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.

Reductions and inductions

Just as a completely positive (CPTP) map can always be considered the reduction of unitary evolution on a system with an initially unentangled auxiliary, quantum instruments are the reductions of projective measurement with a conditional unitary, and also reduce to CPTP maps and POVMs when ignore measurement outcomes and state evolution, respectively. In John Smolin's terminology, this is an example of "going to the Church of the Larger Hilbert space".

As a reduction of projective measurement and conditional unitary

Any quantum instrument on a system ${\displaystyle {\mathcal {S}}}$ can be modeled as projective measurement on ${\displaystyle {\mathcal {S}}}$ and (jointly) an uncorrelated auxiliary ${\displaystyle {\mathcal {A}}}$ followed by a unitary conditional on the measurement outcome. Let ${\displaystyle \eta }$ (with ${\displaystyle \eta >0}$ and ${\displaystyle \mathrm {Tr} \,\eta =1}$) be the normalized initial state of ${\displaystyle {\mathcal {A}}}$, let ${\displaystyle \{\Pi _{i}\}}$ (with ${\displaystyle \Pi _{i}=\Pi _{i}^{\dagger }=\Pi _{i}^{2}}$ and ${\displaystyle \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}}$) be a projective measurement on ${\displaystyle {\mathcal {SA}}}$, and let ${\displaystyle \{U_{i}\}}$ (with ${\displaystyle U_{i}^{\dagger }=U_{i}^{-1}}$) be unitaries on ${\displaystyle {\mathcal {SA}}}$. Then one can check that

${\displaystyle {\mathcal {E}}_{i}(\rho ):=\mathrm {Tr} _{\mathcal {A}}\left(U_{i}\Pi _{i}(\rho \otimes \eta )\Pi _{i}U_{i}^{\dagger }\right)}$

defines a quantum instrument. Furthermore, one can also check that any choice of quantum instrument ${\displaystyle \{{\mathcal {E}}_{i}\}}$ can be obtained with this construction for some choice of ${\displaystyle \eta }$ and ${\displaystyle \{U_{i}\}}$.

In this sense, a quantum instrument can be thought of as the reduction of a projective measurement combined with a conditional unitary.

Reduction to CPTP map

Any quantum instrument ${\displaystyle \{{\mathcal {E}}_{i}\}}$ immediately induces a completely positive trace-preserving (CPTP) map, i.e., a quantum channel:

${\displaystyle {\mathcal {E}}(\rho ):=\sum _{i}{\mathcal {E}}_{i}(\rho )}$

This can be thought of as the overall effect of the measurement on the quantum system if the measurement outcome is thrown away.

Reduction to POVM

Any quantum instrument ${\displaystyle \{{\mathcal {E}}_{i}\}}$ immediately induces a positive operator-valued measurement (POVM):

${\displaystyle M_{i}:=\sum _{a}K_{a}^{(i)\dagger }K_{a}^{(i)}}$

where ${\displaystyle K_{a}^{(i)}}$ are any choice of Kraus operators for the ${\displaystyle {\mathcal {E}}_{i}}$,

${\displaystyle {\mathcal {E}}_{i}(\rho )=\sum _{a}K_{a}^{(i)}\rho K_{a}^{(i)\dagger }.}$

The Kraus operators ${\displaystyle K_{a}^{(i)}}$ are not uniquely determined by the CP maps ${\displaystyle {\mathcal {E}}_{i}}$, but the above definition of the POVM elements ${\displaystyle M_{i}}$ is the same for any choice. The POVM can be thought of as the measurement of the quantum system if the information about how the system is affected by the measurement is thrown away.

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