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arXiv:0912.3120

Entangled Brachistochrone: Minimum Time to reach Target Entangled State

Arun Kumar Pati, Pradhan, Biswajit, Pankaj Agrawal

Published Dec 16, 2009• 0 citations
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Abstract

We address the question: Given an arbitrary initial state and a general physical interaction what is the minimum time for reaching a target entangled state? We show that the minimum time is inversely proportional to the quantum mechanical uncertainty in the non-local Hamiltonian. We find that the presence of initial entanglement helps to minimize the waiting time. Furthermore, we find that in a bi-local rotating frame the entangling capability is actually a geometric quantity. We give an universal bound for the time average of entanglement rate for general quantum systems. The time average of entanglement rate does not depend on the particular Hamiltonian, rather on the fluctuation in the Hamiltonian. There can be infinite number of nonlocal Hamiltonians which may give same average entanglement rate. We also prove a composition law for minimum time when the system evolves under a composite Hamiltonian.

DOI
10.48550/arxiv.0912.3120

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Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum brachistochrone problem. Given an initial and a final state of a composite system we consider the amount of entanglement associated with the brachistochrone evolution between those states, showing that entanglement is an essential resource to achieve the alluded time-optimal quantum evolution.
We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrödinger–Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.
We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrodinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.
We establish a relation between the geometric time-energy uncertainty and multipartite entanglement. In particular, we show that the time-energy uncertainty relation is bounded below by the geometric measure of multipartite entanglement for an arbitrary quantum evolution of any multipartite system. The product of the time-averaged speed of the quantum evolution and the time interval of the evolution is bounded below by the multipartite entanglement of the target state. This relation holds for pure as well as for mixed states. We provide examples of physical systems for which the bound reaches close to saturation.
We discuss upper bounds on the rate at which unitary evolution governed by a nonlocal Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian HHH coupling two finite dimensional particles AAA and BBB, the entangling rate is shown to be upper bounded by $c\text{ }\text{log}(d)\ensuremath{\Vert}H\ensuremath{\Vert}$, where ddd is the smallest dimension of the interacting particles, $\ensuremath{\Vert}H\ensuremath{\Vert}$ is the operator norm of HHH, and ccc is a constant close to 1. Under certain restrictions on the initial state we prove an analogous upper bound for the ancilla-assisted entangling rate with a constant ccc that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate---a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.