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 $\{{\mathcal {E}}_{x}\}_{x\in X}$ denote a collection of trace-non-increasing completely positive maps, such 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 an instrument ${\mathcal {I}}$ , the maps ${\mathcal {E}}_{x}$ 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 {tr} ({\mathcal {E}}_{x}(\rho )).

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

\rho _{x}={\frac {{\mathcal {E}}_{x}(\rho )}{\operatorname {tr} ({\mathcal {E}}_{x}(\rho ))}}

If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections ${\textstyle |x\rangle \langle x|\in {\mathcal {B}}(\mathbb {C} ^{|X|})}$ , then the action of an instrument ${\mathcal {I}}$ is given by a quantum channel ${\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|X|})$ with

{\mathcal {I}}(\rho ):=\sum _{x}{\mathcal {E}}_{x}(\rho )\otimes \vert x\rangle \langle x|

Here ${\mathcal {H}}_{1}$ and ${\mathcal {H}}_{2}\otimes \mathbb {C} ^{|X|}$ 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

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