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.

Definition

Let $X$ be the countable set describing the outcomes of a measurement and $\{\mathcal{E}_x \}_{x\in X}$ a collection of subnormalized completely positive maps, given in such a way that the sum of all $\mathcal{E}_x$ is trace preserving, i.e. ${\textstyle \operatorname{tr}\left(\sum_x\mathcal{E}_x(\rho)\right)=\operatorname{tr}(\rho) }$ for all positive operators $\rho$.

Now for describing a quantum measurement by a instrument $\mathcal{I}$, the maps $\mathcal{E}_x$ are used to model the mapping from an input state $\rho$ to the outputstate of a measurement conditioned on an classical measurement outcome $x$. Thereby the probability of measuring an specific outcome $x$ on a state $\rho$ is given by

${\displaystyle p(x|\rho)=\operatorname{tr}(\mathcal{E}_x(\rho)) }$.

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

${\displaystyle \rho_x=\frac{\mathcal{E}_x(\rho)}{\operatorname{tr}(\mathcal{E}_x(\rho))} }$

If the measurement outcomes are recorded in a classical register, i.e. this can be modelled by a set of orthonormal projections ${\textstyle |x\rangle\langle x| \in \mathcal{B}(\mathbb{C}^{|x|}) }$ , the action of a instrument $\mathcal{I}$ is given by an channel $\mathcal{I}:\mathcal{B}(\mathcal{H}_1) \rightarrow \mathcal{B}(\mathcal{H}_2)\otimes \mathcal{B}(\mathbb{C}^{|x|})$ with

${\displaystyle \mathcal{I}(\rho):= \sum_x \mathcal{E}_x \left( \rho\right)\otimes \vert x \rangle \langle x| }$

Here $\mathcal{H}_1$ and $\mathcal{H}_2$ are the Hilbert spaces corresponding to the input and the output quantum system of a measurement.

A quantum instrument is more general than a quantum operation because it records the outcome $x$ of which operator acted on the state. 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