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Quantum signal splitting as entanglement due to three-spin interactions

Dmitry Mozyrsky, Vladimir Privman A

Published Sep 16, 1996• 1 citation
OpenAlex

Abstract

The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t + \\Delta t. For possible applications in quantum computing, we derive an explicit Hamiltonian to accomplish this process; it turns out to involve only three-spin x,y-component interactions.

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The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t + \Delta t. For possible applications in quantum computing, we derive an explicit Hamiltonian to accomplish this process; it turns out to involve only three-spin x,y-component interactions.
The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system (qubit, spin) into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t+Δt. In essence, the restriction on the initial target states is exchanged for uncertainty as to which two of the three qubits retain copies of the initial source state.
We derive an explicit Hamiltonian for copying the basis up and down states of a quantum two-state system—a qubit—onto n “copy ” qubits (n ≥ 1) initially all prepared in the down state. In terms of spin components, for spin- 1 2 particle spin states, the resulting Hamiltonian involves n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a quantum-computing controlled-NOT gate. PACS numbers: 03.65.Bz, 85.30.St. – 1 – Interest in quantum computing [1-27] has boosted studies of quantum mechanics of two-state systems such as the spin states of spin- 1 2 particles. We will use “spin ” to indicate a two-state system in this context. The “binary ” up and down spin states are of particular significance and
We have found a quantum cloning machine that optimally duplicates the entanglement of a pair of ddd-dimensional quantum systems prepared in an arbitrary isotropic state. It maximizes the entanglement of formation contained in the two copies of any maximally entangled input state, while preserving the separability of unentangled input states. Moreover, it cannot increase the entanglement of formation of isotropic states. For large ddd, the entanglement of formation of each clone tends to one-half the entanglement of the input state, which corresponds to a classical behavior. Finally, we investigate a local entanglement cloner, which yields entangled clones with one-fourth the input entanglement in the large-ddd limit.
Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.