Device-independent tests of quantum channels
We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis 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 with any 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 and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: i) any dihedrally-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 any erasure, depolarizing, universal cloning, and universal transposition channels).
Any physical experiment is based upon the observation of correlations among events at various points in space and time, along with some assumptions about the underlying physics. Naturally, in order to be operational any such assumption must have been tested as a hypothesis in a previous experiment. Ultimately, to break an otherwise circular argument, experiments involving no further assumptions are required – that is, device-independent tests.
Formally, a hypothesis consists of a circuit circuit (), which is usually assumed to have a global causal structure (following special relativity), and its components, which are usually assumed to be governed by classical or quantum theories and thus representable by channels.
Denoting a hypothesis (circuit) by , the set of correlations compatible with is denoted by . Then, hypothesis is falsified, along with any other hypothesis such that , as soon as the observed correlation does not belong to (This inclusion relation induces an ordering among channels which is reminiscent of that introduced by Shannon Sha58 () among classical channels). Therefore, from the theoretical viewpoint, the problem of falsifying a hypothesis can be recast selftest () as that of characterising the set of compatible correlations.
Since (discrete, memoryless) classical channels are by definition input-output correlations (conditional probabilities), the characterisation of is trivial in classical theory as it is a polytope easily related to the correlation defining the channel. On the contrary, the problem is far from trivial in quantum theory: due to the existence of superpositions of states and effects, the set can be strictly convex.
In this work we address the problem of device-independent tests of quantum channels, in particular the characterization of the set of -inputs/-outputs correlations obtainable through an arbitrary given channel , upon the input of an arbitrary preparation and the measurement of an arbitrary POVM , that is
An alternative formulation for the problem considered here can be given in terms of a “game” involving two parties: an experimenter, claiming to be able to prepare quantum states, feed them through some quantum channel , and then perform measurements on the output, and a skeptical theoretician, willing to trust observed correlations only. If the experimenter produces some correlations lying outsides of , then the theoretician must conclude that the actual channel is not worse than at producing correlations, but this is not sufficient to support the experimenter’s claim. Indeed, in order to convince the theoretician, the experimenter must produce the entire set : in fact, it is sufficient to produce a set of correlations whose convex hull contains . Then, the theoretician must conclude that whatever channel the experimenter actually has is at least as good as at producing correlations, and the experimenter’s claim is accepted.
It is hence clear that the problem of device-independent tests of quantum channels induces a preordering relation among quantum channels: if and only if . (The order also depends upon and , but for compactness we drop the indexes whenever they are clear from the context). In order to characterize such preorder, for any given channel , we need to i) provide the experimenter with all the states and measurements generating the extremal correlations of , and ii) provide the theoretician with a full closed-form characterization of the set of compatible correlations.
As a preliminary result, we find that the sets coincide for any -dimensional unitary and dephasing channels, for any , , and (this is an immediate consequence of a remarkable result by Frenkel and Weiner FW15 ().) Upon considering only the binary case , our first result is to show that any correlation on the boundary of is achieved by a pair of commuting pure states – irrespective of whether is a commutativity-preserving channel. Then, we derive the complete closed-form characterization of 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 cloning Wer98 (), and universal transposition BDPS03 () channels.
Upon specifying as the -dimensional identity channel , one recovers device-independent dimension tests analogous to those discussed in Refs. GBHA10 (); HGMBAT12 (); ABCB11 (); DPGA12 (), in which case the aforementioned ordering induced by the inclusion is of course total. However, the completeness of our characterization of implies that our framework detects all correlations incompatible with the given hypothesis, unlike Refs. GBHA10 (); HGMBAT12 (); ABCB11 (); DPGA12 (); CBB15 () where the set of correlations is tested only along an arbitrarily chosen direction.
Let us provide a preview of some consequences of our results:
Any Pauli channel is compatible with if and only if
any amplitude-damping channel with and is compatible with if and only if
any -dimensional erasure channel for some pure state is compatible with if and only if
any -dimensional depolarizing channel is compatible with if and only if
the -dimensional universal optimal cloning Wer98 () channel is compatible with if and only if
any -dimensional universal optimal transposition BDPS03 () channel is compatible with if and only if
This paper is structured as follows. We will introduce our framework and discuss the case of unitary and trace class channels in Section II. For the binary case, introduced in Section III, we will solve the problem for any qubit dihedrally-covariant channel in Section IV, and for any arbitrary-dimensional universally-covariant commutativity-preserving channel in Section V. In Section VI we will provide a natural geometrical interpretation of our results, and in Section VII we will summarize our results and present further outlooks.
Ii General results
We will make use of standard definitions and results in quantum information theory Wil11 (). Since is convex for any and , the hyperplane separation theorem BV04 (); Bus12 () states that if and only if there exists an real matrix such that
where , and
We call a channel witness and its threshold value for channel .
