Contents

Quantum instrument

Definition[edit]

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

The state after a measurement with the specific outcome ${\displaystyle x}$ is given by

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

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (July 2009) (Learn how and when to remove this message)

• 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

This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.