Brachistochrone of entanglement for spin chains

Abstract

We analytically investigate the role of entanglement in time-optimal state evolution as an application of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly coupled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entanglement plays a role for W and GHZ initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.

Quantum brachistochrone

“…of a few qubits (e.g. [22,23,38–40]), but the problem becomes increasingly difficult as the dimensionality N of the Hilbert space H and the number of control variables become large. Thus, numerical approaches are necessary.…”

“…We give some simplest examples. Other examples related to quantum information processing are found in [20,22,23,42,43,70].…”

“…The examples include quantum error correction [1], quantum metrology [2], quantum cooling [3,4], quantum battery [5] and gates in quantum computers [6], because noise from the environment degrades quantum states over time. Time-optimal quantum control [7,8] has been studied for systems with fast local control [9][10][11][12][13][14][15] and with close-level controllable couplings [16][17][18][19], for spin chains [20][21][22][23], for the Zermelo navigation problem [24][25][26][27] and for linear networks [28]. Related interests include quantum speed limit [29][30][31][32][33][34][35][36] (see also [37]) developed from the time-energy uncertainty relation.…”

“…A close relation between time complexity and gate complexity has been suggested by numerical analysis of QBs [43]. Simple systems modelling quantum information device have been analysed by the QB formalism [21][22][23]. The QB formalism also leads to a wider perspective to shortcuts to adiabaticity (STA) [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58] (see also [59]).…”

Minimum-Time Quantum Control and the Quantum Brachistochrone Equation

“…The ability to control quantum systems lies at the heart of various quantum technologies, such as quantum computation [1-3], quantum state preparation [4][5][6][7][8], quantum metrology [9][10][11][12][13], shortcut to adiabaticity [14][15][16][17], measurementbased state stabilization [18], and dynamical decoupling [13,19], among others.…”