Physics:Quantum limit - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:40Z

Restore missing Quantum reference definitions

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	where <math>k_p=\omega_p/c</math>, <math>\omega_p</math> is the light frequency, <math>j=\dots,-1,0,1,\dots</math> is the pulse number and <math>\hat{\phi}_j</math> is the initial (random) phase of the <math>j</math>-th pulse. We assume that the mean value of all these phases is equal to zero, <math>\langle\hat{\phi}_j\rangle=0</math>, and their root mean square (RMS) uncertainty <math>(\langle\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2}</math> is equal to <math>\Delta\phi</math>.		where <math>k_p=\omega_p/c</math>, <math>\omega_p</math> is the light frequency, <math>j=\dots,-1,0,1,\dots</math> is the pulse number and <math>\hat{\phi}_j</math> is the initial (random) phase of the <math>j</math>-th pulse. We assume that the mean value of all these phases is equal to zero, <math>\langle\hat{\phi}_j\rangle=0</math>, and their root mean square (RMS) uncertainty <math>(\langle\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2}</math> is equal to <math>\Delta\phi</math>.

	The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using ''e.g.'' homodyne or heterodyne detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh /> and references therein), or other such read-out techniques.		The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using ''e.g.'' homodyne or heterodyne detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh>
			{{cite journal
			\|last=Danilishin\|first=S. L.
			\|author2=Khalili F. Ya.
			\|title=Quantum Measurement Theory in Gravitational-Wave Detectors
			\|journal=Living Reviews in Relativity
			\|year=2012
			\|volume=15
			\|issue=5
			\|page=60
			\|doi=10.12942/lrr-2012-5
			\|doi-access=free
			\|pmid=28179836
			\| pmc=5256003
			\|arxiv = 1203.1706 \|bibcode = 2012LRR....15....5D }}
			</ref> and references therein), or other such read-out techniques.

	In this example, light pulse phase <math>\hat\phi_j</math> serves as the readout observable <math>\mathcal{O}</math> of the Meter. Then we suppose that the phase <math>\hat{\phi}_j^{\mathrm{refl}}</math> measurement error introduced by the detector is much smaller than the initial uncertainty of the phases <math>\Delta\phi</math>. In this case, the initial uncertainty will be the only source of the position measurement error:		In this example, light pulse phase <math>\hat\phi_j</math> serves as the readout observable <math>\mathcal{O}</math> of the Meter. Then we suppose that the phase <math>\hat{\phi}_j^{\mathrm{refl}}</math> measurement error introduced by the detector is much smaller than the initial uncertainty of the phases <math>\Delta\phi</math>. In this case, the initial uncertainty will be the only source of the position measurement error:
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	==References and notes==		==References and notes==

	== References ==		= References =

	<references/>		<references/>

WikiHarold: Use Quantum See also index module

2026-05-23T22:22:09Z

Use Quantum See also index module

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	Note that due to an overloading of the word "limit", the classical limit is ''not'' the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit). In "classical limit", "limit" is used in the sense of a limiting process. (Note that there is no ''simple rigorous'' mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)		Note that due to an overloading of the word "limit", the classical limit is ''not'' the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit). In "classical limit", "limit" is used in the sense of a limiting process. (Note that there is no ''simple rigorous'' mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)

	==See also==		== See also ==
	*Classical limit		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	*Heisenberg limit
	*Ultrarelativistic limit

	==References and notes==		==References and notes==

	~~== See also ==~~
	~~{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}~~
	== References ==		== References ==

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	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
	{{Quantum article nav\|previous=Physics:Quantum Zeno effect\|previous label=Zeno effect\|next=Physics:Quantum information theory\|next label=Information theory}}		{{Quantum article nav\|previous=Physics:Quantum Zeno effect\|previous label=Zeno effect\|next=Physics:Quantum information theory\|next label=Information theory}}

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	{{Short description\|Limit on measurements at quantum scales}}		{{Short description\|Limit on measurements at quantum scales}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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	{{Short description\|Limit on measurements at quantum scales}}~~Book I~~		{{Short description\|Limit on measurements at quantum scales}}

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	~~~~{{Short description\|Limit on measurements at quantum scales}}		{{Short description\|Limit on measurements at quantum scales}}Book I

