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arXiv:2110.06499 · Journal of Physics A Mathematical and Theoretical

Transmission of coherent information at the onset of interactions

Emily Kendall, Barbara Šoda, Achim Kempf

Published Apr 29, 2022• 2 citations
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Abstract

Abstract In this work, we investigate the parameters governing the rate at which a quantum channel arises at the onset of an interaction between two systems, A and B . In particular, when system A is pre-entangled with an ancilla, A ~ , we quantify the early-time transmission of pre-existing entanglement by calculating the leading order change in coherent information of the complementary channel ( A → B ′). We show that, when A and B are initially unentangled and B is pure, there is no change in coherent information to first order, while the leading (second) order change is divergent. However, this divergence may be regulated by embedding the conventional notion of coherent information into what we call the family of n -coherent informations, defined using n -Rényi entropies. We find that the rate of change of the n -coherent information at the onset of the interaction is governed by a quantity, which we call the n -exposure, which captures the extent to which the initial coherent information of A with A ~ is exposed to or ‘seen by’ the interaction Hamiltonian between A and B . We give examples in qubit systems and in the light–matter interaction.

DOI
10.1088/1751-8121/ac6bce

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Abstract In this work, we investigate the parameters governing the rate at which a quantum channel arises at the onset of an interaction between two systems, A and B . In particular, when system A is pre-entangled with an ancilla, A ~ , we quantify the early-time transmission of pre-existing entanglement by calculating the leading order change in coherent information of the complementary channel ( A → B ′). We show that, when A and B are initially unentangled and B is pure, there is no change in coherent information to first order, while the leading (second) order change is divergent. However, this divergence may be regulated by embedding the conventional notion of coherent information into what we call the family of n -coherent informations, defined using n -Rényi entropies. We find that the rate of change of the n -coherent information at the onset of the interaction is governed by a quantity, which we call the n -exposure, which captures the extent to which the initial coherent information of A with A ~ is exposed to or ‘seen by’ the interaction Hamiltonian between A and B . We give examples in qubit systems and in the light–matter interaction.
Abstract At the onset of an interaction between two initially independent systems, each system tends to experience an increase in its n -Rényi entropies, such as its von Neumann entropy ( n = 1) and its mixedness ( n = 2). We here ask which properties of a system determine how quickly its Rényi entropies increase and, therefore, how sensitive the system is to becoming entangled. We find that the rate at which the n -Rényi entropy increases in an interaction is determined by a quantity which we term the n -fragility of the system. The 2-fragility is closely related to the notion of 2-norm coherence, in that it too quantifies the extent to which a density matrix is ‘off-diagonal’ with respect to the eigenbasis of a reference operator. Nevertheless, the 2-fragility is not a coherence monotone in the resource theoretic sense since it depends also on the eigenvalues of the reference operator. It is this additional sensitivity to the eigenvalues of the reference operator, here the interaction Hamiltonian, which enables the 2-fragility to quantify the rate of entropy production in interactions. We give an example using the light–matter interaction and we anticipate applications to the study of the rates at which two systems exchange classical and quantum information when starting to interact.
Abstract Quantum information, in the form of entanglement with an ancilla, can be transmitted to a third system through interaction. Here, we investigate this process of entanglement transmission perturbatively in time. Using the entanglement monotone negativity, we determine how the proclivity of an interaction to either generate, transfer or lose entanglement depends on the choice of Hamiltonians and initial states. These three proclivities are captured by Hamiltonian- and state-dependent quantities that we call negativity susceptibility, negativity transmissibility and negativity vulnerability respectively. These notions could serve, for example, as cost functions in quantum technologies such as machine-learned quantum error correction.
A quantum channel physically is a unitary interaction between an information carrying system and an environment, which is initialized in a pure state before the interaction. Conventionally, this state, as also the parameters of the interaction, is assumed to be fixed and known to the sender and receiver. Here, following the model introduced by us earlier [1], we consider a benevolent third party, i.e., a helper, controlling the environment state, and show how the helper’s presence changes the communication game. In particular, we define and study the classical capacity of a unitary interaction with helper, in two variants: one where the helper can only prepare separable states across many channel uses, and one without this restriction. Furthermore, two even more powerful scenarios of pre-shared entanglement between helper and receiver, and of classical communication between sender and helper (making them conferencing encoders) are considered.
An exactly soluble non-linear interaction Hamiltonian is proposed to study fundamental properties of the entanglement dynamics for a coupled non-linear oscillators. The time-evolved state is obtained analytically for initial products of two coherent and two number states and relevant informations are extracted from the dynamics of various quantities like subsystem linear and Von Neumann entropies, quadrature mean values, variances and Q-functions. We determined the re-coherence time scales and found among the interaction terms present in the Hamiltonian the one responsible for the entanglement in both cases. We identify the existence of two regimens for the entanglement dynamics in the case of initially coherent states: the short time, phase spread regimen where the entropy rises monotonically and the self-interference regimen where the entropy oscillates and re-coherence phenomenon can be observed. We also found that the break time from the first regimen to the second one becomes longer, as well as the re-coherence and reversibility times, as the Planck's constant becomes much smaller than a typical action in phase space.
This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems R and Q, after which Q is subjected to a dynamical evolution given by the superoperator EQ{\mathit{E}}^{\mathit{Q}}EQ. Two interesting quantities can be defined for this process: the entanglement fidelity Fe{\mathit{F}}_{\mathit{e}}Fe​ and the entropy exchange Se{\mathit{S}}_{\mathit{e}}Se​. It turns out that neither of these quantities depends in any way on the system R, but only on the initial state and dynamical evolution of Q. Fe{\mathit{F}}_{\mathit{e}}Fe​ and Se{\mathit{S}}_{\mathit{e}}Se​ are related to various other fidelities and entropies and are connected by an inequality reminiscent of the Fano inequality of classical information theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes. \textcopyright{} 1996 The American Physical Society.
We define and analyse the concept of entanglement production during the evolution of a general quantum mechanical dissipative system. While it is important to minimise entropy production in order to achieve thermodynamical efficiency, maximising the rate of change of entanglement is important in quantum information processing. Quantitative relations are obtained between entropy and entanglement productions, under specific assumptions detailed in the text. We apply these to the processes of dephasing and decay of correlations between two initially entangled qubits. Both the Master equation treatment as well as the higher Hilbert space analysis are presented. Our formalism is very general and contains as special cases many reported individual instance of entanglement dynamics, such as, for example, the recently discovered notion of the sudden death of entanglement.