Insights from quantum information theory show that correlation measures based on quantum entropy are fundamental tools that reveal the entanglement structure of multipartite states. In that spirit, Groisman, Popescu, and Winter [Phys. Rev. A 72, 032317 (2005)PLRAAN1050-294710.1103/PhysRevA.72.032317] showed that the quantum mutual information I(A;B) quantifies the minimal rate of noise needed to erase the correlations in a bipartite state of quantum systems AB. Here, we investigate correlations in tripartite systems ABE. In particular, we are interested in the minimal rate of noise needed to apply to the systems AE in order to erase the correlations between A and B given the information in system E, in such a way that there is only negligible disturbance on the marginal BE. We present two such models of conditional decoupling, called deconstruction and conditional erasure cost of tripartite states ABE. Our main result is that both are equal to the conditional quantum mutual information I(A;B|E)-establishing it as an operational measure for tripartite quantum correlations.
We define the deconstruction cost of a tripartite quantum state on systems
ABE as the minimum rate of noise needed to apply to the
AE systems, such that there is negligible disturbance to the marginal state on the
BE systems, while the system
A of the resulting state is locally recoverable from the
E system alone. We refer to such actions as deconstruction operations and protocols implementing them as state deconstruction protocols. State deconstruction generalizes Landauer erasure of a single-party quantum state as well the erasure of correlations of a two-party quantum state. We find that the deconstruction cost of a tripartite quantum state on systems
ABE is equal to its conditional quantum mutual information (CQMI)
I(A;B∣E), thus giving the CQMI an operational interpretation in terms of a state deconstruction protocol. We also define a related task called conditional erasure, in which the goal is to apply noise to systems
AE in order to decouple system
A from systems
BE, while causing negligible disturbance to the marginal state of systems
BE. We find that the optimal rate of noise for conditional erasure is also equal to the CQMI
I(A;B∣E). State deconstruction and conditional erasure lead to operational interpretations of the quantum discord and squashed entanglement, which are quantum correlation measures based on the CQMI. We find that the quantum discord is equal to the cost of simulating einselection, the process by which a quantum system interacts with an environment, resulting in selective loss of information in the system. The squashed entanglement is equal to half the minimum rate of noise needed for deconstruction and/or conditional erasure if Alice has available the best possible system
E to help in the deconstruction and/or conditional erasure task.
We show that the minimal rate of noise needed to catalytically erase the entanglement in a bipartite quantum state is given by the regularized relative entropy of entanglement. This offers a solution to the central open question raised in [Groisman et al., Phys. Rev. A 72, 032317 (2005)PLRAAN1050-294710.1103/PhysRevA.72.032317] and complements their main result that the minimal rate of noise needed to erase all correlations is given by the quantum mutual information. We extend our discussion to the tripartite setting where we show that an asymptotic rate of noise given by the regularized relative entropy of recovery is sufficient to catalytically transform the state to a locally recoverable version of the state.
The decoupling technique was originally developed for information-theoretical purposes. It describes the conditions under which the correlations in a bipartite state disappear if one part undergoes an evolution separated from the other. In the past years there has been enormous progress in understanding the foundations of statistical mechanics from first principles of quantum mechanics. By use of the decoupling technique we are able to reproduce and generalize major results of this development and to approach open problems. As a first application of the decoupling technique we generalize the result of [Popescu et al., Nat. Phys. 2, 754-758 (2006)] about the apparent validity of the postulate of equal a priori probabilities to states which may be correlated to a reference. We express it in a form which allows to apply recent results about random two-qubit interactions. We give a criterion for the apparent validity of the postulate which is tight up to differences between different entropy measures. Similarly, we generalize the result of [Linden et al., Phys. Rev. E 79, 061103 (2009)] about the independence of the temporal average of a quantum mechanical system of its initial state to initial states which may be correlated to a reference. We develop a criterion for whether a quantum-mechanical system has “forgotten” about its initial state which is tight up to differences between different entropy measures. We find that comparing two local entropies of just one particular state tells us whether generic initial states of the system have already evolved to the same state or not. After developing new bounds on the times which are necessary for entropy changes, we are able to provide lower bounds on the times which are necessary for a system to become independent of its initial state. We discover an intimate connection between a system becoming independent of its initial state and a loss of an observer’s knowledge about the state of the system. As a further application of the decoupling technique we find sufficient conditions under which a system stays close to its initial state for all times, thereby extending the result of [Gogolin et al., Phys. Rev. Lett. 106, 040401 (2011)].
Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.
We give an operational definition of the quantum, classical, and total amounts of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work (noise) that is required to erase (destroy) the correlations: for the total correlation, we have to erase completely, for the quantum correlation we have to erase until a separable state is obtained, and the classical correlation is the maximal correlation left after erasing the quantum correlations. In particular, we show that the total amount of correlations is equal to the quantum mutual information, thus providing it with a direct operational interpretation. As a by-product, we obtain a direct, operational, and elementary proof of strong subadditivity of quantum entropy.
Given a bipartite quantum system, correlations between its subsystems can be understood as information that each one carries about the other. In order to give a model-independent description of secure information disposal, we propose the paradigm of private quantum decoupling, corresponding to locally reducing correlations in a given bipartite state without transferring them to the environment. In this framework, the concept of private local randomness naturally arises as a resource, and total correlations get divided into eliminable and ineliminable ones. We prove upper and lower bounds on the amount of ineliminable correlations present in an arbitrary bipartite state, and show that, in tripartite pure states, ineliminable correlations satisfy a monogamy constraint, making apparent their quantum nature. A relation with entanglement theory is provided by showing that ineliminable correlations constitute an entanglement parameter. In the limit of infinitely many copies of the initial state provided, we compute the regularized ineliminable correlations to be measured by the coherent information, which is thus equipped with a new operational interpretation. In particular, our results imply that two subsystems can be privately decoupled if their joint state is separable.