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}

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	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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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;">
-A '''quantum limit''' in physics is a limit on measurement accuracy at quantum scales.<ref name=QMeas_Br_Kh>
+'''limit''' a quantum limit in physics is a limit on measurement accuracy at quantum scales. Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring quantum states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes. The usage of the term standard quantum limit or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical observable of a system under study that does not commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause. A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp. So any measurement is a result of interaction and that acts in both ways. This is known as back action (quantum) of the Meter on the system under measurement. This one is known as measurement imprecision or measurement noise. The equality is reached if the system is in a minimum uncertainty state. The terms "quantum limit" and "standard quantum limit" are sometimes used interchangeably. Usually, "quantum limit" is a general term which refers to any restriction on measurement due to quantum effects, while the "standard quantum limit" in any given context refers to a quantum limit which is ubiquitous in that context. Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement.
 </div>

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	{{Short description\|Limit on measurements at quantum scales}}		{{Short description\|Limit on measurements at quantum scales}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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	\|publisher=Cambridge University Press		\|publisher=Cambridge University Press
	\|isbn=978-0-521-48413-8		\|isbn=978-0-521-48413-8
	}}</ref> Depending on the context, the limit may be absolute (such as the ~~[[Physics:~~Heisenberg limit~~\|Heisenberg limit]]~~), or it may only apply when the experiment is conducted with naturally occurring [[Physics:Quantum state\|quantum state]]s (e.g. the '''standard quantum limit''' in interferometry) and can be circumvented with advanced state preparation and measurement schemes.		}}</ref> Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring [[Physics:Quantum state\|quantum state]]s (e.g. the '''standard quantum limit''' in interferometry) and can be circumvented with advanced state preparation and measurement schemes.

	The usage of the term '''standard quantum limit''' or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical ~~[[Physics:Observable\|~~observable]] of a system under study that does not ~~[[Commutator\|~~commute]] with itself at different times leads to such limits. In short, it is the ~~[[Uncertainty principle\|~~Heisenberg uncertainty principle]] that is the cause.		The usage of the term '''standard quantum limit''' or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical observable of a system under study that does not commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause.
	A more detailed explanation would be that any measurement in [[Physics:Quantum mechanics\|quantum mechanics]] involves at least two parties, an Object and a Meter. The former is the system whose observable, say <math>\hat x</math>, we want to measure. The latter is the system we couple to the Object in order to infer the value of <math>\hat x</math> of the Object by recording some chosen observable, <math>\hat{\mathcal{O}}</math>, of this system, ''e.g.'' the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as ''indirect'' measurements (see pp. 38–42 of <ref name=QMeas_Br_Kh />). So any measurement is a result of interaction and that acts in both ways. Therefore, the Meter acts on the Object during each measurement, usually via the quantity, <math>\hat{\mathcal{F}}</math>, conjugate to the readout observable <math>\hat{\mathcal{O}}</math>, thus perturbing the value of measured observable <math>\hat x</math> and modifying the results of subsequent measurements. This is known as ~~[[Physics:Back action (quantum)\|~~back action (quantum)]] of the Meter on the system under measurement.		A more detailed explanation would be that any measurement in [[Physics:Quantum mechanics\|quantum mechanics]] involves at least two parties, an Object and a Meter. The former is the system whose observable, say <math>\hat x</math>, we want to measure. The latter is the system we couple to the Object in order to infer the value of <math>\hat x</math> of the Object by recording some chosen observable, <math>\hat{\mathcal{O}}</math>, of this system, ''e.g.'' the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as ''indirect'' measurements (see pp. 38–42 of <ref name=QMeas_Br_Kh />). So any measurement is a result of interaction and that acts in both ways. Therefore, the Meter acts on the Object during each measurement, usually via the quantity, <math>\hat{\mathcal{F}}</math>, conjugate to the readout observable <math>\hat{\mathcal{O}}</math>, thus perturbing the value of measured observable <math>\hat x</math> and modifying the results of subsequent measurements. This is known as back action (quantum) of the Meter on the system under measurement.

	At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty, <math>\delta\hat{\mathcal{O}}</math>, additive to and independent of the value of the measured quantity <math>\hat x</math>. This one is known as ''measurement imprecision'' or ''measurement noise''. Because of the ~~[[Uncertainty principle\|~~Heisenberg uncertainty principle]], this imprecision cannot be arbitrary and is linked to the back-action perturbation by the ~~[[Uncertainty principle\|~~uncertainty relation]]:		At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty, <math>\delta\hat{\mathcal{O}}</math>, additive to and independent of the value of the measured quantity <math>\hat x</math>. This one is known as ''measurement imprecision'' or ''measurement noise''. Because of the Heisenberg uncertainty principle, this imprecision cannot be arbitrary and is linked to the back-action perturbation by the uncertainty relation:

