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Bulletin of the American Physical Society

Coherence cost for measurement and computation under conservation laws

H. Tajima, Naoto Shiraishi, Keiji Saito, Hiroshi Nagaoka

Published Mar 5, 2020• 0 citations
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Uncertainty relations are one of the fundamental principles in physics. It began as a fundamental limitation in quantum mechanics, and today the word {\it uncertainty relation} is a generic term for various trade-off relations in nature. In this letter, we improve the Kennard-Robertson uncertainty relation, and clarify how much coherence we need to implement quantum measurement under conservation laws. Our approach systematically improves and reproduces the previous various refinements of the Kennard-Robertson inequality. As a direct consequence of our inequalities, we improve a well-known limitation of quantum measurements, the Wigner-Araki-Yanase-Ozawa theorem. This improvement gives an asymptotic equality for the necessary and sufficient amount of coherence to implement a quantum measurement with the desired accuracy under conservation laws.

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Coherence cost for violating conservation laws

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In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.

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Uncertainty relations are one of the fundamental principles in physics. It began as a fundamental limitation in quantum mechanics, and today the word {\it uncertainty relation} is a generic term for various trade-off relations in nature. In this letter, we improve the Kennard-Robertson uncertainty relation, and clarify how much coherence we need to implement quantum measurement under conservation laws. Our approach systematically improves and reproduces the previous various refinements of the Kennard-Robertson inequality. As a direct consequence of our inequalities, we improve a well-known limitation of quantum measurements, the Wigner-Araki-Yanase-Ozawa theorem. This improvement gives an asymptotic equality for the necessary and sufficient amount of coherence to implement a quantum measurement with the desired accuracy under conservation laws.
In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics.To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment.This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest.In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation.We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error.These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality.We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.