For a pair of quantum channels with the same input space, we show that the possibility of approximation of one channel by post-processings of the other channel can be characterized by comparing the success probabilities for the two ensembles obtained as outputs for any ensemble on the input space coupled with an ancilla. This provides an operational interpretation to a natural extension of Le Cam's deficiency to quantum channels. In particular, we obtain a version of the randomization criterion for quantum statistical experiments. The proofs are based on some properties of the diamond norm and its dual, which are of independent interest.
For a pair of quantum channels with the same input space, we show that the possibility of approximation of one channel by post-processings of the other channel can be characterized by comparing the success probabilities for the two ensembles obtained as outputs for any ensemble on the input space coupled with an ancilla. This provides an operational interpretation to a natural extension of Le Cam's deficiency to quantum channels. In particular, we obtain a version of the randomization criterion for quantum statistical experiments. The proofs are based on some properties of the diamond norm and its dual, which are of independent interest.
We present a general theory of comparison of quantum channels, concerning with the question of simulability or approximate simulability of a given quantum channel by allowed transformations of another given channel. We introduce a modification of conditional min-entropies, with respect to the set F of allowed transformations, and show that under some conditions on F, these quantities characterize approximate simulability. If F is the set of free superchannels in a quantum resource theory of processes, the modified conditional min-entropies form a complete set of resource monotones. If the transformations in F consist of a preprocessing and a postprocessing of specified forms, approximate simulability is also characterized in terms of success probabilities in certain guessing games, where a preprocessing of a given form can be chosen and the measurements are restricted. These results are applied to several specific cases of simulability of quantum channels, including postprocessings, preprocessings and processing of bipartite channels by LOCC superchannels and by partial superchannels, as well as simulability of sets of quantum measurements. These questions are first studied in a general setting that is an extension of the framework of general probabilistic theories (GPT), suitable for dealing with channels. Here we prove a general theorem that shows that approximate simulability can be characterized by comparing outcome probabilities in certain tests. This result is inspired by the classical Le Cam randomization criterion for statistical experiments and contains its finite dimensional version as a special case.
Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate t-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders t, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel t-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping noise), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest in moment calculations.
The classical randomization criterion is an important result of statistical decision theory. Recently, a quantum analogue has been proposed, giving equivalent conditions for two sets of quantum states, ensuring existence of a quantum channel mapping one set close to the other, in L1-distance. In the present paper, we extend these concepts in several ways. First, sets of states are replaced by channels and randomization is performed by either post- or pre-composition with another channel. The L1-distance is replaced by the diamond norm. Secondly, the maps are not required to be completely positive, but positivity is given by an admissible family of convex cones. It is shown that the randomization theorems, generalizing both quantum and classical randomization criteria, can be proved in the framework of base section norms, including the diamond norm and its dual. The theory of such norms is developed in the Appendix.
A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.
We study the estimation of an unknown quantum channel E with input dimension d1, output dimension d2 and Kraus rank at most r. We establish a connection between the query complexities in two models: (i) access to E, and (ii) access to a random dilation of E. Specifically, we show that for parallel (possibly coherent) testers, access to dilations does not help. This is proved by constructing a local tester that uses n queries to E yet faithfully simulates the tester with n queries to a random dilation. As application, we show that:
- O(rd1d2/ε2) queries to E suffice for channel tomography to within diamond norm error ε.
Moreover, when rd2=d1, we show that the Heisenberg scaling O(1/ε) can be achieved, even if E is not a unitary channel:
- O(min{d12.5/ε,d12/ε2}) queries to E suffice for channel tomography to within diamond norm error ε, and O(d12/ε) queries suffice for the case of Choi state trace norm error ε.
- O(min{d11.5/ε,d1/ε2}) queries to E suffice for tomography of the mixed state E(∣0⟩⟨0∣) to within trace norm error ε.
We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.