Information measures acquire operational meaning through coding theorems. The quantum conditional mutual information (QCMI) is nonnegative due to strong subadditivity, yet a direct connection with channel coding has remained elusive. In this work, we propose a quantum communication task-conditional quantum communication-that fills this gap. We show that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, is given by half the QCMI. This result naturally extends the classical key generation capacity of Csiszár and Ahlswede to the quantum domain. We place our model within the family tree of quantum protocols and compute the conditional capacity for several example channels. Our results provide new insights for code design in reliable quantum information processing.
Information measures acquire operational meaning through coding theorems. The quantum conditional mutual information (QCMI) is nonnegative due to strong subadditivity, yet a direct connection with channel coding has remained elusive. In this work, we propose a quantum communication task-conditional quantum communication-that fills this gap. We show that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, is given by half the QCMI. This result naturally extends the classical key generation capacity of Csiszár and Ahlswede to the quantum domain. We place our model within the family tree of quantum protocols and compute the conditional capacity for several example channels. Our results provide new insights for code design in reliable quantum information processing.
The transmission of classical information over a classical channel gave rise to the classical capacity theorem with the optimal rate in terms of the classical mutual information. Despite classical information being a subset of quantum information, the rate of the quantum capacity problem is expressed in terms of the coherent information, which does not mathematically generalize the classical mutual information. Additionally, there are multiple capacity theorems with distinct formulas when dealing with transmitting information over a noisy quantum channel. This leads to the question of what constitutes a mathematically accurate quantum generalization of classical mutual information and whether there exists a quantum task that directly extends the classical capacity problem. In this paper, we address these inquiries by introducing a quantity called the generalized information, which serves as a mathematical extension encompassing both classical mutual information and coherent information. We define a transmission task, which includes as specific instances both classical information and quantum information capacity problems, and show that the transmission capacity of this task is characterized by the generalized information.
Lieb and Ruskai's strong subadditivity theorem, which shows that the conditional mutual information must be nonnegative, is fundamental in quantum theory. It has numerous applications, such as in quantum error correction. When the mutual information is zero, the Petz recovery map can be used to reconstruct the quantum channel. When the mutual information is small, one seeks to define an optimal recovery channel. To this end, a mathematical characterization of the mutual information is desirable. We address this problem by providing an exact characterization of the mutual information, along with characterizations for other entropies. Our controls are sharp, leaving no room for improvement, in the sense that we provide equalities, regardless of whether the mutual information (or remainder) is small or large. We transform the definitions of these entropies into a summation of explicitly constructed terms, and the definition of each term obviously demonstrates the desired positivity/convexity/concavity. The summation converges rapidly and absolutely in a chosen elementary norm.
If a sender and a receiver lack precise knowledge about the communication line that connects them, designing a scheme to reliably transmit information becomes more challenging. This has been studied in classical and quantum information theory in the context of compound channel models and arbitrarily varying channel models. However, a fully quantum version of system uncertainty allows for an even more challenging coding scenario with entangled channel uses. This type of model has previously been investigated for classical and quantum capacity. Here, we address the problem of entanglement-assisted capacity in the presence of such system uncertainty. We find that, under the assumption of a finite environment dimension, it is equal to a corresponding compound capacity. Intriguingly, our results imply that in certain fully quantum arbitrarily varying channel models, the entanglement-assisted capacity can be positive while the classical capacity is equal to zero, a phenomenon that does not occur in regular single-channel coding.
The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H ( B ) -H ( E )—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. We proved achievability of the coherent information for quantum data transmission in Corollary 22.2.1, but the roundabout path that we followed to prove achievability there perhaps does not give much insight into the structure of a quantum code that achieves the coherent information. Our approach in this chapter is different and should shed more light on this structure. Specifically, we show how to make coherent versions of the private classical codes from the previous chapter.
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every α∈[1/2,1), yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.