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 ${\displaystyle X}$ be a countable set describing the outcomes of a measurement, and let ${\displaystyle \{{\mathcal {E}}_{x}\}_{x\in X}}$ denote a collection of trace-non-increasing completely positive maps, such that the sum of all ${\displaystyle {\mathcal {E}}_{x}}$ is trace-preserving, i.e. ${\textstyle \operatorname {tr} \left(\sum _{x}{\mathcal {E}}_{x}(\rho )\right)=\operatorname {tr} (\rho )}$ for all positive operators ${\displaystyle \rho }$.

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

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

The state after a measurement with the specific outcome ${\displaystyle x}$ 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, 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 ${\displaystyle {\mathcal {I}}}$ is given by a quantum channel ${\displaystyle {\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|X|})}$ with

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

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