Although Eq. (2) generally allows one to detect some conditional probability distributions not belonging to for any arbitrarily fixed witness , here our aim is to detect any such . Direct application of Eq. (2) is impractical, as one would need to consider all of the infinitely many witnesses . Notice however that Eq. (2) can be rewritten through negation by stating that if and only if for any witness one has
We then have our first preliminary result.
A channel is compatible with conditional probability distribution if and only if
Let us start by considering an arbitrary -dimensional unitary channel , for some unitary with . If , the maximization in Eq. (3) is trivial, since the input labels can all be encoded on orthogonal states, so that any conditional probability distribution can in fact be obtained. However, if , the evaluation of the witness threshold for any witness is far from obvious. The solution immediately follows from a recent, remarkable result by Frenkel and Weiner FW15 (). It turns out that is attained on extremal conditional probability distributions compatible with the exchange of a classical -level system, namely, those where or for any and , and such that for at most different values of . Frenkel and Weiner’s result hence guarantees that the threshold can be provided in closed form since, for any and , the number of such extremal classical conditional probabilities is finite, i.e., the set is a polytope. Any probability lying outside can thus be detected by testing the violation of Eq. (4) for a finite number of witnesses , corresponding to the faces of the polytope. Moreover, the set of distributions compatible with any -dimensional unitary channel coincides with the set of distributions compatible with any -dimensional dephasing channel .
At the opposite end of the unitary channels, there sit trace-class channels for some arbitrary but fixed state . In this case, no information about (the input label) can be communicated. Of course, the set of correlations achievable through any trace-class channel does not depend on the particular choice of : a trace-class channel simply means that no communication is available. For any trace-class channel and any witness , it immediately follows that the threshold is achieved by conditional probabilities such that for a single value of , and therefore is given by . As a consequence, the set is a polytope with vertices, and any probability lying outside can be detected by testing the violation of Eq. (4) for a finite number of witnesses .
Iii Binary conditional probability distribution
In the remainder of this work we will consider the case where is a binary input-output conditional probability distributions (i.e. ).
First, we show that it suffices to consider diagonal or anti-diagonal witnesses with positive entries summing up to one. Indeed, for any witness , the witness , where and is such that is independent of , leaves Eq. (4) invariant for any conditional probability distribution and channel , since .
By taking for any and , the witness is diagonal, anti-diagonal, or has a single non-null column. We first consider the latter case. Clearly, the maximum in Eq. (3) is attained when is a vertex of the polytope of probabilities compatible with any trace-type channel , and therefore Eq. (4) is always verified. Then we consider the case of diagonal and anti-diagonal witnesses. By taking one recovers the normalization condition , thus proving the statement.
Therefore, upon denoting with the diagonal and anti-diagonal witnesses given by
where , one has the following preliminary result.
The maximum in Eq. 4 is attained for a diagonal or anti-diagonal witness, namely
Any extremal distribution in Eq. (3) can be represented by states and and a POVM such that . Since is diagonal or anti-diagonal, Eq. (3) represents the maximum probability of success in the discrimination of states with prior probabilities given by the non-null entries of , in the presence of noise , namely
It is a well-known fact Hel76 () that the solution of the optimization problem over POVMs is given as a function of the Helstrom matrix defined as
where denotes the operator -norm.
It is easy to see that without loss of generality one can take and such that . Indeed, let be a basis of eigenvectors of the Helstrom matrix . The complete dephasing channel on the basis is such that
where and therefore . By applying channel we have the following identity
Therefore, the encoding performs as well as the encoding , and thus without loss of generality we can take the supremum in Eq. (5) over commuting encodings only.
Moreover, one can see that without loss of generality one can take to be orthogonal pure states. Indeed, let be a spectral decomposition of . Due to the convexity of the trace norm we have
Then we have the following preliminary result.
The maximum in Eq. (3) is given by an orthonormal pure encoding, namely
and by an orthogonal POVM such that is the projector on the positive part of and .
Here, for any pure state we denote with the corresponding projector.
Iv Dihedrally covariant qubit channel
Let us start with the case where is a qubit channel, i.e. . Since Pauli matrices span the space of qubit Hermitian operators, any qubit state can be parametrized in terms of Pauli matrices, i.e.
where and are the vectors of Pauli matrices and their real coefficients, respectively. Analogously, any qubit channel can be parametrized in terms of Pauli matrices, i.e.
where and .
With such a parametrization assumes a very simple form given by
Notice that this expression is the maximum between two strategies. The first one is given by the trivial POVM and thus corresponds to trivial guessing. The second one can be further simplified by means of the following substitutions. Let be a polar decomposition of matrix with and unitaries and diagonal and positive-semidefinite with eigenvalues (accordingly ). By unitary invariance of the -norm one has
By defining one has
where denotes the Moore-Penrose pseudoinverse. By explicit computation it follows that , and therefore vectors and are orthogonal. Then for any optimal one has that is also optimal, since . Therefore we have
The maximum in Eq. (8) is a quadratically constrained quadratic optimization problem, which is known to be NP-hard in general. However, has a simple geometrical interpretation: it is the maximum Euclidean distance of vector and ellipsoid . This interpretation suggests symmetries under which the optimization problem becomes feasible. In particular, we take vector to be parallel to one of the axis of the ellipsoid , namely (up to irrelevant permutations of the computational basis).
