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arXiv:cond-mat/0307679 · Physical review. B, Condensed matter

Asymptotic von Neumann measurement strategy for solid-state qubits

Frank K. Wilhelm

Published Aug 19, 2003• 21 citations
arXivHTMLPDF

Abstract

A measurement on a macroscopic quantum system does not, in general, lead to a projection of the wave function in the basis of the detector as predicted by von Neumann's postulate. Hence, it is a question of fundamental interest, how the preferred basis onto which the state is projected is selected out of the macroscopic Hilbert space of the system. Detector-dominated von Neumann measurements are also desirable for both quantum computation and verification of quantum mechanics on a macroscopic scale. The connection of these questions to the predictions of the spin-boson model is outlined. I propose a measurement strategy, which uses the entanglement of the qubit with a weakly damped harmonic oscillator. It is shown that the degree of entanglement controls the degree of renormalization of the qubit and identify that this is equivalent to the degree to which the measurement is detector dominated. This measurement very rapidly decoheres the initial state, but the thermalization is slow. The implementation in Josephson quantum bits is described and it is shown that this strategy also has practical advantages for the experimental realization.

DOI
10.1103/physrevb.68.060503

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A measurement on a macroscopic quantum system does not, in general, lead to a projection of the wave function in the basis of the detector as predicted by von Neumann's postulate. Hence, it is a question of fundamental interest, how the preferred basis onto which the state is projected is selected out of the macroscopic Hilbert space of the system. Detector-dominated von Neumann measurements are also desirable for both quantum computation and verification of quantum mechanics on a macroscopic scale. The connection of these questions to the predictions of the spin-boson model is outlined. I propose a measurement strategy, which uses the entanglement of the qubit with a weakly damped harmonic oscillator. It is shown that the degree of entanglement controls the degree of renormalization of the qubit and identify that this is equivalent to the degree to which the measurement is detector dominated. This measurement very rapidly decoheres the initial state, but the thermalization is slow. The implementation in Josephson quantum bits is described and it is shown that this strategy also has practical advantages for the experimental realization.
We set out a general protocol for steering the state of a quantum system from an arbitrary initial state toward a chosen target state by coupling it to auxiliary quantum degrees of freedom.The protocol requires multiple repetitions of an elementary step: During each step, the system evolves for a fixed time while coupled to auxiliary degrees of freedom (which we term "detector qubits") that have been prepared in a specified initial state.The detectors are discarded at the end of the step, or equivalently, their state is determined by a projective measurement with an unbiased average over all outcomes.The steering harnesses backaction of the detector qubits on the system, arising from entanglement generated during the coupled evolution.We establish principles for the design of the system-detector coupling that ensure steering of a desired form.We illustrate our general ideas using both few-body examples (including a pair of spins-1/2 steered to the singlet state) and a many-body example (a spin-1 chain steered to the Affleck-Kennedy-Lieb-Tasaki state).We study the continuous time limit in our approach and discuss similarities to (and differences from) drive-and-dissipation protocols for quantum state engineering.Our protocols are amenable to implementations using present-day technology.Obvious extensions of our analysis include engineering of other many-body phases in one and higher spatial dimensions, adiabatic manipulations of the target states, and the incorporation of active error correction steps.
We set out a general protocol for steering the state of a quantum system from an arbitrary initial state towards a chosen target state by coupling it to auxiliary quantum degrees of freedom. The protocol requires multiple repetitions of an elementary step: during each step the system evolves for a fixed time while coupled to auxiliary degrees of freedom (which we term 'detector qubits') that have been prepared in a specified initial state. The detectors are discarded at the end of the step, or equivalently, their state is determined by a projective measurement with an unbiased average over all outcomes. The steering harnesses back-action of the detector qubits on the system, arising from entanglement generated during the coupled evolution. We establish principles for the design of the system-detector coupling that ensure steering of a desired form. We illustrate our general ideas using both few-body examples (including a pair of spins-1/2 steered to the singlet state) and a many-body example (a spin-1 chain steered to the Affleck-Kennedy-Lieb-Tasaki state). We study the continuous time limit in our approach and discuss similarities to (and differences from) drive-and-dissipation protocols for quantum state engineering. Our protocols are amenable to implementations using present-day technology. Obvious extensions of our analysis include engineering of other many-body phases in one and higher spatial dimensions, adiabatic manipulations of the target states, and the incorporation of active error correction steps.
We discuss the fact that there is a crucial contradiction within Von Neumann's theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in NNN spin-1/2 systems (1≤N<+∞1 \leq N < +\infty1≤N<+∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann's projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann's projective measurement. Therefore, Von Neumann's theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann's theory. We investigate whether Von Neumann's theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck's constant (ℏ\hbarℏ) to a new constant (ℏ/2\hbar/\sqrt{2}ℏ/2​). It may be said that a new type of the quantum theory early approaches Newton's theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch's algorithm.
In this article, we introduce a new form of quantum selective measurement in which the von Neumann projection postulate is replaced by quasilinear evolution, governed by a nonlinear generalization of the von Neumann equation. We demonstrate that this equation preserves the equivalence of quantum ensembles and, consequently, satisfies the no-signalling principle, ensuring consistency with both quantum mechanics and Einstein causality. Our approach eliminates the need for instantaneous, discontinuous state collapse and provides a unified description of the postmeasurement quantum state reduction as a form of quantum state evolution. Notably, it does not require invoking concepts such as the quantum state assigned to a classical apparatus. At the same time, the stochastic character of selective measurement and the Born rule remain unchanged. We present several numerical solutions of the evolution equation for quasilinear selective measurement in two-level quantum systems and compare them with the standard von Neumann projection. The results demonstrate agreement between the two measurement schemes in their fundamental properties. Furthermore, we investigate phenomena associated with the structural instability of the evolution equation and identify very narrow parameter regions in which the outcomes deviate from those predicted by the von Neumann projection. These regions may offer opportunities to test the proposed approach experimentally. Finally, using specific analytical solutions, we discuss the Stern-Gerlach experiment within the framework of quasilinear measurement.
We analyze the operation of a switching-based detector that probes a qubit's observable that does not commute with the qubit's Hamiltonian, leading to a nontrivial interplay between the measurement and free-qubit dynamics. In order to obtain analytical results and develop intuitive understanding of the different possible regimes of operation, we use a theoretical model where the detector is a quantum two-level system that is constantly monitored by a macroscopic system. We analyze how to interpret the outcome of the measurement and how the state of the qubit evolves while it is measured. We find that the answers to the above questions depend on the relation between the different parameters in the problem. In addition to the traditional strong-measurement regime, we identify a number of regimes associated with weak qubit-detector coupling. An incoherent detector whose switching time is measurable with high accuracy can provide high-fidelity information, but the measurement basis is determined only upon switching of the detector. An incoherent detector whose switching time can be known only with low accuracy provides a measurement in the qubit's energy eigenbasis with reduced measurement fidelity. A coherent detector measures the qubit in its energy eigenbasis and, under certain conditions, can provide high-fidelity information.
The quantum evolution of an individual solid-state qubit during the process of its continuous measurement can be described by the recently developed Bayesian formalism. In contrast to the conventional ensemble-averaged formalism, it takes into account the measurement record (in a way similar to the standard Bayesian analysis) and therefore is able to consider individual realizations of the measurement process. The formalism provides testable experimental predictions and can be used for the analysis of a quantum feedback control of solid-state qubits. The Bayesian formalism can be also applied to the continuous measurement of entangled qubits; in particular, it shows how to create a fully entangled pair of qubits without their direct interaction, just by measuring them with an equally coupled detector.