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arXiv:2308.04499 · Physical Review A

Quantum partial information decomposition

S. J. van Enk

Published Dec 13, 2023• 3 citations
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Abstract

Partial information decomposition (PID) takes one step beyond Shannon's theory in decomposing the information two variables, $A$ and $B$, possess about a third variable, $\mathcal{T}$, into distinct parts: unique, shared (or redundant), and synergistic information. Here we show how these concepts can be defined in a quantum setting. We apply a quantum PID to scrambling in quantum many-body systems, for which a quantum-theoretic description has been proven to be productive. Unique information in particular provides a finer description of scrambling than does the so-called tri-information.

DOI
10.1103/physreva.108.062415

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Partial information decomposition (PID) takes one step beyond Shannon's theory in decomposing the information two variables, AAA and BBB, possess about a third variable, T\mathcal{T}T, into distinct parts: unique, shared (or redundant), and synergistic information. Here we show how these concepts can be defined in a quantum setting. We apply a quantum PID to scrambling in quantum many-body systems, for which a quantum-theoretic description has been proven to be productive. Unique information in particular provides a finer description of scrambling than does the so-called tri-information.
We extend the classical ideas of the Partial Information Decomposition (PID) to the quantum domain and quantify unique, redundant, and synergistic quantum information. We show that unique information plays the central role in quantum error correction codes: any erasure-correctable subset of encoding qubits must contain zero unique information. Synergistic information between two disjoint subsets of encoding qubits appears when a logical operation is supported on the whole set but not on the subsets separately. In a different application of our PID, we show that redundant quantum information is the crucial feature of the decoherence mechanism proposed by Zurek \textit{et al.} to explain how the classical world emerges out of the quantum world.
The partial information decomposition (PID) framework is an extension of classical information theory that enables a systematic analysis of the structure of information in multipartite systems. We advance a simple extension of this framework enabling the construction of multipartite quantum PIDs (QPIDs); generalizing recent, exclusively bipartite, extensions. We construct two QPIDs of the Holevo quantity. First a naive QPID based on the classical minimum mutual information. Second a sophisticated QPID enabled by our development of an operational partial order on quantum ensembles in terms of ensemble (inter-)convertability. Finally, as a concrete example of how QPIDs can be applied, we investigate objectivity scenarios; using our Holevo QPIDs to directly quantify objective information in extended graph states. These findings ultimately establish a foundation upon which generic multipartite QPIDs can be systematically constructed, introduce an operational partial order on ensembles, advance QPIDs as interesting objects of study in their own right, and demonstrate their immediate applicability to problems in quantum information.
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional C∗C^*C∗-algebras throughout, modeling finite quantum systems that may have superselection sectors.
This paper answers Bell’s question: What does quantum information refer to? It is about quantum properties represented by subspaces of the quantum Hilbert space, or their projectors, to which standard (Kolmogorov) probabilities can be assigned by using a projective decomposition of the identity (PDI or framework) as a quantum sample space. The single framework rule of consistent histories prevents paradoxes or contradictions. When only one framework is employed, classical (Shannon) information theory can be imported unchanged into the quantum domain. A particular case is the macroscopic world of classical physics whose quantum description needs only a single quasiclassical framework. Nontrivial issues unique to quantum information, those with no classical analog, arise when aspects of two or more incompatible frameworks are compared.
Consider a situation in which a quantum system is secretly prepared in a state chosen from the known set of states. We present a principle that gives a definite distinction between the operations that preserve the states of the system and those that disturb the states. The principle is derived by alternately applying a fundamental property of classical signals and a fundamental property of quantum ones. The principle can be cast into a simple form by using a decomposition of the relevant Hilbert space, which is uniquely determined by the set of possible states. The decomposition implies the classification of the degrees of freedom of the system into three parts depending on how they store the information on the initially chosen state: one storing it classically, one storing it nonclassically, and the other one storing no information. Then the principle states that the nonclassical part is inaccessible and the classical part is read-only if we are to preserve the state of the system. From this principle, many types of no-cloning, no-broadcasting, and no-imprinting conditions can easily be derived in general forms including mixed states. It also gives a unified view on how various schemes of quantum cryptography work. The principle helps one to derive optimum amount of resources (bits, qubits, and ebits) required in data compression or in quantum teleportation of mixed-state ensembles.
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable X i has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X 0 and X 1 redundantly share with Y, what X 0 uniquely shares with Y, what X 1 uniquely shares with Y, and finally what X 0 and X 1 synergistically share with Y. Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique information. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PID’s meaning—interpretations not present in PID’s definition but that, we argue, need to be explicit. Specifically, the use of a consistent PID quantified using a secret key agreement rate naturally induces a directional interpretation of the PID. We further reveal a surprising connection between third-order connected information, two-way secret key agreement rate, and synergy. We also consider difficulties which arise with a popular PID measure in light of the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.