Quantum Mutual Information Research Articles

Entanglement-assisted private communication over quantum broadcast channels

We consider entanglement-assisted (EA) private communication over a quantum broadcast channel, in which there is a single sender and multiple receivers. We divide the receivers into two sets: the decoding set and the malicious set. The decoding set and the malicious set can either be disjoint or can have a finite intersection. For simplicity, we say that a single party Bob has access to the decoding set and another party Eve has access to the malicious set, and both Eve and Bob have access to the pre-shared entanglement with Alice. The goal of the task is for Alice to communicate classical information reliably to Bob and securely against Eve, and Bob can take advantage of pre-shared entanglement with Alice. In this framework, we establish a lower bound on the one-shot EA private capacity. When there exists a quantum channel mapping the state of the decoding set to the state of the malicious set, such a broadcast channel is said to be degraded. We establish an upper bound on the one-shot EA private capacity in terms of smoothed min- and max-entropies for such channels. In the limit of a large number of independent channel uses, we prove that the EA private capacity of a degraded quantum broadcast channel is given by a single-letter formula. Finally, we consider two specific examples of degraded broadcast channels and find their capacities. In the first example, we consider the scenario in which one part of Bob’s laboratory is compromised by Eve. We show that the capacity for this protocol is given by the conditional quantum mutual information of a quantum broadcast channel, and so we thus provide an operational interpretation to the dynamic counterpart of the conditional quantum mutual information. In the second example, Eve and Bob have access to mutually exclusive sets of outputs of a broadcast channel.

Conditional Decoupling of Quantum Information.

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.

Calculation of quantum discord in higher dimensions for X- and other specialized states

Quantum discord, a kind of quantum correlation based on entropic measures, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. Procedures are available for analytical calculation of discord when one of the parties is a qubit with dimension two and measurements made on it to get that one-way discord. We extend now to systems when both parties are of larger dimension, of interest to qudit-quDit with d, D > 2 or spin chains of spins > 1/2. While recognizing that no universal scheme is feasible, applicable to all density matrices, nevertheless a procedure similar to that for d=2 that works for many mixed-state density matrices remains of interest as shown by recent such applications. We focus on this method that uses unitary operations to describe measurements, reducing them to a compact form so as to minimize the number of variables needed for extremizing the classical correlation, often the most difficult part of the discord calculation. Results are boiled down to a simple recipe for that extremization; for some classes of density matrices, the procedure even gives trivially the final value of the classical correlation without that extremization. A qutrit-qutrit (d=D=3) system is discussed in detail with specific applications to density matrices for whom other calculations involved difficult numerics. Special attention is given to the so-called X-states and Werner and isotropic states when the calculations become particularly simple. An appendix discusses an independent but related question of the systematics of X-states of arbitrary dimension. It forms a second, separate, part of this paper, extending our previous group-theoretic considerations of systematics for qubits now to higher d.

Weak quantum discord

Originally introduced as the difference between two possible forms of quantum mutual information, quantum discord has posteriorly been shown to admit a formulation according to which it measures a distance between the state under scrutiny and the closest projectively measured (non-discordant) state. Recently, it has been shown that quantum discord results in higher values when projective measurements are substituted by weak measurements. This sounds paradoxical since weaker measurements should imply weaker disturbance and, thus, a smaller distance. In this work we solve this puzzle by presenting a quantifier and an underlying interpretation for what we call weak quantum discord. As a by-product, we introduce the notion of symmetrical weak quantum discord.

A Study of the Quantum Correlations in a Tavis-Cummings System

The quantum correlations of two qubits in a microcavity with dissipation and a single quantized mode of the electromagnetic field are studied. In order to study our physical system, the Tavis-Cummings model and the formalism of the master equation of the density operator under the Born-Markov approximations will be used. The cavity mode will be coupled with an external pulsed laser and the master equation of the density operator that describes the evolution of the system will be solved numerically. The concurrence, the quantum mutual information, and the quantum discord are determined for the system of two qubits. The correlations are studied as a function of the dimensionless quantity $\tau = gt $ when the intensity of the laser cavity coupling varies, and two initial conditions are considered in the weak coupling regime in which the dynamic evolution of these correlations was studied.

Adaptation of the Alicki—Fannes—Winter method for the set of states with bounded energy and its use

We describe a modification of the Alicki-Fannes-Winter method which allows to prove uniform continuity on the set of quantum states with bounded energy of any locally almost affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. The asymptotic continuity of the relative entropy of entanglement and of its regularization under the energy constraint on one subsystem is proved. Channel-independent continuity bounds for the quantum mutual information at the output of a local channel and for the output Holevo quantity under the input energy constraint are obtained.

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks.

We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of electroencephalogram or functional magnetic resonance imaging measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z_{2}, mean field superfluid to Mott insulator, and a Berzinskii-Kosterlitz-Thouless crossover.

