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arXiv:quant-ph/0608218 · Physical Review A

Strength of interaction for information distribution

Takayuki Miyadera, Hideki Imai

Published Dec 7, 2006• 4 citations
arXivHTMLPDF

Abstract

Let us consider two quantum systems: system $A$ and system $B$. Suppose that some classical information is encoded to quantum states of the system $A$ and we distribute this information to both systems by making them interact with each other. We show that it is impossible to achieve this goal perfectly if the strength of interaction between the quantum systems is smaller than a quantity that is determined by noncommutativity between a Hamiltonian of the system $A$ and the states (density operators) used for the information encoding. This is a consequence of a generalized Winger-Araki-Yanase theorem which enables us to treat conserved quantities other than additive ones.

DOI
10.1103/physreva.74.064302

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Let us consider two quantum systems: system AAA and system BBB. Suppose that some classical information is encoded to quantum states of the system AAA and we distribute this information to both systems by making them interact with each other. We show that it is impossible to achieve this goal perfectly if the strength of interaction between the quantum systems is smaller than a quantity that is determined by noncommutativity between a Hamiltonian of the system AAA and the states (density operators) used for the information encoding. This is a consequence of a generalized Winger-Araki-Yanase theorem which enables us to treat conserved quantities other than additive ones.
The Wigner-Araki-Yanase (WAY) theorem is a well-known theorem regarding limitations of quantum measurement in the presence of additive conservation laws. Under the assumptions of the von Neumann measurement model, for which the system conserved quantity LSL_{S}LS​ is bounded, given a conserved total additive system plus apparatus quantity LSAL_{SA}LSA​, the measurement operator ESE_{S}ES​ must commute with LSL_{S}LS​. Prior proofs have exploited the properties of unitary evolution constrained by momentum conserving operations that tend to obscure the physical nature of the WAY theorem and as well lead to bounds on performance. As it is generally agreed that momentum is always exactly conserved in measurement, we instead develop a general angular momentum conserving model of measurement. This model is shown to lead to a simple explanation of the major implications of the WAY theorem and provides exact results of the effects of measurement based on the apparatus model. This is shown by both tracing the apparatus from the density matrix and also via a system-only channel model based on Kraus operators.
The Wigner-Araki-Yanase (WAY) theorem can be understood as a result in the resource theory of asymmetry asserting the impossibility of perfectly simulating, via symmetric processing, the measurement of an asymmetric observable unless one has access to a state that is perfectly asymmetric, that is, one whose orbit under the group action is a set of orthogonal states. The simulation problem can be characterized information-theoretically by considering how well both the target observable and the resource state can provide an encoding of an element of the symmetry group. Leveraging this information-theoretic perspective, we show that the WAY theorem is a consequence of the no-programming theorem for projective measurements. The connection allows us to clarify the conceptual content of the theorem and to deduce some interesting generalizations.