	:<math>\Delta {\mathcal{O}} \Delta {\mathcal{F}} \geqslant \hbar/2\,,</math>		:<math>\Delta {\mathcal{O}} \Delta {\mathcal{F}} \geqslant \hbar/2\,,</math>

	where <math>\Delta a = \sqrt{\langle\hat a^2\rangle-\langle\hat a\rangle^2}</math> is a standard deviation of observable <math>a</math> and <math>\langle\hat a\rangle</math> stands for ~~[[Expectation value\|~~expectation value]] of <math>a</math> in whatever [[Physics:Quantum state\|quantum state]] the system is. The equality is reached if the system is in a ''minimum uncertainty state''. The consequence for our case is that the more precise is our measurement, ''i.e'' the smaller is <math>\Delta \mathcal{\delta O}</math>, the larger will be perturbation the Meter exerts on the measured observable <math>\hat x</math>. Therefore, the readout of the meter will, in general, consist of three terms:		where <math>\Delta a = \sqrt{\langle\hat a^2\rangle-\langle\hat a\rangle^2}</math> is a standard deviation of observable <math>a</math> and <math>\langle\hat a\rangle</math> stands for expectation value of <math>a</math> in whatever [[Physics:Quantum state\|quantum state]] the system is. The equality is reached if the system is in a ''minimum uncertainty state''. The consequence for our case is that the more precise is our measurement, ''i.e'' the smaller is <math>\Delta \mathcal{\delta O}</math>, the larger will be perturbation the Meter exerts on the measured observable <math>\hat x</math>. Therefore, the readout of the meter will, in general, consist of three terms:

	:<math>\hat{\mathcal{O}} = \hat x_{\mathrm{free}} + \delta \hat{\mathcal{O}} + \delta \hat x_{BA}[\hat{\mathcal{F}}]\,,</math>		:<math>\hat{\mathcal{O}} = \hat x_{\mathrm{free}} + \delta \hat{\mathcal{O}} + \delta \hat x_{BA}[\hat{\mathcal{F}}]\,,</math>
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	===Displacement measurement===		===Displacement measurement===
	Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a ~~[[General position\|~~general position]] measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body <math>M</math>. The position <math>x</math> of <math>M</math> is probed periodically with time interval <math>\vartheta</math>. We assume mass <math>M</math> large enough to neglect the displacement inflicted by the pulses regular (classical) ~~[[Astronomy:Radiation pressure\|~~radiation pressure]] in the course of measurement process.		Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body <math>M</math>. The position <math>x</math> of <math>M</math> is probed periodically with time interval <math>\vartheta</math>. We assume mass <math>M</math> large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.

	[[File:Position measurement using reflected light.png\|thumb\|Simplified scheme of optical measurement of mechanical object position]]		[[File:Position measurement using reflected light.png\|thumb\|Simplified scheme of optical measurement of mechanical object position]]
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	</math>\|{{EquationRef\|1}}}}		</math>\|{{EquationRef\|1}}}}

	where <math>k_p=\omega_p/c</math>, <math>\omega_p</math> is the light frequency, <math>j=\dots,-1,0,1,\dots</math> is the pulse number and <math>\hat{\phi}_j</math> is the initial (random) phase of the <math>j</math>-th pulse. We assume that the mean value of all these phases is equal to zero, <math>\langle\hat{\phi}_j\rangle=0</math>, and their ~~[[Root mean square\|~~root mean square]] (RMS) uncertainty <math>(\langle\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2}</math> is equal to <math>\Delta\phi</math>.		where <math>k_p=\omega_p/c</math>, <math>\omega_p</math> is the light frequency, <math>j=\dots,-1,0,1,\dots</math> is the pulse number and <math>\hat{\phi}_j</math> is the initial (random) phase of the <math>j</math>-th pulse. We assume that the mean value of all these phases is equal to zero, <math>\langle\hat{\phi}_j\rangle=0</math>, and their root mean square (RMS) uncertainty <math>(\langle\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2}</math> is equal to <math>\Delta\phi</math>.

	The reflected pulses are detected by a phase-sensitive device (the ~~[[Engineering:Phase detector\|~~phase detector]]). The implementation of an optical phase detector can be done using ''e.g.'' ~~[[Homodyne detection\|~~homodyne]] or ~~[[Optical heterodyne detection\|~~heterodyne]] detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh /> and references therein), or other such read-out techniques.		The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using ''e.g.'' homodyne or heterodyne detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh /> and references therein), or other such read-out techniques.

