Abstract In classical information theory, uncommon information refers to the amount of information that is not shared between two messages, and it admits an operational interpretation as the minimum communication cost required to exchange the messages. Extending this notion to the quantum setting, quantum uncommon information is defined as the amount of quantum information necessary to exchange two quantum states. While the value of uncommon information can be computed exactly in the classical case, no direct method is currently known for calculating its quantum analogue. Prior work has primarily focused on deriving upper and lower bounds for quantum uncommon information. In this work, we propose a new approach for estimating these bounds by utilizing the quantum Donsker–Varadhan representation and implementing a gradient-based optimization method. Our results suggest a pathway toward efficient approximation of quantum uncommon information using variational techniques grounded in quantum neural architectures.
Abstract In classical information theory, uncommon information refers to the amount of information that is not shared between two messages, and it admits an operational interpretation as the minimum communication cost required to exchange the messages. Extending this notion to the quantum setting, quantum uncommon information is defined as the amount of quantum information necessary to exchange two quantum states. While the value of uncommon information can be computed exactly in the classical case, no direct method is currently known for calculating its quantum analogue. Prior work has primarily focused on deriving upper and lower bounds for quantum uncommon information. In this work, we propose a new approach for estimating these bounds by utilizing the quantum Donsker–Varadhan representation and implementing a gradient-based optimization method. Our results suggest a pathway toward efficient approximation of quantum uncommon information using variational techniques grounded in quantum neural architectures.
Abstract In classical information theory, channel capacity quantifies the maximum number of messages that can be reliably transmitted using shared information. An equivalent concept, termed uncommon information, represents the number of messages required to be exchanged to completely share all information in common. However, this equivalence does not extend to quantum information theory. Specifically, quantum uncommon information (QUI) is operationally defined as the minimal amount of entanglement required for the quantum communication task of quantum state exchange (QSE), where two parties exchange quantum states to share all quantum messages in common. Currently, an analytical closed-form expression for the QUI remains undetermined. In this work, by investigating underlying characterization of the QUI, we derive improved bounds on it. To obtain these bounds, we develop a subspace exchange strategy that leverages a common subspace of two parties to identify the unnecessary qubits for exchange. We also consider a referee-assisted exchange, wherein a referee aids two parties in efficiently performing the QSE. Our bounds provide more precise estimations for the QUI. Furthermore, we demonstrate that the subspace technique is a versatile tool for characterizing uncommon information not only in the bipartite scenario but also in various multi-partite ones.
If two parties share an unknown quantum state, one can ask how much quantum communication is needed for party A to send her share to party B. Recently, it was found that the number of qubits which should be sent is given by the conditional entropy. This quantifies the notion of partial information, and it can even be negative. Here, we not only demand that A send her state to B, but additionally, B should send his state to A. Paradoxically, we find that requiring that the parties perform this additional task can lower the amount of quantum communication required. This primitive, which we call quantum state exchange, can be used to quantify the notion of uncommon information, since the two parties only need to send each other the parts of their state they don't hold in common. In the classical case, the concept of uncommon information follows trivially from the concept of partial information. We find that for quantum states, this is not so. We prove upper and lower bounds for the uncommon information and find optimal protocols for several classes of states.
Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (Rényi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension d with bounded Thompson metric, our theory establishes a copy complexity of O(∣Θ(U)∣d/ε2) for QNE with a quantum circuit parameter set Θ(U), which has minimax optimal dependence on the accuracy ε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O(∣Θ(U)∣polylog(d)/ε2). Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.
Abstract We propose a method of quantum machine learning called quantum mutual information neural estimation (QMINE) for estimating von Neumann entropy and quantum mutual information, which are fundamental properties in quantum information theory. The QMINE proposed here basically utilizes a technique of quantum neural networks (QNNs), to minimize a loss function that determines the von Neumann entropy, and thus quantum mutual information, which is believed more powerful to process quantum datasets than conventional neural networks due to quantum superposition and entanglement. To create a precise loss function, we propose a quantum Donsker-Varadhan representation (QDVR), which is a quantum analog of the classical Donsker-Varadhan representation. By exploiting a parameter shift rule on parameterized quantum circuits, we can efficiently implement and optimize the QNN and estimate the quantum entropies using the QMINE technique. Furthermore, numerical observations support our predictions of QDVR and demonstrate the good performance of QMINE.
We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum channels, and then investigate the properties of such quantities. These are the fully quantum generalizations of the analogous quantities for bipartite classical functions that have found many applications recently, in particular for proving communication complexity lower bounds. Our definition is strongly tied to the quantum state redistribution task. Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite channel on a given state. This generalizes a result of Braverman and Rao to quantum protocols, and even strengthens the classical result in a bounded round scenario. Also, this provides an analogue of the Schumacher source compression theorem for interactive quantum protocols, and answers a question raised by Braverman. We also discuss some potential applications to quantum communication complexity lower bounds by specializing our definition for classical functions and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new evidence suggesting that the bounded round quantum communication complexity of the disjointness function is Ω(n/M + M), for M-message protocols. This would match the best known upper bound.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.