Physics:Quantum instrument - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:40Z

Restore missing Quantum reference definitions

← Older revision		Revision as of 00:31, 24 May 2026
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	</math> is given by a quantum channel <math>		</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\|})		\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> with<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>

	<math display="block">		<math display="block">
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	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.		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 ==		= References =
	{{reflist}}		{{reflist}}

WikiHarold: Clean Quantum page image and red links

2026-05-23T23:34:41Z

Clean Quantum page image and red links

← Older revision		Revision as of 23:34, 23 May 2026
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	<div style="width:300px;">		<div style="width:300px;">
	[[File:~~File not found~~.~~png~~\|thumb\|280px\|~~No image available~~.]]		[[File:Quantum_instrument_concept_map.svg\|thumb\|280px\|instrument in the Quantum Collection.]]
	</div>		</div>

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	Let <math>		Let <math>
	X		X
	</math> be a ~~[[Countable set\|~~countable set]] describing the outcomes of a quantum measurement, and let <math>		</math> be a countable set describing the outcomes of a quantum measurement, and let <math>
	\{\mathcal{E}_x \}_{x\in X}		\{\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>		</math> denote a collection of trace-non-increasing completely positive maps, such that the sum of all <math>
	\mathcal{E}_x		\mathcal{E}_x
	</math> is trace-preserving, i.e. <math display="inline">		</math> is trace-preserving, i.e. <math display="inline">
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	== Reductions and inductions ==		== 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]]".		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 ===		=== As a reduction of projective measurement and conditional unitary ===
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	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>		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.		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 ===		=== Reduction to CPTP map ===
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	=== Reduction to POVM ===		=== Reduction to POVM ===

	Any quantum instrument <math>\{\mathcal{E}_i\}</math> immediately induces a positive operator-valued measurement ([[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>		:<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>		where <math>K_a^{(i)}</math> are any choice of Kraus operators for <math>\mathcal{E}_i</math>,<ref name=Busch2016></ref>

WikiHarold: Expand short Quantum intro

2026-05-23T23:02:15Z

Expand short Quantum intro

← Older revision		Revision as of 23:02, 23 May 2026
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~{{Short description\|Mathematical abstraction of~~ quantum ~~measurement~~		A quantum instrument is a Book I topic in the Quantum Collection. It is a mathematical description of a quantum measurement that records both the probability of each classical outcome and the corresponding change in the quantum state. Unlike a simple observable, an instrument keeps track of measurement back-action, state update, and conditional dynamics. Quantum instruments are used in open systems, generalized measurements, quantum operations, feedback, tomography, and information theory. They make precise the idea that measurement is a physical process, not only a readout.
	}}
	~~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>~~
	</div>		</div>

Harold: Add missing image fallback to Quantum header

2026-05-17T22:33:17Z

Add missing image fallback to Quantum header

← Older revision		Revision as of 22:33, 17 May 2026
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	<div style="width:300px;">		<div style="width:300px;">
	~~<!--~~ No ~~lead~~ image available ~~in existing page~~. ~~-->~~		[[File:File not found.png\|thumb\|280px\|No image available.]]
	</div>		</div>

Harold: Restore Quantum article header template

2026-05-17T21:52:42Z

Restore Quantum article header template

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- {{Short description|Mathematical abstraction of quantum measurement
+{{Short description|Quantum physics topic}}
 }}
 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==

imported>WikiHarold: change

2026-01-27T16:29:45Z

change

← Older revision	Revision as of 16:29, 27 January 2026
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imported>WikiHarold: change

2026-01-27T16:29:45Z

change

New page

{{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}}