The sum of the Holevo quantity (that bounds the capacity of quantum channels to transmit classical information about an observable) and the quantum discord (a measure of the quantumness of correlations of that observable) yields an observable-independent total given by the quantum mutual information. This split naturally delineates information about quantum systems accessible to observers – information that is redundantly transmitted by the environment – while showing that it is maximized for the quasi-classical pointer observable. Other observables are accessible only via correlations with the pointer observable. We also prove an anti-symmetry property relating accessible information and discord. It shows that information becomes objective – accessible to many observers – only as quantum information is relegated to correlations with the global environment, and, therefore, locally inaccessible. The resulting complementarity explains why, in a quantum Universe, we perceive objective classical reality while flagrantly quantum superpositions are out of reach.
The sum of the Holevo quantity (that bounds the capacity of quantum channels to transmit classical information about an observable) and the quantum discord (a measure of the quantumness of correlations of that observable) yields an observable-independent total given by the quantum mutual information. This split naturally delineates information about quantum systems accessible to observers – information that is redundantly transmitted by the environment – while showing that it is maximized for the quasi-classical pointer observable. Other observables are accessible only via correlations with the pointer observable. We also prove an anti-symmetry property relating accessible information and discord. It shows that information becomes objective – accessible to many observers – only as quantum information is relegated to correlations with the global environment, and, therefore, locally inaccessible. The resulting complementarity explains why, in a quantum Universe, we perceive objective classical reality while flagrantly quantum superpositions are out of reach.
Our everyday reality is characterized by objective information-information that is selected and amplified by the environment that interacts with quantum systems. Many observers can accurately infer that information indirectly by making measurements on fragments of the environment. The correlations between the system, S, and a fragment, F, of the environment, E, is often quantified by the quantum mutual information, or the Holevo quantity, which bounds the classical information about S transmittable by a quantum channel F. The latter is a quantum mutual information but of a classical-quantum state where measurement has selected outcomes on S. The measurement generically reflects the influence of the remaining environment, E/F, but can also reflect hypothetical questions to deduce the structure of SF correlations. Recently, Touil et al. examined a different Holevo quantity, one from a quantum-classical state (a quantum S to a measured F). As shown here, this quantity upper bounds any accessible classical information about S in F and can yield a tighter bound than the typical Holevo quantity. When good decoherence is present-when the remaining environment, E/F, has effectively measured the pointer states of S-this accessibility bound is the accessible information. For the specific model of Touil et al., the accessible information is related to the error probability for optimal detection and, thus, has the same behavior as the quantum Chernoff bound. The latter reflects amplification and provides a universal approach, as well as a single-shot framework, to quantify records of the missing, classical information about S.
Two classically identical expressions for the mutual information generally differ when the systems involved are quantum. This difference defines the quantum discord. It can be used as a measure of the quantumness of correlations. Separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord, thus showing that absence of entanglement does not imply classicality. We relate this to the quantum superposition principle, and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system, i.e., the pointer states.
We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types. We describe properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi–Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical. Two possible definitions of the quantum discord for states with infinite one-way classical correlation are proposed and analysed. The generalized versions of Koashi–Winter and Xi–Lu–Wang–Li relations are used to obtain advanced continuity bounds for the Holevo information at the outputs of a channel and its complementary channel (as functions of a channel for a given ensemble of input states), for the Holevo capacity and the unregularized private capacity of a quantum channel depending either on the input dimension or on the input energy bound. We also discuss the properties of quantum channels which are “doppelgangers” of the monotonicity of the quantum discord and the entropy reduction of a local measurement under quantum channels acting on an unmeasured subsystem.
How strongly can one system be correlated with another? In the classical world, this basic question concerning correlative capacity has a very satisfying answer: The ``effective size'' of the marginal system, as quantified by the Shannon entropy, sets a tight upper bound to the correlations, as quantified by the mutual information. Although in the quantum world bipartite correlations, like their classical counterparts, are also well quantified by mutual information, the similarity ends here: The correlations in a bipartite quantum system can be twice as large as the marginal entropy. In the paradigm of quantum discord, the correlations are split into classical and quantum components, and it was conjectured that both the classical and quantum correlations are (like the classical mutual information) bounded above by each subsystem's entropy. In this work, by exploiting the interplay between entanglement of formation, mutual information, and quantum discord, we disprove that conjecture. We further indicate a scheme to restore harmony between quantum and classical correlative capacities. The results illustrate dramatically the asymmetric nature of quantum discord and highlight some subtle and unusual features of quantum correlations.
We establish bounds on quantum correlations in many-body systems. They reveal what sort of information about a quantum system can be simultaneously recorded in different parts of its environment. Specifically, independent agents who monitor environment fragments can eavesdrop only on amplified and redundantly disseminated-hence, effectively classical-information about the decoherence-resistant pointer observable. We also show that the emergence of classical objectivity is signaled by a distinctive scaling of the conditional mutual information, bypassing hard numerical optimizations. Our results validate the core idea of quantum Darwinism: objective classical reality does not need to be postulated and is not accidental, but rather a compelling emergent feature of quantum theory that otherwise-in the absence of decoherence and amplification-leads to "quantum weirdness." In particular, a lack of consensus between agents that access environment fragments is bounded by the information deficit, a measure of the incompleteness of the information about the system.
Quantum discord measures quantum correlation by comparing the quantum mutual information with the maximal amount of mutual information accessible to a quantum measurement. This paper analyzes the properties of diagonal discord, a simplified version of discord that compares quantum mutual information with the mutual information revealed by a measurement that correspond to the eigenstates of the local density matrices. In contrast to the optimized discord, diagonal discord is easily computable; it also finds connections to thermodynamics and resource theory. Here we further show that, for the generic case of non-degenerate local density matrices, diagonal discord exhibits desirable properties as a preferable discord measure. We employ the theory of resource destroying maps [Liu/Hu/Lloyd, PRL 118, 060502 (2017)] to prove that diagonal discord is monotonically nonincreasing under the operation of local discord nongenerating qudit channels, d>2, and provide numerical evidence that such monotonicity holds for qubit channels as well. We also show that it is continuous, and derive a Fannes-like continuity bound. Our results hold for a variety of simple discord measures generalized from diagonal discord.