Measurement disturbance and conservation laws in quantum mechanics

Abstract

We consider measurement disturbance for pairs of observables in the presence of a conservation law, providing conditions under which non-disturbance can be achieved. This is done by analysing the fixed point structure of the measurement channel. From here several extensions of the Wigner-Araki-Yanase theorem are found.

Covariant catalysis requires correlations and good quantum reference frames degrade little

“…A particularly interesting obstruction arising from conservation laws is the impossibility of measuring exactly (in the sense of von Neumann) an observable that does not commute with a conserved quantity. This statement is known as Wigner-Araki-Yanase (WAY) theorem [58][59][60], and has recently been re-interpreted and generalized in several ways (see, for example, [10,13,15,[61][62][63][64][65]). From the point of view of the resource theory of asymmetry, the theorem may be understood as saying that the measurement apparatus implementing an approximate measurement of an observable that does not commute with a conserved quantity must include a subsystem that serves as a quantum reference frame.…”

Thermodynamically free quantum measurements

“…We say that an arbitrary channel Φ acting in H (fully) conserves the Hamiltonian H if all moments of energy are invariant under its action, i.e., tr[H k Φ( )] = tr[H k ] for all states and k ∈ N [26]. But in finite dimensions, and for bistochastic channels, if the first moment is conserved (k = 1), then all moments will be conserved (k > 1).…”

“…( 1) we see that it is only the pointer observable, and not the instrument that measures it, which uniquely determines the instrument acting in the system-we do demand that the Z-channel J X conserves energy. As shown in [26], this constraint demands that Z commutes with H A . Such commutation is known as the the Yanase condition [39,40] which was first introduced in the context of the Wigner-Araki-Yanase theorem [41][42][43].…”

“…However, recall that a thermal channel always leaves the Gibbs state invariant, i.e., I X (π β ) = π β . Since the Gibbs state is full-rank (or faithful), then as shown in [64] a thermal E-instrument will disturb all observables not commuting with E. Additionally, if F is not thermal, i.e., if F does not commute with H S , then as shown in [26] non-disturbance will be possible only if F also commutes with ∆H S := I * X (H S ) − H S , where I * X is the "Heisenberg picture" dual of the E-channel I X .…”

“…For example, a thermal instrument is shown to only measure observables that commute with the Hamiltonian, disturbs all observables not commuting with the measured observable, can never act as a non-trivial state preparation device, and always thermalises the measured system when performing a complete measurement, thereby destroying all the information contained therein. The demarcation of measurements that are not thermodynamically free is a first step towards a resource-theoretic quantification of the thermodynamic cost of measurements: for example, if a measurement is not thermodynamically free then its cost can be quantified by the free energy or coherence/asymmetry that must be initially present in the probe so as to approximately implement the desired measurement while still maintaining energy conservation for the measurement process [23][24][25][26]. This approach will also allow for such cost-quantification to be independent of the measured system's initial state, instead depending only on the properties of the desired measurement.…”