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arXiv:quant-ph/0501031 · Physical Review A

Quantum nondemolition measurement saturates fidelity trade-off

Ladislav Mišta, Radim Filip

Published Sep 30, 2005• 15 citations
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Abstract

Quantum mechanics sets a bound between the quality of estimation from measurement of a single copy of a $d$-level system and the degree of disturbance of this system. We show that ideal quantum nondemolition (QND) measurement fed by a properly prepared ancilla saturates this bound. The ratio between the quality of estimation and the degree of disturbance can be continuously controlled by preparation of the ancilla. Further, we show that also nonideal QND measurement achieves this bound if a suitable measurement on the ancilla is performed. Finally, we propose a feasible scheme for such a measurement for optical two-level systems that is based on the controlled-NOT gate.

DOI
10.1103/physreva.72.034307

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Quantum mechanics sets a bound between the quality of estimation from measurement of a single copy of a ddd-level system and the degree of disturbance of this system. We show that ideal quantum nondemolition (QND) measurement fed by a properly prepared ancilla saturates this bound. The ratio between the quality of estimation and the degree of disturbance can be continuously controlled by preparation of the ancilla. Further, we show that also nonideal QND measurement achieves this bound if a suitable measurement on the ancilla is performed. Finally, we propose a feasible scheme for such a measurement for optical two-level systems that is based on the controlled-NOT gate.
High-fidelity qubit measurements play a crucial role in quantum computation, communication, and metrology. In recent experiments, it has been shown that readout fidelity can be improved by performing repeated quantum nondemolition (QND) readouts of a qubit's state through an ancilla. For a qubit encoded in a two-level system, the fidelity of such schemes is limited by the fact that a single error can destroy the information in the qubit. On the other hand, if a bosonic system is used, this fundamental limit can be overcome by utilizing higher levels such that a single error still leaves states distinguishable. In this work, we present a robust readout scheme which leverages both repeated QND readouts and higher-level encodings to asymptotically suppress the effects of mode relaxation and individual measurement infidelity. We calculate the measurement fidelity in terms of general experimental parameters, provide an information-theoretic description of the scheme, and describe its application to several encodings, including cat and binomial codes.
Summary form only given. The present techniques allow one to perform continuous or discrete quantum measurements at or even beyond the standard "quantum noise" level, even on single quantum objects. In this context, Kip Thorne and Vladimir Braginsky introduced in the seventies the concept of "Quantum Non Demolition (QND) measurement", which was initially meant to be applied to mechanical harmonic oscillators designed for detecting gravitational waves, the so-called "Weber's bars". But the QND ideas encountered a real success in quantum optics, where quantum fluctuations are readily accessible, and can also be circumvented. We review the state of the art of QND measurements, in this broad sense of sub-shot-noise optical information processing. Most of the initial objectives of QND measurements have now been successfully reached through proof-of-principle experiments. The domain is now opening on future perspectives related to quantum information processing (e.g. quantum cryptography), either with "discrete" or "continuous" quantum variables. In this context the close relationship between QND measurements and controlled-NOT (CNOT) gates are discussed and illustrated.
We derive the optimal trade-off between the estimation fidelity and the fidelity of the state after the estimation for a single copy of an unknown pure state of a ddd-level system belonging to a specific subset of the set of all pure states. The set of states to be considered here is formed by all pure states of a ddd-level system produced by ddd-independent phase shifts of some reference state. We also propose a measurement scheme realizing such an optimal partial estimation of multiple phases in which the trade-off between the fidelities can be controlled by the state of the ancilla.