This configuration corresponds to a -covariant channel , where is the dihedral group of the symmetries of a line segment, consisting of two reflections and a -rotation. This configuration is depicted in Fig. 1.
In particular, a qubit channel is -covariant if and only if there exist unitary representations and of such that
Up to unitaries, the most general unitary representation of in is given by
where and are reflections and is a -rotation. We take and . Then by explicit computation we have
where we used the fact that for any . Therefore, covariance expressed by Eq. (9) is equivalent to the requirement , namely .
Under the assumption of -covariance, we take without loss of generality and . If also , we further take without loss of generality . Then, as formally proved in the Appendix, the maximum Euclidean distance in Eq. (8) can be explicitly computed, leading to the following result.
(the limit should be considered if ), or in their intersection given by
We can then state our first main result, formally proved in the Appendix, namely a complete and closed-form characterization of the set of conditional probability distributions compatible with any qubit -covariant channel .
Any given binary conditional probability distribution is compatible with any given qubit -covariant channel if and only if
As applications of Theorem 1, let us explicitly characterize the sets of binary conditional probability distributions compatible with two relevant examples of qubit -covariant channels: the Pauli and amplitude-damping channels.
Any Pauli channel can be written as , where are the Pauli matrices. One has that and , thus and and the maximum in Eq. (10) is attained for . Thus, upon applying Theorem 1, one has the following result.
Any given binary conditional probability distribution is compatible with the Pauli channel if and only if
Any amplitude-damping channel can be written as , where and . As shown in the Appendix, one has that and , , and thus the maximum in Eq. (10) is attained for or . Thus, upon applying Theorem 1, one has the following result, formally proved in the Appendix.
Any given binary conditional probability distribution is compatible with the amplitude-damping channel if and only if
V Universally-covariant commutativity-preserving channels
Let us now move to the arbitrary dimensional case. We trade generality regarding the dimension for generality regarding the symmetry of the channel, and assume universal covariance. A channel is universally covariant if and only if there exist unitary representations and of the special unitary group with , such that for every state one has
From universal covariance it immediately follows that any orthonormal pure encoding attains the witness threshold in Eq. (5). Indeed, for any orthonormal pure states let be the unitary such that . Then one has
where the second equality follows from Eq. (11), and the third from the invariance of trace distance under unitary transformations. Then we have the following result.
The witness threshold of any universally covariant channel is given by
The optimal encoding is given by any pair of orthonormal pure states.
Equation (12) has a simple dependence on in the case when channel is commutativity preserving, i.e. whenever . Notice that it suffices to check commutativity preservation for pure states, indeed a channel is commutativity preserving if and only if whenever . Necessity is trivial, and sufficiency follows by assuming , and considering a simultaneous spectral decompositions of and . Then one has
where the last inequality follows from the fact that . For a universally covariant channel , it immediately follows from Eq. (11) that it suffices to check commutativity preservation for an arbitrary pair of orthogonal pure states.
In this case and admit a common basis of eigenvectors , and thus a spectral decomposition of the Helstrom matrix is given by
where and are the half-sum and half-difference of the -th eigenvectors of and , respectively. Therefore Eq. (12) becomes
Then, the optimization problem in Eq. (4) becomes piece-wise linear, thus the maximum is attained on the intersections of the piece-wise components given by when such values belongs to the domain , or on its extrema. We can then provide our second main result, namely a complete closed-form characterization of the set of conditional probability distributions compatible with any arbitrary-dimensional universally-covariant commutativity-preserving channel .
Any given binary conditional probability distribution is compatible with any given arbitrary-dimensional universally-covariant commutativity-preserving channel if and only if
for any such that .
As applications of Theorem 2, let us explicitly compute the binary conditional probability distributions compatible with any erasure, depolarizing, universal optimal cloning, and universal optimal transposition channels. As discussed before, commutativity preservation can be immediately verified for all of these channels by checking that .
Any erasure channel can be written as , where is some pure state. One can compute that and , thus upon applying Theorem 2 one has the following Corollary.
Any given binary conditional probability distribution is compatible with the erasure channel if and only if
Any depolarizing channel can be written as . One can compute that and , thus upon applying Theorem 2 one has the following Corollary.
Any given binary conditional probability distribution is compatible with the depolarizing channel if and only if
The universal optimal cloning channel can be written as . By explicit computation one has
and therefore , thus the universal optimal cloning is a commutativity preserving channel. One can compute that and , thus upon applying Theorem 2 one has the following Corollary.
Any given binary conditional probability distribution is compatible with the universal optimal cloning channel if and only if
The universal transposition channel can be written as . One can compute that and , thus upon applying Theorem 2 one has the following Corollary.
Any given binary conditional probability distribution is compatible with the universal transposition channel if and only if