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{{Short description|Mathematical abstraction of quantum measurement
}}
In [[Physics:Quantum physics|quantum physics]], a '''quantum instrument''' is a mathematical description of a quantum measurement, capturing both the [[Physics:Classical physics|classical]] and [[Physics:Quantum physics|quantum]] outputs.<ref name=Alter2001>{{cite book| first1=Orly | last1=Alter | first2=Yoshihisa | last2=Yamamoto | title=Quantum Measurement of a Single System | location=New York | publisher=Wiley | year=2001 | doi=10.1002/9783527617128 | isbn=9780471283089 }}</ref> It can be equivalently understood as a [[Quantum channel|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.<ref name=Jordan2024>{{cite book| first1=Andrew N. | last1=Jordan | first2=Irfan A. | last2=Siddiqi | title=Quantum Measurement: Theory and Practice | publisher=Cambridge University Press | year=2024 | isbn= 978-1009100069}}
</ref>

==Definition==
Let <math>
X
</math> be a [[Countable set|countable set]] describing the outcomes of a quantum measurement, and let <math>
\{\mathcal{E}_x \}_{x\in X}
</math> denote a collection of trace-non-increasing [[Completely positive map|completely positive map]]s, such that the sum of all <math>
\mathcal{E}_x
</math> is trace-preserving, i.e. <math display="inline">
\operatorname{tr}\left(\sum_x\mathcal{E}_x(\rho)\right)=\operatorname{tr}(\rho)
</math> for all positive operators <math>
\rho.</math>

Now for describing a measurement by an instrument <math>
\mathcal{I}
</math>, the maps <math>
\mathcal{E}_x
</math> are used to model the mapping from an input state <math>
\rho
</math> to the output state of a measurement conditioned on a classical measurement outcome <math>
x
</math>. Therefore, the probability that a specific measurement outcome <math>
x
</math> occurs on a state <math>
\rho
</math> is given by<ref name=":0" /><ref name=Busch2016>{{cite book | last1=Busch | first1=Paul | last2=Lahti | first2= Pekka | last3=Pellonpää | first3=Juha-Pekka | last4=Ylinen | first4=Kari | title=Quantum measurement | publisher =Springer| date=2016 | volume=23 | pages=261--262 | isbn=978-3-319-43387-5 | doi=10.1007/978-3-319-43389-9}}
</ref>
<math display="block">
p(x|\rho)=\operatorname{tr}(\mathcal{E}_x(\rho)).
</math>

The state after a measurement with the specific outcome <math>
x
</math> is given by<ref name=":0" /><ref name=Busch2016></ref>

<math display="block">
\rho_x=\frac{\mathcal{E}_x(\rho)}{\operatorname{tr}(\mathcal{E}_x(\rho))}.
</math>

If the measurement outcomes are recorded in a classical register, whose states are modeled by a set of orthonormal projections <math display="inline">
|x\rangle\langle x| \in \mathcal{B}(\mathbb{C}^{|X|})
</math> , then the action of an instrument <math>
\mathcal{I}
</math> is given by a quantum channel <math>
\mathcal{I}:\mathcal{B}(\mathcal{H}_1) \rightarrow \mathcal{B}(\mathcal{H}_2)\otimes \mathcal{B}(\mathbb{C}^{|X|})
</math> with<ref name=Jordan2024></ref>

<math display="block">
\mathcal{I}(\rho):=
\sum_x \mathcal{E}_x
( \rho)\otimes \vert x \rangle \langle x|.
</math>

Here <math>
\mathcal{H}_1
</math> and <math>
\mathcal{H}_2 \otimes \mathbb{C}^{|X|}
</math> are the Hilbert spaces corresponding to the input and the output systems of the instrument.

== Reductions and inductions ==

Just as a completely positive trace preserving (CPTP) map can always be considered as 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.<ref name=Busch2016></ref> In [[Biography:John A. Smolin|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 <math>\mathcal{S}</math> can be modeled as a projective measurement on <math>\mathcal{S}</math> and (jointly) an uncorrelated auxiliary <math>\mathcal{A}</math> followed by a unitary ''conditional'' on the measurement outcome.<ref name=":0">{{Cite journal |last=Ozawa |first=Masanao |year=1984 |title=Quantum measuring processes of continuous observables |url=https://doi.org/10.1063/1.526000 |journal=Journal of Mathematical Physics |volume=25 |pages=79-87}}</ref><ref name=Busch2016></ref> Let <math>\eta</math> (with <math>\eta > 0</math> and <math>\mathrm{Tr} \, \eta =1</math>) be the normalized initial state of <math>\mathcal{A}</math>, let <math>\{\Pi_i\}</math> (with <math>\Pi_i = \Pi_i^\dagger = \Pi_i^2</math> and <math>\Pi_i \Pi_j = \delta_{ij} \Pi_i</math>) be a projective measurement on <math>\mathcal{SA}</math>, and let <math>\{U_i\}</math> (with <math>U_i^\dagger = U_i^{-1}</math>) be unitaries on <math>\mathcal{SA}</math>. Then one can check that
:<math>\mathcal{E}_i (\rho) := \mathrm{Tr}_{\mathcal{A}}\left(U_i\Pi_i(\rho\otimes\eta)\Pi_i U_i^\dagger\right)</math>
defines a quantum instrument.<ref name=Busch2016></ref> Furthermore, one can also check that any choice of quantum instrument <math>\{\mathcal{E}_i\}</math> can be obtained with this construction for some choice of <math>\eta</math> and <math>\{U_i\}</math>.<ref name=Busch2016></ref>

In this sense, a quantum instrument can be thought of as the ''[[Quantum entanglement#Reduced density matrices|reduction]]'' of a projective measurement combined with a conditional unitary.

=== Reduction to CPTP map ===

Any quantum instrument <math>\{\mathcal{E}_i\}</math> immediately induces a CPTP map, i.e., a quantum channel:<ref name=Busch2016></ref>
:<math>\mathcal{E} (\rho) := \sum_i \mathcal{E}_i(\rho).</math>
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 <math>\{\mathcal{E}_i\}</math> immediately induces a positive operator-valued measurement ([[POVM]]):
:<math>M_i := \sum_a K_a^{(i)\dagger} K_a^{(i)}</math>
where <math>K_a^{(i)}</math> are any choice of Kraus operators for <math>\mathcal{E}_i</math>,<ref name=Busch2016></ref>
:<math>\mathcal{E}_i (\rho) = \sum_a K_a^{(i)}\rho K_a^{(i)\dagger}.</math>
The Kraus operators <math>K_a^{(i)}</math> are not uniquely determined by the CP maps <math>\mathcal{E}_i</math>, but the above definition of the POVM elements <math>M_i</math> is the same for any choice.<ref name=Busch2016></ref> 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 ==
{{reflist}}

{{DEFAULTSORT:Quantum Instrument}}
[[Category:Quantum mechanics]]

{{Sourceattribution|Quantum instrument}}

Physics:Quantum instrument - Revision history

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