Wigner-Araki-Yanase Theorem for Continuous and Unbounded Conserved Observables

The Wigner-Araki-Yanase (WAY) theorem states that additive conservation laws imply the commutativity of exactly implementable projective measurements and the conserved observables of the system. Known proofs of this theorem are only restricted to bounded or discrete-spectrum conserved observables of the system and are not applicable to unbounded and continuous observables like a momentum operator. In this Letter, we present the WAY theorem for possibly unbounded and continuous conserved observables under the Yanase condition, which requires that the probe positive operator-valued measure should commute with the conserved observable of the probe system. As a result of this WAY theorem, we show that exact implementations of the projective measurement of the position under momentum conservation and of the quadrature amplitude using linear optical instruments and photon counters are impossible. We also consider implementations of unitary channels under conservation laws and find that the conserved observable $L_S$ of the system commutes with the implemented unitary $U_S$ if $L_S$ is semibounded, while $U_S^{†}{L}_S{U}_S$ can shift up to possibly nonzero constant factor if the spectrum of $L_S$ is upper and lower unbounded. We give simple examples of the latter case, where $L_S$ is a momentum operator.

Figure 1

It is well known that position and momentum cannot be measured without error simultaneously. However, in natural settings like the above schematic, position measurements will be performed under the momentum conservation law. If the WAY theorem is correct for unbounded operators, the position itself cannot be measured without error under such natural settings.

Figure 2

A system-environment model that implements a channel $\mathrm{\Psi }$.

Figure 3

A measurement model that implements a POVM ${\mathsf{E}}_S$.

Figure 4

Measurement model with passive optical operations and photon-counting measurements.