Approximation of mutual information in a bipartite quantum state under single-party decoherence

We consider the problem of approximate calculation of quantum mutual information in a bipartite quantum state where one of subsystems goes through a decoherence process (quantum channel). The approximation employs a reduced density matrix of the subsystem affected by decoherence and the initial value of mutual information before decoherence. We test the approximation on sets of randomly generated qubit-qudit mixed states for three types of qubit decoherence channels: dephasing, depolarizing, and damping. We consider how approximation accuracy is related to decoherence type, decoherence strength, and the dimension of the qudit subsystem.

Parametrically driven hybrid qubit-photon systems: Dissipation-induced quantum entanglement and photon production from vacuum

We consider a dissipative evolution of parametrically-driven qubits-cavity system under the periodical modulation of coupling energy between two subsystems, which leads to the amplification of counterrotating processes. We reveal a very rich dynamical behavior of this hybrid system. In particular, we find that the energy dissipation in one of the subsystems can enhance quantum effects in another subsystem. For instance, optimal cavity decay assists to stabilize entanglement and quantum correlations between qubits even in the steady state and to compensate finite qubit relaxation. On the contrary, energy dissipation in qubit subsystem results in the enhanced photon production from vacuum for strong modulation, but destroys both quantum concurrence and quantum mutual information between qubits. Our results provide deeper insights to nonstationary cavity quantum electrodynamics in context of quantum information processing and might be of importance for dissipative quantum state engineering.

Data processing inequality and open quantum systems: Beyond Markov states

Using a tripartite framework consisting of an open quantum system, its environment, and a passive reference system, in this study we discuss quantum mutual information (QMI) and data processing inequality (DPI). Without any restriction on the initial correlations, the necessary and sufficient conditions for the decrease or increase as well as for the conservation of QMI are obtained for any joint unitary evolution of the open quantum system and its environment. In the special case of the input Markov states, it has been shown that as long as the tripartite input state is a Markov state then DPI holds for every joint unitary evolution even in the presence of initial correlations encoded in the input. We also point out that the converse of the last statement, that is, sufficiency of the existence of a local quantum channel or of the fulfillment of DPI for the input Markov state does not hold in general and this fact is exhibited by counterexamples.

Role of coherence during classical and quantum decoherence

The total correlations in a bipartite quantum system are measured by the quantum mutual information $\mathcal{I}$, which consists of quantum discord and classical correlation. However, recent results in quantum information shows that coherence, which is a part of total correlation, is more general and more fundamental than discord. The role of coherence in quantum resource theories is worthwhile to investigate. We first study the relation between quantum discord and coherence by reducing the difference between them. And then, we consider the dynamics of quantum discord, classical correlations and quantum coherence under incoherent quantum channels. We discover that coherence indicate the behavior of quantum discord (classical correlation) for times $t<\bar t$, and indicate the decoherence of classical correlation (quantum discord) for times $t>\bar t$. What is more, the coherence frozen and decay indicate the quantum discord and classical correlation frozen and decay respectively.

Second law of thermodynamics with quantum memory

We design a heat engine with multi-heat-reservoir, ancillary system and quantum memory. We then derive an inequality related with the second law of thermodynamics, and give a new limitation about the work gain from the engine by analyzing the entropy change and quantum mutual information change during the process. In addition and remarkably, by combination of two independent engines and with the help of the entropic uncertainty relation with quantum memory, we find that the total maximum work gained from those two heat engines should be larger than a quantity related with quantum entanglement between the ancillary state and the quantum memory. This result provides a lower bound for the maximum work extracted, in contrast with the upper bound in the conventional second law of thermodynamics. However, the validity of this inequality depends on whether the maximum work can achieve the upper bound.

Non-Markovianity measure based on the relative entropy of coherence in an extended space

An alternative non-Markovianity measure for open quantum processes is proposed, which takes advantage of the nonincreasing property of relative entropy of coherence under the incoherent completely positive and trace-preserving maps in the extended Hilbert space constituted by the open system and its ancillary. By applying the proposed measure to some typical noisy channels, we find that for phase damping and amplitude damping channels it is equivalent to the three previous measures of non-Markovianity, i.e., the measures based on the quantum trace distance, dynamical divisibility, and quantum mutual information. For the random unitary channel, however, these measures do not coincide exactly, and the proposed measure in the witness of Markovianity is more general than the measures based on quantum trace distance and dynamical divisibility but overlaps partly with the measure based on quantum mutual information.

Multiparty quantum mutual information: An alternative definition

Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems. This may, however, no longer be true for three or more party quantum systems. In this paper, we propose an alternative definition of multipartite information taking into account the shared information between two and more parties. It is non-negative, observes monotonicity under partial trace as well as completely positive maps, and equals multipartite information measure in literature for pure states. We then define multiparty quantum discord, and give some examples. Interestingly, we observe that quantum discord increases when measurement is performed on a large number of subsystems. Consequently, the symmetric quantum discord, which involves measurement on all parties, reveals the maximal quantumness. This raises question on the interpretation of measured mutual information as classical correlation.

