arXiv:1508.04249v1 [quant-ph] 18 Aug 2015
Quantum Measurements As a Control Resource Alexander N. Pechen∗ Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel Steklov Mathematical Institute, Gubkina str. 8, Moscow 119991, Russia Abstract We discuss the use of back-action of quantum measurements as a resource for controlling quantum systems and review its application to optimal approximation of quantum anti-Zeno effect.
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Introduction
Quantum measurements are commonly used to extract information about the measured system [1] that can then be used to control the system via feedback [2, 3]. However, quantum measurements also typically affect the system through the measurement’s back-action that can be directly used to manipulate the system dynamics even if the measurement results are not recorded [4, 5, 6, 7, 8, 9]. A manifestation of such control is the quantum anti-Zeno effect, where continuous observation of certain time-dependent operators steers the system dynamics along a predefined pathway [10, 11]. The use of the anti-Zeno effect requires continuous monitoring of the system that sometimes can be hard to realize. We review the proposed in [7] general scheme for optimal control by discrete quantum measurements and consider as an application optimal approximation of the anti-Zeno effect by a finite number of measurements. ∗
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Optimal Control by Quantum Measurements
A non-selective Pmeasurement of an observable Q with the spectral decomposition Q = i qi Pi , where qi is an eigenvalue and Pi is the corresponding P projector, transforms the system density matrix ρi into MQ (ρi ) := i Pi ρi Pi . Measuring the observables Q1 , . . . , QN evolves the system state ρi into ρfinal = MQN ◦ MQN −1 · · · ◦ MQ1 (ρi )
(1)
This induced by non-selective quantum measurements transformation of the system state can be used to optimize some system related properties. One class of practically important problems can be described by maximizing the expectation value of a target operator O JN [Q1 , . . . , QN ] = Tr[ρfinal O] → max
(2)
The control goal is to find optimal observables Q∗1 , . . . , Q∗N such that their sequential measurement maximizes JN . The back-action of quantum measurements can be supplemented by coherent control u(t) acting between the measurements to produce a unitary system dynamics governed by the equation ρ(t) ˙ = −i[H0 −µu(t), ρ(t)], where H0 is the free system Hamiltonian and µ is the dipole moment. We denote Ui (ρ) = Ui ρUi† , where Ui is the unitary evolution between the i-th and (i+1)th measurements. Then the system density matrix after N measurements will be ρfinal = UN ◦ MQN ◦ UN −1 ◦ MQN −1 ◦ · · · ◦ U1 ◦ MQ1 ◦ U0 (ρi ) The density matrix ρfinal depends on u(t) and Q1 , . . . QN and determines the objective functional J[u(t), Q1 , . . . , QN ] = Tr [ρfinal O] → max where both the standard coherent field u(t) and the observables Q1 , . . . , QN are treated as controls to be optimized [7].
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Optimal Approximation of Anti-Zeno Dynamics
According to quantum anti-Zeno effect, under continuous measurement of the projection operator E(t) = U (t)EU † (t), where E is a projector leaving 2
the initial state unchanged and U (t) a unitary operator such that U (0) = 1, the probability of finding E(t) = 1 at any t ∈ [0, T ] is one [10]. Thus the evolution of the continuously monitored system will follow the prescribed pathway E(t). Continuous quantum measurements may sometimes be hard to realize and optimal approximations of the anti-Zeno dynamics by a finite number of measurements may be desirable. In [7] an optimal approximation of anti-Zeno effect by any finite number of measurements was obtained for two-level systems. As an example, Fig. 1 shows on the Bloch sphere the Stokes vectors for ten optimal measurements which approximate the antiZeno dynamics with E = |0ih0| and E(T ) = |1ih1|.
Figure 1: (From Ref. [7]. Copyright (2006) by the American Physical Society.) Optimal approximation of the anti-Zeno dynamics in a twolevel system by ten quantum measurements. The initial system state is ρi = 12 (I + a0 σ) = E and the desired target state ρT = 12 (I + wT σ) = E(T ), where σ = (σx , σy , σz ). Left plot shows the Bloch sphere and ten unit norm vectors wi = bi /kbi k which determine the observables (projectors) Q∗i = 21 (I + wi σ) such that their sequential measurement optimally approximates the anti-Zeno dynamics. State of the system after i-th measurement is ρi = 21 (I + bi σ), where the vectors bi (kbi k < 1, thus each ρi is mixed) are plotted on the right plot.
Acknowledgments This work was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.
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