditional probabilitydistribution p is compatible with the universal transposi-tion channel T d if and only if | p | − p | | ≤ d + 1 , | p | − p | | − | p | − p | | ≤ . The results of Corollaries 1, 2, 3, 4, 5, and 6 are sum-marized in Table I.
VI. CARTESIAN REPRESENTATION
In this Section we provide a geometrical interpretationof our results. Binary conditional probability distribu-tions are represented by 2 × R . However, due to the nor-malization constraint P j p j | i = 1 for any i , they all liein a bidimensional affine subspace. A natural Cartesianparametrization of such a subspace is given by p j | i = p ( x, y ) = 12 (cid:20)(cid:18) (cid:19) + x (cid:18) − − (cid:19) + y (cid:18) − − (cid:19)(cid:21) , (13)and binary conditional probability distributions form thesquare | x ± y | ≤
1, whose 4 vertices are the right-stochastic matrices with all entries equal to 0 or 1.As it is clear from Eq. (13): X p ∈ S ( X ) P ~λ | p | − p | | ≤ max k ∈ [1 , | λ + λ k ) − |A λ ( √ p | p | − √ p | p | ) ≤ λ E λd | p | − p | | ≤ λ D λd ( | p | − p | | ≤ λ | p | − p | | −| p | − p | | ≤ dλ − λ + dλ C d | p | − p | | ≤ dd +1 T d ( | p | − p | | ≤ d +1 | p | − p | | −| p | − p | | ≤ Table I. Complete closed-form characterization of the set S ( X ) of binary conditional probability distributions compat-ible with channel X , for X given by the Pauli channel P ~λ , theamplitude damping channel A λ , the erasure channel E λd , thedepolarizing channel D λd , the universal 1 → C d , and the universal transposer T d , as given by Corollaries 1,2, 3, 4, 5, and 6, respectively. • a permutation of the states { ρ , ρ } corresponds tothe transformation ( x, y ) → ( x, − y ); • a permutation of the effects { π , π } correspondsto the transformation ( x, y ) → ( − x, − y ); • a permutation of the states { ρ , ρ } and ef-fects { π , π } corresponds to the transformation( x, y ) → ( − x, y ).Therefore, for any channel X , the set S ( X ) of binaryconditional probability distributions compatible with X is symmetric for reflections around the x or y axes (i.e.,it is D -covariant).As a consequence of our previous results, the sets S ( U d ) and S ( F λd ) of conditional probability distribu-tions compatible with any unitary and dephasing chan-nels U d and F λd coincide with the square | x ± y | ≤
1, forany d and any λ . The set S ( T ) of conditional probabil-ity distributions compatible with any trace-class channel T coincide with the segment x ∈ [ − , y = 0.With the parametrization in Eq. (13), the sets ofbinary conditional probability distributions compatiblewith any Pauli, amplitude-damping, erasure, depolariz-ing, universal 1 → VII. CONCLUSIONS AND OUTLOOK
In this work, we developed a device-independentframework for testing quantum channels. The problemwas framed as a game involving an experimenter, claim-ing to be able to produce some quantum channel, and atheoretician, willing to trust observed correlations only. X p ( x, y ) ∈ S ( X ) P ~λ | y | ≤ max k ∈ [1 , | λ + λ k ) − |A λ (cid:16)p − y − x + y − p y − x + y (cid:17) ≤ λ E λd | y | ≤ λ D λd ( | y | ≤ λ | y | −| x | ≤ dλ − λ + dλ C d | y | ≤ dd +1 T d ( | y | ≤ d +1 | y | −| x | ≤ Table II. Cartesian parametrization of the set S ( X ) of binaryconditional probability distributions compatible with chan-nel X , for X given by the Pauli channel P ~λ , the amplitudedamping channel A λ , the erasure channel E λd , the depolariz-ing channel D λd , the universal 1 → C d , andthe universal transposer T d . The optimal strategy consists of i) all the input statesand measurements generating the extremal correlationsthat the experimenter needs to produce, and ii) a fullclosed-form characterization of the correlations compati-ble with the claim, that the theoretician needs to comparewith the observed correlations. For binary correlations,we explicitly derived the optimal strategy for the caseswhere the claimed channel is a dihedrally-covariant qubitchannel, such as any Pauli and amplitude-damping chan-nels, or an arbitrary-dimensional universally-covariantcommutativity-preserving channel, such as any erasure,depolarizing, universal cloning, and universal transposi-tion channels.Natural generalisation of our results include relaxingthe restriction of binary correlations, that is m = n = 2,and extending the characterization of S nm ( X ) to otherclasses of channels. An interesting generalisation wouldconsist of letting the POVM { π y } depend upon an inputnot known during the preparation of { ρ x } , as is the casein quantum random access codes. Moreover, the setupin Eq. (1) could be modified to allow for entanglementalongside X , or many parallel or sequential uses of chan-nel X .We conclude by remarking that our results are par-ticularly suitable for experimental implementation. Forany channel X an experimenter claims to be able to pro-duce, our framework only requires them to prepare or-thogonal pure input states and perform orthogonal mea-surements in order to fully characterize S ( X ) and thusdevice-independently test X . DATA ACCESSIBILITY
This work does not have any experimental data. ( a ) ( b )( c ) ( d ) x xx xy yy y ∆ y ∆ y ∆ y ∆ x ∆ y ∆ y Figure 2. Cartesian representation of the space of binary con-ditional probability distributions p . The outer white squaredenotes the polytope of all binary conditional probability dis-tributions. The inner yellow region denotes the sets S ( X ) ofconditional probability distributions compatible with: (a) theerasure channel X = E λd (for ∆ y = λ ) and the universal opti-mal 1 → X = C d (for ∆ y = dd +1 ); (b) thePauli channel X = P ~λ (for ∆ y = max k ∈ [1 , | λ + λ k ) − | ); (c) the depolarizing channel X = D λd (for ∆ x = d − d (1 − λ )and ∆ y = λ ) and the universal optimal transposition chan-nel T d (for ∆ x = d − d +1 and ∆ y = d +1 ); (d) the amplitude-damping channel A λ (for ∆ y = λ and ∆ y = √ λ ). COMPETING INTERESTS
The authors declare no competing interests.
