Entanglement and the speed of evolution of multi-partite quantum systems

Abstract

There exists an interesting relationship between entanglement and the time evolution of composite quantum systems: quantum entanglement enhances the 'speed' of evolution of certain quantum states, as measured by the time needed to reach an orthogonal state. Previous research done on this subject has been focused upon comparing extreme cases (highly entangled states versus separable states) or upon bi-partite systems. In the present contribution we explore the aforementioned connection (between entanglement and time evolution) in the cases of two-qubits and N-qubits systems, taking into account states of intermediate entanglement. In particular, we investigate a family of energetically symmetric states of low entanglement that saturate the quantum speed bound. We show that, as the number of qubits increases, very little entanglement is needed to reach the quantum speed limit.

Classical analogue of the statistical operator

“…It is important to mention that the classical partial tracelike operation does not commute with the operation of constructing the density matrix ρ from a probability density . That is, if one starts from the joint probability density (x y), computes the marginal probability density for subsystem A, A (x) = N 2 (x y), and then constructs its associated density matrixρ 2 A (x x ) = A (x) A (x ), thisρ A is not, in general, going to coincide with the marginal density matrix ρ A given by (14). In fact, the marginal probability density A (x) does not contain any information concerning the correlations between subsystems A and B, while the marginal density matrix ρ A (obtained via a partial trace operation conducted upon the joint density matrix ρ) does.…”

Tightening Quantum Speed Limits for Almost All States

“…The earliest derivation of minimal time of evolution was in 1945 by Mandelstam and Tamm [4] with the aim of operationalising the famous (but oft misunderstood) timeenergy uncertainty relations [5-9] ∆t ≥ /∆E, relating the standard deviation of energy with the time it takes to go from one state to another. QSLs were originally derived for the unitary evolution of pure states [10][11][12]; since then they have been generalized to the case of mixed states [13][14][15][16], nonunitary evolution [17][18][19], and multi-partite systems [20][21][22][23][24].Extending their original scope, their significance has evolved from fundamental physics to practical relevance, defining the limits of the rate of information transfer [25] and processing [26], entropy production [27], precision in quantum metrology [28] and time-scale of quantum optimal control [29]. For example, in [30], the authors use QSLs to calculate the maximal rate of information transfer along a spin chain; similarly, Reich et al show that optimization algorithms and QSLs can be used together to achieve quantum control over a large class of physical systems [31].…”

Entanglement and the speed of evolution of two interacting qubits

“…The ω = 0 case of the Hamiltonian can also be studied analytically and corresponds to a system constituted by non-interacting parts [18]. It is useful to provide a brief summary of its main features, in order to compare them with the ones corresponding to the interacting case investigated in the present contribution.…”

“…These are separable states of non-interacting qubits where only one of the qubits actually evolves and the remaining qubits are in eigenstates of their corresponding Hamiltonians. This is a marginal case where, from the dynamical point of view, one effectively deals with a single-qubit system, and the composite character of the whole system plays no physical role (see [18] and references therein).…”