The Quantum Formalization Drive

A quantum channel can carry classical information at a rate given by its Holevo capacity $C_1$ for one use. If a sender may encode messages in states that are entangled across $m$ uses, the capacity per use could in principle exceed $C_1$, and the ultimate classical capacity then requires a limit $m\to\infty$ (Holevo; Schumacher–Westmoreland). Whether this…

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations.…

We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be…

We present a family of additive quantum error-correcting codes whose capacities exceeds that of quantum random coding (hashing) for very noisy channels. These codes provide non-zero capacity in a depolarizing channel for fidelity parameters $f$ when $f> .80944$. Random coding has non-zero capacity only for $f>.81071$; by analogy to the classical Shannon coding…

We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A…

Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine…

We present two approaches for transmitting classical information over quantum broadcast channels. The first technique is a quantum generalization of the superposition coding scheme for the classical broadcast channel. We use a quantum simultaneous nonunique decoder and obtain a proof of the rate region stated in [Yard et al., IEEE Trans. Inf. Theory 57…

The one-shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold. In this work, we show that this capacity is well approximated by a relative-entropy-type measure defined via hypothesis testing. Combined…

Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user and broadcast channels, under the presumption of two minimum output entropy conjectures. Despite…

A bipartite quantum state $\rho$ shared by Alice and Bob is $(n_A,n_B)$-extendible if there is a state on $n_A$ copies of Alice's system and $n_B$ copies of Bob's system whose reduced state on every pair (one Alice copy, one Bob copy) is $\rho$. Extendibility (also called shareability) measures how much the correlations of $\rho$ can be shared among many parties. Only…

Quantum error correction protects quantum information from noise by storing it in a code and undoing the damage with a recovery operation. The textbook theory, the Knill–Laflamme conditions, characterizes codes that can be corrected perfectly. Real noise rarely admits perfect correction, and codes that are only approximately correctable can perform better…

Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the "partial information" one system needs conditioned on it's prior information. It turns out to be given by an extremely simple formula, the conditional entropy. In the…

It is proven that recently introduced states with perfectly secure bits of cryptographic key (private states representing secure bit) [K. Horodecki et al., Phys. Rev. Lett. 94, 160502 (2005)] as well as its multipartite and higher dimension generalizations always represent distillable entanglement. The corresponding lower bounds on distillable entanglement…

Consider many instances of an arbitrary quadripartite pure state of four quantum systems ABCD. Alice holds the AC part of each state, Bob holds B, while D represents all other parties correlated with ABC. Alice is required to redistribute the C systems to Bob while asymptotically preserving the overall purity. We prove that this is possible using Q qubits…

In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call "squashed…

In finite dimensional quantum information, a state of a full matrix algebra can be restricted to a smaller algebra of observables. The resulting entropy depends on which observables are retained. For a maximal commutative subalgebra, the retained observables are equivalent to measurements in a fixed orthonormal basis. The restriction has an ordinary…