AUTHOR’S CONTRIBUTIONS
All authors equally contributed to the originalideas, analytical derivations, and final writing of thismanuscript, and gave final approval for publication.
ACKNOWLEDGEMENTS
We are grateful to Alessandro Bisio, Antonio Ac´ın, Gi-acomo Mauro D’Ariano, and Vlatko Vedral for valuablediscussions and suggestions.
FUNDING
M. D. acknowledges support from the Singapore Min-istry of Education Academic Research Fund Tier 3(Grant No. MOE2012-T3-1-009). F. B acknowledgessupport from the JSPS KAKENHI, No. 26247016.
Appendix A: Proofs
In this Section we prove those results reported in theprevious Sections for which the proof, being lengthy andnot particularly insightful, had only been outlined. Thenumbering of statements follows that of the previous Sec-tions.
Lemma 4.
The witness threshold W ( X , w ± ( ω )) of anyqubit D -covariant channel X is given by Eq. (7) where ∆( ω ) = ( d q c ω d − d , if | ω | < d − d d c ,d + c | ω | , otherwise.Proof. Under the assumption of D -covariance, takewithout loss of generality ~c = (0 , , c ) T . Then with-out loss of generality we take d ≥ d and c ≥
0. If c = 0 without loss of generality we also take d ≥ d .First notice that ~y ∗ , which attains the maximum inEq. (7), lies in the yz plane. Indeed, any ellipse obtainedas the intersection of the ellipsoid | D − ~y | ≤ z axis is, up to a z rotation, a subsetof the ellipse obtained as the intersection of the ellipsoid | D − ~y | ≤ yz plane.The generic vector on the boundary of the yz ellipsecan be parametrized as ~y = , ± d s − z d , z ! T , with z ∈ [ − d , d ], and thus the maximum Euclideandistance in Eq. (8) is given by∆( ω ) = max z ∈ [ − d ,d ] s d (cid:18) − z d (cid:19) + ( z − ωc ) . (A1)By explicit computation one has d ∆( ω ) dz = (cid:20) d (cid:18) − z d (cid:19) + ( z − ωc ) (cid:21) − (cid:20)(cid:18) − d d (cid:19) z − c ω (cid:21) , which is zero for z ∗ = c d ωd − d , and d ∆( ω ) dz (cid:12)(cid:12)(cid:12) z = z ∗ = " d s d (cid:18) c ω d − d (cid:19) − ( d − d ) , namely z ∗ attains the maximum in Eq. (A1) whenever d ≥ d . Therefore the maximum is attained by z = z ∗ iff − d < z ∗ ≤ d , namely when | ω | < d − d d c , and by z = ± d otherwise. By replacing z ∗ and ± d in Eq. (A1)the statement follows. Theorem 1.
Any given binary conditional probabilitydistribution p is compatible with any given qubit D -covariant channel X if and only if max ω ∈ Ω ( p · w ± ( ω ) − W (cid:0) X , w ± ( ω ) (cid:1) ) ≤ , where Ω := { , ± ω , ± ω , ± } ∩ [ − , .Proof. The function f ± ( ω ) := p T · w ± ( ω ) − W ( X , ω )is the minimum of continuous functions g ± ( ω ) := p T · w ± ( ω ) − (1+ | ω | ) and h ± ( ω ) := p T · w ± ( ω ) − (1+∆( ω )).Therefore, max ω ∈ [ − , f ± ( x ) is attained by those valuesof ω maximizing g ± ( ω ) or h ± ( ω ), or in the intersectionsof g ± ( ω ) and h ± ( ω ).The function g ± ( ω ) is piece-wise linear and attains itsmaximum on [ − ,
1] in 0. The function h ± ( ω ) is quasi-concave continuous with a continuous derivative. Indeed2 dh ± ( ω ) dω = ± ( p | − p | ) − d c ω √ ( d − d )( d − d + c ω ) , if | ω | < d − d d c , ± ( p | − p | ) − sgn( ω ) c , otherwise , is continuous and2 d h ± ( ω ) dω = − d c ( d − d ) [ ( d − d )( d − d + c ω ) ] / , if | ω | < d − d d c , , if | ω | > d − d d c . is non positive. Therefore h ± ( ω ) attains its maximum on[ − ,
1] in 0, ±
1, or in the zero ± ω of its first derivative.Due to the piece-wise linearity of g ± ( ω ) and the quasi-concavity of h ± ( ω ), since g ± (0) ≥ h ± (0) and g ± ( ± ≤ h ± ( ±
1) one has that g ± ( ω ) and h ± ( ω ) intersect in ex-actly two points ± ω ∈ [ − , Corollary 2.
Any given binary conditional probabilitydistribution p is compatible with the amplitude-dampingchannel A λ if and only if (cid:0) √ p | p | − √ p | p | (cid:1) ≤ λ. Proof.
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