	In this example, light pulse phase <math>\hat\phi_j</math> serves as the readout observable <math>\mathcal{O}</math> of the Meter. Then we suppose that the phase <math>\hat{\phi}_j^{\mathrm{refl}}</math> measurement error introduced by the detector is much smaller than the initial uncertainty of the phases <math>\Delta\phi</math>. In this case, the initial uncertainty will be the only source of the position measurement error:		In this example, light pulse phase <math>\hat\phi_j</math> serves as the readout observable <math>\mathcal{O}</math> of the Meter. Then we suppose that the phase <math>\hat{\phi}_j^{\mathrm{refl}}</math> measurement error introduced by the detector is much smaller than the initial uncertainty of the phases <math>\Delta\phi</math>. In this case, the initial uncertainty will be the only source of the position measurement error:
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	</math>		</math>

	where we assumed all contributions to our ~~[[Measurement uncertainty\|~~measurement uncertainty]] statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence <math>\Delta x_{\rm meas}(t_{j+1}) = \Delta x_{\rm meas}(t_{j}) \equiv \Delta x_{\rm meas} = \Delta\phi/(2k_p)</math>.		where we assumed all contributions to our measurement uncertainty statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence <math>\Delta x_{\rm meas}(t_{j+1}) = \Delta x_{\rm meas}(t_{j}) \equiv \Delta x_{\rm meas} = \Delta\phi/(2k_p)</math>.

	Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:		Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:
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	===Usage in quantum optics===		===Usage in quantum optics===
	In the context of ~~[[Astronomy:Interferometry\|~~interferometry]] or other optical measurements, the standard quantum limit usually refers to the minimum level of [[Physics:Quantum noise\|quantum noise]] which is obtainable without ~~[[Physics:Squeezed coherent state\|~~squeezed states]].<ref>		In the context of interferometry or other optical measurements, the standard quantum limit usually refers to the minimum level of [[Physics:Quantum noise\|quantum noise]] which is obtainable without squeezed states.<ref>
	{{cite journal		{{cite journal
	\|last1=Jaekel \|first1=M. T.		\|last1=Jaekel \|first1=M. T.
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	}}</ref>		}}</ref>

	There is additionally a quantum limit for ~~[[Phase noise\|~~phase noise]], reachable only by a ~~[[Physics:Laser\|~~laser]] at high noise frequencies.		There is additionally a quantum limit for phase noise, reachable only by a laser at high noise frequencies.

	In ~~[[Physics:Spectroscopy\|~~spectroscopy]], the shortest wavelength in an X-ray spectrum is called the quantum limit.<ref>		In spectroscopy, the shortest wavelength in an X-ray spectrum is called the quantum limit.<ref>
	{{cite journal		{{cite journal
	\|last1=Piston \|first1=D. S.		\|last1=Piston \|first1=D. S.
	\|year=1936		\|year=1936
	\|title=The Polarization of X-Rays from Thin Targets		\|title=The Polarization of X-Rays from Thin Targets
	\|journal=~~[[Physics:~~Physical Review~~\|Physical Review]]~~		\|journal=Physical Review
	\|volume=49 \|issue=4 \|pages=275–279		\|volume=49 \|issue=4 \|pages=275–279
	\|bibcode= 1936PhRv...49..275P		\|bibcode= 1936PhRv...49..275P
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	==Misleading relation to the classical limit==		==Misleading relation to the classical limit==
	Note that due to an overloading of the word "limit", the ~~[[Physics:Classical limit\|~~classical limit]] is ''not'' the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the ~~[[Astronomy:Armstrong limit\|~~Armstrong limit]]). In "classical limit", "limit" is used in the sense of a ~~[[Limit (mathematics)\|~~limiting process]]. (Note that there is no ''simple rigorous'' mathematical limit which fully recovers ~~[[Physics:Classical mechanics\|~~classical mechanics]] from quantum mechanics, the [[Ehrenfest theorem]] notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)		Note that due to an overloading of the word "limit", the classical limit is ''not'' the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit). In "classical limit", "limit" is used in the sense of a limiting process. (Note that there is no ''simple rigorous'' mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)

	==See also==		==See also==
	*~~[[Physics:~~Classical limit~~\|Classical limit]]~~		*Classical limit
	*~~[[Physics:Heisenberg limit\|~~Heisenberg limit]]		*Heisenberg limit
	*[[Ultrarelativistic limit]]		*Ultrarelativistic limit

	==References and notes==		==References and notes==