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arXiv:2004.07281 · Physical Review A

Protective measurement of a qubit by a qubit probe

Maximilian Schlosshauer

Published Apr 15, 2020• 0 citations
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Abstract

We study the protective measurement of a qubit by a second qubit acting as a probe. Consideration of this model is motivated by the possibility of its experimental implementation in multiqubit systems such as trapped ions. In our scheme, information about the expectation value of an arbitrary observable of the system qubit is encoded in the rotation of the state of the probe qubit. We describe the structure of the Hamiltonian that gives rise to this measurement and analyze the resulting dynamics under a variety of realistic conditions, such as noninfinitesimal measurement strengths, repeated measurements, non-negligible intrinsic dynamics of the probe, and interactions of the system and probe qubits with an environment. We propose an experimental realization of our model in an ion trap. The experiment may be performed with existing technology and makes use of established experimental methods for the engineering and control of Hamiltonians for quantum gates and quantum simulations of spin systems.

DOI
10.1103/physreva.101.042113

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We study the protective measurement of a qubit by a second qubit acting as a probe. Consideration of this model is motivated by the possibility of its experimental implementation in multiqubit systems such as trapped ions. In our scheme, information about the expectation value of an arbitrary observable of the system qubit is encoded in the rotation of the state of the probe qubit. We describe the structure of the Hamiltonian that gives rise to this measurement and analyze the resulting dynamics under a variety of realistic conditions, such as noninfinitesimal measurement strengths, repeated measurements, non-negligible intrinsic dynamics of the probe, and interactions of the system and probe qubits with an environment. We propose an experimental realization of our model in an ion trap. The experiment may be performed with existing technology and makes use of established experimental methods for the engineering and control of Hamiltonians for quantum gates and quantum simulations of spin systems.
We report experimental implementation of various types of qubit channels using an individual trapped ion. We analyzed experimental data and we performed tomographic reconstruction of quantum channels based on these data. Specifically, we studied phase damping channels, where the damping acts either in the xy-plane of the Bloch sphere or in an arbitrary plane that includes the origin of the Bloch sphere. We also experimentally realized and consequently analyzed quantum channels that in addition to phase damping affect also a polarization rotation. We used three reconstruction schemes for estimation of quantum channels from experimental data: (1) a linear inverse method, (2) a maximum likelihood estimation, and (3) a constrained maximum likelihood estimation. We took into account realistic experimental conditions where imperfect test-state preparations and biased measurements are incorporated into the estimation schemes. As a result we found that imperfections present in the process of preparation of test states and as well as in measurements of the considered ion trap system do not limit the control of the implementation of the desired channel. Even imperfect preparation of test state and subsequent measurements still provide sufficient resources for the complete quantum-channel tomography.
Making measurements on single quantum systems is considered difficult, almost impossible if the state is a-priori unknown. Protective measurements suggest a possibility to measure single quantum systems and gain some new information in the process. Protective measurement is described, both in the original and generalized form. The degree to which the system and the apparatus remain entangled in a protective measurement, is assessed. A possible experimental test of protective measurements is discussed.
In a recent experiment, Barreiro et al (2011 Nature 470 486) demonstrated the fundamental building blocks of an open-system quantum simulator with trapped ions. Using up to five ions, dynamics were realized by sequences that combined single- and multi-qubit entangling gate operations with optical pumping. This enabled the implementation of both coherent many-body dynamics and dissipative processes by controlling the coupling of the system to an artificial, suitably tailored environment. This engineering was illustrated by the dissipative preparation of entangled two- and four-qubit states, the simulation of coherent four-body spin interactions and the quantum non-demolition measurement of a multi-qubit stabilizer operator. In this paper, we present the theoretical framework of this gate-based ('digital') simulation approach for open-system dynamics with trapped ions. In addition, we discuss how within this simulation approach, minimal instances of spin models of interest in the context of topological quantum computing and condensed matter physics can be realized in state-of-the-art linear ion-trap quantum computing architectures. We outline concrete simulation schemes for Kitaev's toric code Hamiltonian and a recently suggested color code model. The presented simulation protocols can be adapted to scalable and two-dimensional ion-trap architectures, which are currently under development.
The most typical ingredient of topologically protected quantum states is magnetic fluxes. In a system of spins, complex-valued interaction parameters give rise to a flux, if their phases do not add up to zero along a closed loop. Here we apply periodic driving, a powerful tool for quantum engineering, to a trapped-ion quantum simulator in order to generate such spin-spin interactions. We consider a simple driving scheme, consisting of a repeated series of locally quenched fields, and demonstrate the feasibility of this approach by studying the dynamics of a small system. An emblematic hallmark of the flux, accessible in experiments, is the appearance of chiral spin currents. Strikingly, we find that in parameter regimes where, in the absence of fluxes, phonon excitations dramatically reduce the fidelity of the spin model simulation, the spin dynamics remains widely unaffected by the phonons when fluxes are present. Our work provides a realistic experimental recipe to engineer the minimal building block of a topological quantum system with a currently existing ion trap apparatus.
We study protective quantum measurements in the presence of an environment and decoherence. We consider the model of a protectively measured qubit that also interacts with a spin environment during the measurement. We investigate how the coupling to the environment affects the two characteristic properties of a protective measurement, namely (i) the ability to leave the state of the system approximately unchanged and (ii) the transfer of information about expectation values to the apparatus pointer. We find that even when the interaction with the environment is weak enough not to lead to appreciable decoherence of the initial qubit state, it causes a significant broadening of the probability distribution for the position of the apparatus pointer at the conclusion of the measurement. This washing out of the pointer position crucially diminishes the accuracy with which the desired expectation values can be measured from a readout of the pointer. We additionally show that even when the coupling to the environment is chosen such that the state of the system is immune to decoherence, the environment may still detrimentally affect the pointer readout.
Unsharp positive operator-valued measurements allow a variety of measurement applications which minimally disrupt the state of the quantum system. Experimental schemes are proposed for implementing unsharp measurements on the qubit levels of a trapped ion. The schemes rely on introducing weak entanglement between the state of a target ion and that of an auxiliary ion, using standard ion-trap quantum logic operations, and then realizing an unsharp measurement through projective measurement on the auxiliary atom. We analyze common sources of error and their effect on different applications of unsharp measurements.