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Physical Review Letters

Reversing Measurement and Probabilistic Quantum Error Correction

Masato Koashi, Masahito Ueda

Published Mar 22, 1999• 119 citations
Publisher

Abstract

We consider a probabilistic reversing operation that returns the measured system to its original state by means of a physical process, and derive a trade-off relation between the unsharpness of the measurement and the best efficiency of the reversing operation. Such a reversing operation is shown to serve as a probabilistic quantum error correction, which will be useful when the numbers of qubits and gate operations are limited.

DOI
10.1103/physrevlett.82.2598

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We consider a probabilistic reversing operation that returns the measured system to its original state by means of a physical process, and derive a trade-off relation between the unsharpness of the measurement and the best efficiency of the reversing operation. Such a reversing operation is shown to serve as a probabilistic quantum error correction, which will be useful when the numbers of qubits and gate operations are limited.
A quantum measurement is logically reversible if the premeasurement density operator of the measured system can be calculated from the postmeasurement density operator and from the outcome of the measurement. This paper analyzes why many quantum measurements are logically irreversible, shows how to make them logically reversible, and discusses reversing measurement that returns the postmeasurement state to the premeasurement state by another measurement (physical reversibility). Reversing measurement and unitarily reversible quantum operation are compared from the viewpoint of error correction in quantum computation.
A quantum measurement is logically reversible if the premeasurement density operator of the measured system can be calculated from the postmeasurement density operator and from the outcome of the measurement. This paper analyzes why many quantum measurements are logically irreversible, shows how to make them logically reversible, and discusses reversing measurement that returns the postmeasurement state to the premeasurement state by another measurement (physical reversibility). Reversing measurement and unitarily reversible quantum operation are compared from the viewpoint of error correction in quantum computation. 1.
Quantum operations provide a general description of the state changes allowed by quantum mechanics. The reversal of quantum operations is important for quantum error–correcting codes, teleportation and reversing quantum measurements. We derive information–theoretic conditions and equivalent algebraic conditions that are necessary and sufficient for a general quantum operation to be reversible. We analyse the thermodynamic cost of error correction and show that error correction can be regarded as a kind of `Maxwell demon', for which there is an entropy cost associated with information obtained from measurements performed during error correction. A prescription for thermodynamically efficient error correction is given.
Quantum measurement is a basic tool to manifest intrinsic quantum effects from fundamental tests to quantum information applications. While a measurement is typically performed to gain information on a quantum state, its role in quantum technology is indeed manifold. For instance, quantum measurement is a crucial process element in measurement-based quantum computation. It is also used to detect and correct errors thereby protecting quantum information in error-correcting frameworks. It is therefore important to fully characterize the roles of quantum measurement encompassing information gain, state disturbance and reversibility, together with their fundamental relations. Numerous efforts have been made to obtain the trade-off between information gain and state disturbance, which becomes a practical basis for secure information processing. However, a complete information balance is necessary to include the reversibility of quantum measurement, which constitutes an integral part of practical quantum information processing. We here establish all pairs of trade-off relations involving information gain, disturbance, and reversibility, and crucially the one among all of them together. By doing so, we show that the reversibility plays a vital role in completing the information balance. Remarkably, our result can be interpreted as an information-conservation law of quantum measurement in a nontrivial form. We completely identify the conditions for optimal measurements that satisfy the conservation for each tradeoff relation with their potential applications. Our work can provide a useful guideline for designing a quantum measurement in accordance with the aims of quantum information processors.
Abstract Quantum computing1—the processing of information according to the fundamental laws of physics—offers a means to solve efficiently a small but significant set of classically intractable problems. Quantum computers are based on the controlled manipulation of entangled quantum states, which are extremely sensitive to noise and imprecision; active correction of errors must therefore be implemented without causing loss of coherence. Quantum error-correction theory2, 3, 4, 5, 6, 7, 8, 9 has made great progress in this regard, by predicting error-correcting ‘codeword’ quantum states. But the coding is inefficient and requires many quantum bits10, 11, 12, which results in physically unwieldy fault-tolerant quantum circuits10, 11, 12, 13, 14, 15, 16, 17, 18. Here I report a general technique for circumventing the trade-off between the achieved noise tolerance and the scale-up in computer size that is required to realize the error correction. I adapt the recovery operation (the process by which noise is suppressed through error detection and correction) to simultaneously correct errors and perform a useful measurement that drives the computation. The result is that a quantum computer need be only an order of magnitude larger than the logic device contained within it. For example, the physical scale-up factor10, 11 required to factorize a thousand-digit number is reduced from 1, 500 to 22, while preserving the original tolerated gate error rate (10−5) and memory noise per bit (10−7). The difficulty of realizing a useful quantum computer is therefore significantly reduced.