Information as a physical quantity

A new physical conception of classical information in quantum mechanical systems is explicated, critically assessed, and formalized in a quantitative measure. Observer-local referential (OLR) information—a shared physical property of entities accessible to a specified observer—is defined on the joint states of composite systems, distinguished from related conceptions of information, and tested against strict criteria that would simultaneously qualify it as a physical state quantity and as a meaningful measure of classical information. It is shown specifically to satisfy these criteria in the technological context of digital computation on quantum-physical substrates, where familiar alternatives—the von Neumann entropy and quantum mutual information (or correlation entropy)— fall short. The OLR conception provides a natural foundation for the fundamental physical description of information processing in digital computing systems, both because it defines information as a physical state quantity—placing it on an equal footing with other physical quantities appearing in such descriptions—and because it captures essential features of information in computational contexts that alternative physical conceptions do not. The OLR information measure enables straightforward and thoroughly physical quantification of digital information in general quantum systems and processes, enabling unambiguous determination of bounds on physical costs of generally noisy digital computation. More generally, OLR information offers a physical foundation for information that does not require the physical to be—ontologically or metaphorically—already informational.

Quantum correlations and coherence between two particles interacting with a thermal bath

Quantum correlations and coherence generated between two free spinless particles in the lattice, interacting with a common quantum phonon bath, are studied. The reduced density matrix is solved using the Markov approach. We show that the bath induces correlations between the particles. The coherence induced by the bath is studied, calculating off-diagonal elements of the density matrix, spatiotemporal dispersion, purity and quantum mutual information. We find a characteristic time-scale pointing out when this coherence is maximum. In addition, a Wigner-like distribution in the phase-space (lattice) is introduced as an indirect indicator of the quantumness of total correlations and coherence induced by the thermal bath. The negative volume of the Wigner function also shows a behavior which is in agreement with the time-scale that we have found. A Gaussian distribution for the profile of particles is not obtained, and interference patterns are observed as the result of bath-induced coherence. As the temperature of the bath vanishes, the ballistic behavior of the tight-binding model is recovered. The geometric quantum discord is calculated, to characterize the nature of the correlations.

A simulator for discrete quantum walks on lattices

In this paper, we present a simulator for two-particle quantum walks on the line and one-particle on a two-dimensional squared lattice. It can be used to investigate the equivalence between the two cases (one- and two-particle walks) for various boundary conditions (open, circular, reflecting, absorbing and their combinations). For the case of a single walker on a two-dimensional lattice, the simulator can also implement the Möbius strip. Furthermore, other topologies for the walker are also simulated by the proposed tool, like certain types of planar graphs with degree up to 4, by considering missing links over the lattice. The main purpose of the simulator is to study the genuinely quantum effects on the global properties of the two-particle joint probability distribution on the entanglement between the walkers/axis. For that purpose, the simulator is designed to compute various quantities such as: the entanglement and classical correlations, (classical and quantum) mutual information, the average distance between the two walkers, different hitting times and quantum discord. These quantities are of vital importance in designing possible algorithmic applications of quantum walks, namely in search, 3-SAT problems, etc. The simulator can also implement the static partial measurements of particle(s) positions and dynamic breaking of the links between certain nodes, both of which can be used to investigate the effects of decoherence on the walker(s). Finally, the simulator can be used to investigate the dynamic Anderson-like particle localization by varying the coin operators of certain nodes on the line/lattice. We also present some illustrative and relevant examples of one- and two-particle quantum walks in various scenarios. The tool was implemented in C and is available on-line at http://qwsim.weebly.com/ .

Non-Markovianity Measure Based on Brukner–Zeilinger Invariant Information for Unital Quantum Dynamical Maps**Supported by the National Natural Science Foundation of China under Grant No. 61505053, the Natural Science Foundation of Hunan Province under Grant No. 2015JJ3092, the Research Foundation of Education Bureau of Hunan Province, China under Grant No. 16B177, the School Foundation from the Hunan University of Arts and Science

A non-Markovianity measure based on Brukner–Zeilinger invariant information to characterize non-Markovian effect of open systems undergoing unital dynamical maps is proposed. The method takes advantage of non-increasing property of the Brukner–Zeilinger invariant information under completely positive and trace-preserving unital maps. The simplicity of computing the Brukner–Zeilinger invariant information is the advantage of the proposed measure because of mainly depending on the purity of quantum state. The measure effectively captures the characteristics of non-Markovianity of unital dynamical maps. As some concrete application, we consider two typical non-Markovian noise channels, i.e., the phase damping channel and the random unitary channel to show the sensitivity of the proposed measure. By investigation, we find that the conditions of detecting the non-Markovianity for the phase damping channel are consistent with the results of existing measures for non-Markovianity, i.e., information flow, divisibility and quantum mutual information. However, for the random unitary channel non-Markovian conditions are same to that of the information flow, but is different from that of the divisibility and quantum mutual information.

Quantum mutual information matrices

For any [Formula: see text]-partite state [Formula: see text], we define its quantum mutual information matrix as an [Formula: see text][Formula: see text][Formula: see text][Formula: see text][Formula: see text] matrix whose [Formula: see text]-entry is given by quantum mutual information [Formula: see text]. Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we define the genuine n-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.

Quantum Mutual Information as a Probe for Many-Body Localization.

We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal-insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from a MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.