Decomposing qubit positive-operator-valued measurements into continuous destructive weak measurements

Abstract

It has been shown that any generalized measurement can be decomposed into a sequence of weak measurements corresponding to a stochastic process. However, the weak measurements may require almost arbitrary unitaries, which are unlikely to be realized by any real measurement device. Furthermore, many measurement processes are destructive, like photon counting procedures that terminate once all photons are consumed. One cannot expect to have full control of the evolution of a state under such destructive measurements, and the possible unitaries allow only a limited set of weak measurements. In this paper, we consider a qubit model of destructive weak measurements, which is a toy version of an optical cavity, in which the state of an electromagnetic field mode inside the cavity leaks out and is measured destructively while the vacuum state |0 leaks in to the cavity. At long times, the state of the qubit inevitably evolves to be |0 , and the only available control is the choice of measurement on the external ancilla system. Surprisingly, this very limited model can still perform an arbitrary projective measurement on the qubit in any basis, where the probability of getting an outcome satisfies the usual Born rule. Combining this method with probabilistic post processing, the result can be extended to any generalized measurement with commuting POVM elements. This implies, among other results, that any two-outcome POVM on a qubit can be decomposed into a sequence of destructive weak measurements by this restricted measurement device.

Qubit positive-operator-valued measurements by destructive weak measurements

“…Hence, for a qubit system, if one can perform any LIPPOVM then all POVMs can be accomplished. We claim that the model of lossy weak measurements [24] can achieve any qubit LIPPOVM and hence any qubit POVM. The following sections will verify the assertion.…”

“…The model of destructive weak measurement in [24] captures the behavior of a destructive strong measurement that leaves the system in a fixed final state after the measurement regardless of the outcome. The only freedom in this model is the ability to choose any projective positive-operator-valued measurement on the ancilla in an adaptive way.…”

“…Recall the model of destructive weak measurements in [24]: we couple a qubit system |ψ ψ| to an ancilla in state |0 by a weak swap defined as…”

“…In Sec. II, we reintroduced the model of destructive weak measurements in [24]. It consists of coupling the system with an ancilla qubit in state |0 by a weak swap operation, followed by a projective POVM on the ancilla.…”

“…Motivated by this example, we developed a model of destructive weak measurements for qubits in [24]. It consists of a weak unitary swap between the system and the ancilla qubit, in the initial state |0 , followed by a projective measurement on the ancilla qubit.…”