Fidelity - Revision history

Nathaniel: Undo revision 967 by Etanjk (talk) The QETLAB function doesn't use squaring; this is addressed in the page's "notes" section.

2024-09-27T16:46:35Z

Undo revision 967 by Etanjk (talk) The QETLAB function doesn't use squaring; this is addressed in the page's "notes" section.

← Older revision		Revision as of 16:46, 27 September 2024
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	\|cvx=yes (concave)}}		\|cvx=yes (concave)}}
	<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:		<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
	: <math>F(\rho,\sigma) := [\mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big)~~]^2~~.</math>		: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>

	Note that, in some sources, "fidelity" refers to the square of this quantity.		Note that, in some sources, "fidelity" refers to the square of this quantity.

Etanjk: The definition of fidelity needs to be squared after tracing according to Josza's paper.

2020-09-14T05:46:55Z

The definition of fidelity needs to be squared after tracing according to Josza's paper.

← Older revision		Revision as of 05:46, 14 September 2020
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	\|cvx=yes (concave)}}		\|cvx=yes (concave)}}
	<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:		<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
	: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>		: <math>F(\rho,\sigma) := [\mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big)]^2.</math>

	Note that, in some sources, "fidelity" refers to the square of this quantity.		Note that, in some sources, "fidelity" refers to the square of this quantity.

Nathaniel: Link to MatsumotoFidelity

2020-06-23T14:43:14Z

Link to MatsumotoFidelity

← Older revision		Revision as of 14:43, 23 June 2020
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	\|name=Fidelity		\|name=Fidelity
	\|desc=Computes the (Uhlmann) fidelity of two density matrices		\|desc=Computes the (Uhlmann) fidelity of two density matrices
	\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]		\|rel=[[MatsumotoFidelity]]<br />[[MaximumOutputFidelity]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]		\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]
	\|upd=November 21, 2014		\|upd=November 21, 2014

Nathaniel: Uhlmann

2020-06-23T14:34:34Z

Uhlmann

← Older revision		Revision as of 14:34, 23 June 2020
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	{{Function		{{Function
	\|name=Fidelity		\|name=Fidelity
	\|desc=Computes the fidelity of two density matrices		\|desc=Computes the (Uhlmann) fidelity of two density matrices
	\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]		\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]		\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]
	\|upd=November 21, 2014		\|upd=November 21, 2014
	\|cvx=yes (concave)}}		\|cvx=yes (concave)}}
	<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:		<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
	: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>		: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>

Nathaniel: added CVX parameter

2014-12-24T16:48:05Z

added CVX parameter

← Older revision		Revision as of 16:48, 24 December 2014
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	\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]		\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]		\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]
	\|upd=November 21, 2014}}		\|upd=November 21, 2014
			\|cvx=yes (concave)}}
	<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:		<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
	: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>		: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>

Nathaniel: Bugfix when dealing with complex matrices

2014-11-27T21:12:13Z

Bugfix when dealing with complex matrices

← Older revision		Revision as of 21:12, 27 November 2014
Line 4:		Line 4:
	\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]		\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]		\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]
	\|upd=November 20, 2014		\|upd=November 21, 2014}}
	~~\|v=0.60~~}}
	<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:		<tt>'''Fidelity'''</tt> is a [[List of functions\|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
	: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>		: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>

Nathaniel at 15:50, 20 November 2014

2014-11-20T15:50:05Z

← Older revision		Revision as of 15:50, 20 November 2014
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	\|name=Fidelity		\|name=Fidelity
	\|desc=Computes the fidelity of two density matrices		\|desc=Computes the fidelity of two density matrices
	\|rel=[[TraceNorm]]		\|rel=[[MaximumOutputFidelity]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]		\|cat=[[List of functions#Norms_and_distance_measures\|Norms and distance measures]]
	\|upd=November 20, 2014		\|upd=November 20, 2014

Nathaniel: Created page with "{{Function |name=Fidelity |desc=Computes the fidelity of two density matrices |rel=TraceNorm |cat=List of functions#Norms_and_distance_measures|Norms and distance measur..."

2014-11-20T15:48:50Z

Created page with "{{Function |name=Fidelity |desc=Computes the fidelity of two density matrices |rel=TraceNorm |cat=List of functions#Norms_and_distance_measures|Norms and distance measur..."

New page

{{Function
|name=Fidelity
|desc=Computes the fidelity of two density matrices
|rel=[[TraceNorm]]
|cat=[[List of functions#Norms_and_distance_measures|Norms and distance measures]]
|upd=November 20, 2014
|v=0.60}}
<tt>'''Fidelity'''</tt> is a [[List of functions|function]] that computes the fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
: <math>F(\rho,\sigma) := \mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big).</math>

Note that, in some sources, "fidelity" refers to the square of this quantity.

==Syntax==
* <tt>FID = Fidelity(RHO,SIGMA)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A density matrix.
* <tt>SIGMA</tt>: A density matrix.

==Examples==
===Pure states===
If $\rho = |v\rangle\langle v|$ and $\sigma = |w\rangle\langle w|$ are both pure states then their fidelity simply equals $\big|\langle v|w \rangle\big|$:
<syntaxhighlight>
>> v = RandomStateVector(4);
>> w = RandomStateVector(4);
>> Fidelity(v*v',w*w')

ans =

0.6486

>> abs(v'*w)

ans =

0.6486
</syntaxhighlight>

===Can be used with CVX===
The fidelity function is a jointly concave function, and it can be used in the objective function or constraints of a CVX optimization problem. For example, the following code computes the [[MaximumOutputFidelity|maximum output fidelity]] of two quantum channels:
<syntaxhighlight>
>> Phi = RandomSuperoperator(3); % generate a random channel
>> Psi = RandomSuperoperator(3); % generate another one
>> cvx_begin sdp quiet
variable rho(3,3) hermitian;
variable sigma(3,3) hermitian;

maximize Fidelity(ApplyMap(rho,Phi),ApplyMap(sigma,Psi))

subject to % the constraints here just force rho and sigma to be density matrices
trace(rho) == 1;
trace(sigma) == 1;
rho >= 0;
sigma >= 0;
cvx_end
>> cvx_optval

cvx_optval =

0.9829
</syntaxhighlight>

Of course, in this case it is more convenient to just use the <tt>[[MaximumOutputFidelity]]</tt> function directly:
<syntaxhighlight>
>> MaximumOutputFidelity(Phi,Psi)

ans =

0.9829
</syntaxhighlight>

{{SourceCode|name=Fidelity}}