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In blind compression of quantum states, a sender Alice is given a specimen of a quantum state $rho$ drawn from a known ensemble (but without knowing what $rho$ is), and she transmits sufficient quantum data to a receiver Bob so that he can decode a near perfect specimen of $rho$. For many such states drawn iid from the ensemble, the asymptotically achievable rate is the number of qubits required to be transmitted per state. The Holevo information is a lower bound for the achievable rate, and is attained for pure state ensembles, or in the related scenario of entanglement-assisted visible compression of mixed states wherein Alice knows what state is drawn. In this paper, we prove a general, robust, lower bound on the achievable rate for ensembles of classical states, which holds even in the least demanding setting when Alice and Bob share free entanglement and a constant per-copy error is allowed. We apply the bound to a specific ensemble of only two states and prove a near-maximal separation between the best achievable rate and the Holevo information for constant error. Since the states are classical, the observed incompressibility is not fundamentally quantum mechanical. We lower bound the difference between the achievable rate and the Holevo information in terms of quantitative limitations to clone the specimen or to distinguish the two classical states.
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There are different inequivalent ways to define the Renyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszar has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Renyi capacity, defined in terms of the sandwiched quantum Renyi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that [ sc(W,R,P)=sup_{alpha>1}frac{alpha-1}{alpha}left[R-chi_{alpha}^*(W,P)right], ] where $chi_{alpha}^*(W,P)$ is the $P$-weighted sandwiched Renyi divergence radius of the image of the channel.
We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and by taking into account energy constraints, we conclude the same for a pure-loss bosonic channel. The method for establishing the aforementioned entropy bound involves identifying an information measure having two key properties: 1) it does not increase under a one-way local operations and classical communication channel from the receiver to the sender and 2) a quantum channel from sender to receiver cannot increase the information measure by more than the maximum output entropy of the channel. This information measure can be understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement.
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships. Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions. The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings.
We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel. Our main result is a capacity theorem that gives a three-dimensional achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically-enhanced father protocol. The classically-enhanced father protocol is more general than other protocols in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The classically-enhanced father protocol also shows an improvement over a time-sharing strategy for the case of a qubit dephasing channel--this result justifies the need for simultaneous coding of classical and quantum information over an entanglement-assisted quantum channel. Our capacity theorem is of a multi-letter nature (requiring a limit over many uses of the channel), but it reduces to a single-letter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.
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