Inblindcompression 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-assistedvisiblecompression of mixed states wherein Alice knows what state is drawn. In this paper, we prove a general and 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 aspecificensemble of only two states and prove a near-maximal separation (saturating the dimension bound in leading order) between the best achievable rate and the Holevo information for constant error. This also implies that the ensemble is incompressible – compression does not reduce the communication cost by much. Since the states areclassical, 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.
Inblindcompression of quantum states, a sender Alice is given a specimen of a quantum stateρdrawn from a known ensemble (but without knowing whatρis), and she transmits sufficient quantum data to a receiver Bob so that he can decode a near perfect specimen ofρ. 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-assistedvisiblecompression of mixed states wherein Alice knows what state is drawn. In this paper, we prove a general and 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 aspecificensemble of only two states and prove a near-maximal separation (saturating the dimension bound in leading order) between the best achievable rate and the Holevo information for constant error. This also implies that the ensemble is incompressible – compression does not reduce the communication cost by much. Since the states areclassical, 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.
We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.
A central question in classical information theory is that of source compression, which is the task where Alice receives a sample from a known probability distribution and needs to transmit it to the receiver Bob with small error. This problem has a one-shot solution due to Huffman, in which the messages are of variable length and the expected length of the messages matches the asymptotic and i.i.d. compression rate of the Shannon entropy of the source. In this work, we consider a quantum extension of above task, where Alice receives a sample from a known probability distribution and needs to transmit a part of a pure quantum state (that is associated to the sample) to Bob. We allow entanglement assistance in the protocol, so that the communication is possible through classical messages, for example using quantum teleportation. The classical messages can have a variable length and the goal is to minimize their expected length. We provide a characterization of the expected communication cost of this task, by giving a lower bound that is near optimal up to some additive factors. A special case of above task, and the quantum analogue of the source compression problem, is when Alice needs to transmit the whole of her pure quantum state. Here we show that there is no one-shot interactive scheme which matches the asymptotic and i.i.d. compression rate of the von Neumann entropy of the average quantum state. This is a relatively rare case in quantum information theory where the cost of a quantum task is significantly different from its classical analogue. Further, we also exhibit similar results for the fully quantum task of quantum state redistribution, employing some different techniques. We show implications for the one-shot version of the problem of quantum channel simulation.
The resource theory of coherence quantifies the quantum properties of states and their role in quantum information processing. Extending coherence theory to ensembles, the natural objects in quantum communication, remains an active research direction, with their operational potential still being explored. In this work, we establish a maximally coherent ensemble in every finite dimension: the continuous ensemble of pure states distributed according to the normalized Haar measure attains the tight upper bound on basis-free relative entropy ensemble coherence. We then prove that ensemble coherence bounds the effective informational advantage of postselective measurements over the Holevo quantity for two equiprobable pure states. Finally, for the qubit B92 QKD protocol, we show that the coherence of Alice's ensemble upper-bounds the secret key rate in the asymptotic regime. These results give ensemble coherence a quantitative operational meaning in quantum communication and cryptography
We study the question of how much classical communication is needed when Alice is given a classical description of a quantum state ∣ψ⟩ for Bob to recover any expectation value ⟨ψ∣M∣ψ⟩ given an observable M with M Hermitian and ∣∣M∣∣op≤1. This task, whose study was initiated by Raz (ACM 1999) and more recently investigated by Gosset and Smolin (TQC 2019), can be thought of as a fully classical version of the pure state case of the well-known classical shadows problem in quantum learning theory. We show how the hardness of these two seemingly distinct problems are connected. We first consider the relative error version of the communication question and prove a lower bound of Ω(2nε−2) on the one-way randomized classical communication, improving upon an additive error lower bound of Ω(2n) as shown by Gosset and Smolin. Notably, we show that this lower bound holds not only for the set of all observables but also when restricted to just the class of Pauli observables. This fact implies a Ω(2n) versus O(poly(n)) separation in the compression size between the relative and additive error settings for non-adaptive Pauli classical shadows with classical memory. Extending this framework, we prove randomized communication lower bounds for other relative error one-way classical communication tasks: an Ω(2nε−2) lower bound when instead Alice is given an observable and Bob is given a quantum state and they are asked to estimate the expectation value, an Ω(nε−2) lower bound when restricted to Paulis, and an Ω(2nε−2) lower bound when Alice and Bob are both given quantum states and asked to estimate the inner product.
A central question in the classical information theory is that of source compression, which is the task where Alice receives a sample from a known probability distribution and needs to transmit it to the receiver Bob with small error. This problem has a one-shot solution due to Huffman, in which the messages are of variable length and the expected length of the messages matches the asymptotic and independent identically distributed (i.i.d.) compression rate of the Shannon entropy of the source. In this paper, we consider a quantum extension of above task, where Alice receives a sample from a known probability distribution and needs to transmit a part of a pure quantum state (that is associated with the sample) to Bob. We allow entanglement assistance in the protocol, so that the communication is possible through classical messages, for example using quantum teleportation. The classical messages can have a variable length, and the goal is to minimize their expected length. We provide a characterization of the expected communication cost of this task, by giving a lower bound that is near optimal up to some additive factors. A special case of above task, and the quantum analogue of the source compression problem, is when Alice needs to transmit the whole of her pure quantum state. Here, we show that there is no one-shot interactive scheme which matches the asymptotic and i.i.d. compression rate of the von Neumann entropy of the average quantum state. This is a relatively rare case in the quantum information theory where the cost of a quantum task is significantly different from its classical analogue. Furthermore, we also exhibit similar results for the fully quantum task of quantum state redistribution, employing some different techniques. We show implications for the one-shot version of the problem of quantum channel simulation.
We consider a generalization of the quantum state redistribution task, where pure multipartite states from an ensemble source are distributed among an encoder, a decoder and a reference system. The encoder, Alice, has access to two quantum systems: system A which she compresses and sends to the decoder, Bob, and the side information system C which she wants to keep at her site. Bob has access to quantum side information in a system B, wants to decode the compressed information in such a way to preserve the correlations with the reference system on average. As figures of merit, we consider both block error (which is the usual one in source coding) and per-copy error (which is more akin to rate-distortion theory), and find the optimal compression rate for the second criterion, and achievable and converse bounds for the first. The latter almost match in general, up to an asymptotic error and an unbounded auxiliary system; for so-called irreducible sources they are provably the same. Full paper forthcoming [1].