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Register a new userOn the minimum output entropy of random orthogonal quantum channels
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We consider sequences of random quantum channels defined using the Stinespring formula with Haar-distributed random orthogonal matrices. For any fixed sequence of input states, we study the asymptotic eigenvalue distribution of the outputs through tensor powers of random channels. We show that the input states achieving minimum output entropy are tensor products of maximally entangled states (Bell states) when the tensor power is even. This phenomenon is completely different from the one for random quantum channels constructed from Haar-distributed random unitary matrices, which leads us to formulate some conjectures about the regularized minimum output entropy.
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We show that the minimum output entropy for all single-mode Gaussian channels is additive and is attained for Gaussian inputs. This allows the derivation of the channel capacity for a number of Gaussian channels, including that of the channel with linear loss, thermal noise, and linear amplification.
Optical channels, such as fibers or free-space links, are ubiquitous in todays telecommunication networks. They rely on the electromagnetic field associated with photons to carry information from one point to another in space. As a result, a complete physical model of these channels must necessarily take quantum effects into account in order to determine their ultimate performances. Specifically, Gaussian photonic (or bosonic) quantum channels have been extensively studied over the past decades given their importance for practical purposes. In spite of this, a longstanding conjecture on the optimality of Gaussian encodings has yet prevented finding their communication capacity. Here, this conjecture is solved by proving that the vacuum state achieves the minimum output entropy of a generic Gaussian bosonic channel. This establishes the ultimate achievable bit rate under an energy constraint, as well as the long awaited proof that the single-letter classical capacity of these channels is additive. Beyond capacities, it also has broad consequences in quantum information sciences.
We introduce a new form for the bosonic channel minimal output entropy conjecture, namely that among states with equal input entropy, the thermal states are the ones that have slightest increase in entropy when sent through a infinitesimal thermalizing channel. We then detail a strategy to prove the conjecture through variational techniques. This would lead to the calculation of the classical capacity of a communication channel subject to thermal noise. Our strategy detects input thermal ensembles as possible solutions for the optimal encoding of the channel, lending support to the conjecture. However, it does not seem to be able to exclude the possibility that other input ensembles can attain the channel capacity.
It is well known in the realm of quantum mechanics and information theory that the entropy is non-decreasing for the class of unital physical processes. However, in general, the entropy does not exhibit monotonic behavior. This has restricted the use of entropy change in characterizing evolution processes. Recently, a lower bound on the entropy change was provided in the work of Buscemi, Das, and Wilde~[Phys.~Rev.~A~93(6),~062314~(2016)]. We explore the limit that this bound places on the physical evolution of a quantum system and discuss how these limits can be used as witnesses to characterize quantum dynamics. In particular, we derive a lower limit on the rate of entropy change for memoryless quantum dynamics, and we argue that it provides a witness of non-unitality. This limit on the rate of entropy change leads to definitions of several witnesses for testing memory effects in quantum dynamics. Furthermore, from the aforementioned lower bound on entropy change, we obtain a measure of non-unitarity for unital evolutions.
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.
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