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