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.
Considerable work has recently been directed toward developing resource theories of quantum coherence. In this Letter, we establish a criterion of physical consistency for any resource theory. This ...criterion requires that all free operations in a given resource theory be implementable by a unitary evolution and projective measurement that are both free operations in an extended resource theory. We show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations as a natural generalization of the physically consistent operations. Necessary and sufficient conditions are derived for the convertibility of qubit states using dephasing-covariant operations, and we show that these conditions also hold for other well-known classes of incoherent operations.
We show that the generalization of the relative entropy of a resource from states to channels is not unique, and there are at least six such generalizations. Then, we show that two of these ...generalizations are asymptotically continuous, satisfy a version of the asymptotic equipartition property, and their regularizations appear in the power exponent of channel versions of the quantum Stein's lemma. To obtain our results, we use a new type of "smoothing" that can be applied to functions of channels (with no state analog). We call it "liberal smoothing" as it allows for more spread in the optimization. Along the way, we show that the diamond norm can be expressed as a max relative entropy distance to the set of quantum channels, and prove a variety of properties of all six generalizations of the relative entropy of a resource.
We find necessary and sufficient conditions to determine the interconvertibility of quantum systems under time-translation covariant evolution, and use it to solve several problems in quantum ...thermodynamics both in the single-shot and asymptotic regimes. It is well known that the resource theory of quantum athermality is not reversible, but in Brandão et al. Phys. Rev. Lett. 111, 250404 (2013) it was claimed that the theory becomes reversible “provided a sublinear amount of coherent superposition over energy levels is available.” Here we show that if a sublinear amount of coherence among energy levels were considered free, then the resource theory of athermality would become trivial. Instead, we show that by considering a sublinear amount of energy to be free, the theory of athermality becomes reversible for the pure-state case. A proof of the same claim for the mixed-state case is still lacking.
What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, ...thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically.
We introduce an axiomatic approach for channel divergences and channel relative entropies that is based on three information-theoretic axioms of monotonicity under superchannels, i.e., generalized ...data processing inequality, additivity under tensor products, and normalization, similar to the approach given for the state domain in Gour and Tomamichel arXiv:2006.11164v1 (2020), arXiv:2006.12408v2 (2020). We show that these axioms are sufficient to give enough structure in the channel domain as well, leading to numerous properties that are applicable to all channel divergences. These include faithfulness, continuity, a type of triangle inequality, and boundedness between the min and max channel relative entropies. In addition, we prove a uniqueness theorem showing that the Kullback-Leibler divergence has only one extension to classical channels. For quantum channels, with the exception of the max relative entropy, this uniqueness does not hold. Instead, we prove the optimality of the amortized channel extension of the Umegaki relative entropy, by showing that it provides a lower bound on all channel relative entropies that reduce to the Kullback-Leibler divergence on classical states. We also introduce the maximal channel extension of a given classical state divergence and study its properties.
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the ...long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d-dimensional Hilbert space, one can construct a complete set of mutually unbiased measurements. Besides their intrinsic link to MUB, we show that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy a non-trivial entropic uncertainty relation similar to MUB.
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