Quantum instrument

In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation. It can be equivalently understood as a quantum channel that takes as input a quantum system and has as its output two systems: a classical system containing the outcome of the measurement and a quantum system containing the post-measurement state.

Definition

Let $X$ be a countable set describing the outcomes of a measurement, and let $\_$ denote a collection of trace-non-increasing completely positive maps, such that the sum of all $\mathcal_x$ is trace-preserving, i.e. $\operatorname\left(\sum_x\mathcal_x(\rho)\right)=\operatorname(\rho)$ for all positive operators $\rho$.

Now for describing a quantum measurement by an instrument $\mathcal$, the maps $\mathcal_x$ are used to model the mapping from an input state $\rho$ to the output state of a measurement conditioned on a classical measurement outcome $x$. Therefore, the probability of measuring a specific outcome $x$ on a state $\rho$ is given by
$p(x, \rho)=\operatorname(\mathcal_x(\rho)).$

The state after a measurement with the specific outcome $x$ is given by

$\rho_x=\frac.$

If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections $, x\rangle\langle x, \in \mathcal(\mathbb^)$ , then the action of an instrument $\mathcal$ is given by a quantum channel $\mathcal:\mathcal(\mathcal_1) \rightarrow \mathcal(\mathcal_2)\otimes \mathcal(\mathbb^)$ with

$\mathcal(\rho):= \sum_x \mathcal_x ( \rho)\otimes \vert x \rangle \langle x, .$

Here $\mathcal_1$ and $\mathcal_2 \otimes \mathbb^$ are the Hilbert spaces corresponding to the input and the output systems of the instrument.

A quantum instrument is an example of a quantum operation in which an "outcome" $x$ indicating which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.

References

* E. Davies, J. Lewis. An operational approach to quantum probability, Comm. Math. Phys., vol. 17, pp. 239–260, 1970.

Distillation of secret key paper

Another paper which uses the concept

Quantum mechanics

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