Distinguishability - Revision history

Rmmateri at 16:33, 13 June 2018

2018-06-13T16:33:17Z

← Older revision		Revision as of 16:33, 13 June 2018
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	\|name=Distinguishability		\|name=Distinguishability
	\|desc=Computes the maximum probability of distinguishing quantum states		\|desc=Computes the maximum probability of distinguishing quantum states
	\|req=[http://cvxr.com/cvx/ ~~cvx~~]		\|req=[http://cvxr.com/cvx/ CVX]
	\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]		\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]
	\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]		\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]

Nathaniel at 17:21, 6 October 2014

2014-10-06T17:21:34Z

← Older revision		Revision as of 17:21, 6 October 2014
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	\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]		\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]
	\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]		\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]
	\|upd=October 3, 2014		\|upd=October 6, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''Distinguishability'''</tt> is a [[List of functions\|function]] that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.		<tt>'''Distinguishability'''</tt> is a [[List of functions\|function]] that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.

Nathaniel at 18:51, 3 October 2014

2014-10-03T18:51:00Z

← Older revision		Revision as of 18:51, 3 October 2014
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	\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]		\|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]
	\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]		\|cat=[[List of functions#Distinguishing_objects\|Distinguishing objects]]
	\|upd=October 2, 2014		\|upd=October 3, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''Distinguishability'''</tt> is a [[List of functions\|function]] that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.		<tt>'''Distinguishability'''</tt> is a [[List of functions\|function]] that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.
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	* <tt>DIST = Distinguishability(X)</tt>		* <tt>DIST = Distinguishability(X)</tt>
	* <tt>DIST = Distinguishability(X,P)</tt>		* <tt>DIST = Distinguishability(X,P)</tt>
			* <tt>[DIST,MEAS] = Distinguishability(X,P)</tt>

	==Argument descriptions==		==Argument descriptions==
			===Input arguments===
	* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.		* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.
	* <tt>P</tt> (optional, default <tt>[1/k, 1/k, ..., 1/k]</tt>, where <tt>k</tt> is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above. All entries must be non-negative, and the entries of this vector must sum to 1.		* <tt>P</tt> (optional, default <tt>[1/k, 1/k, ..., 1/k]</tt>, where <tt>k</tt> is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above. All entries must be non-negative, and the entries of this vector must sum to 1.

			===Output arguments===
			* <tt>DIST</tt>: The maximum probability of distinguishing the states specified by <tt>X</tt>.
			* <tt>MEAS</tt> (optional): A cell containing optimal measurement operators that distinguish the states specified by <tt>X</tt> with probability <tt>DIST</tt>.

	==Examples==		==Examples==

Nathaniel: /* Orthogonal states can be perfectly distinguished */

2014-10-02T17:50:56Z

Orthogonal states can be perfectly distinguished

← Older revision		Revision as of 17:50, 2 October 2014
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	==Examples==		==Examples==
	===Orthogonal states can be perfectly distinguished===		===Orthogonal states can be perfectly distinguished===
	Any number of quantum states can be perfectly distinguished (i.e., distinguished with probability 1) if they are mutually orthogonal. The following code ~~general~~ a random $6\times 6$ unitary matrix (i.e., a matrix with orthogonal pure states as columns) and verifies that those pure states are perfectly distinguishable:		Any number of quantum states can be perfectly distinguished (i.e., distinguished with probability 1) if they are mutually orthogonal. The following code generates a random $6\times 6$ unitary matrix (i.e., a matrix with orthogonal pure states as columns) and verifies that those pure states are perfectly distinguishable:
	<syntaxhighlight>		<syntaxhighlight>
	>> Distinguishability(RandomUnitary(6))		>> Distinguishability(RandomUnitary(6))

Nathaniel at 17:49, 2 October 2014

2014-10-02T17:49:21Z

← Older revision		Revision as of 17:49, 2 October 2014
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	==Argument descriptions==		==Argument descriptions==
	* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.		* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.
	* <tt>P</tt> (optional, default <tt>[1/k, 1/k, ..., 1/k]</tt>, where <tt>k</tt> is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above.		* <tt>P</tt> (optional, default <tt>[1/k, 1/k, ..., 1/k]</tt>, where <tt>k</tt> is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above. All entries must be non-negative, and the entries of this vector must sum to 1.

	==Examples==		==Examples==

Nathaniel at 17:48, 2 October 2014

2014-10-02T17:48:23Z

← Older revision		Revision as of 17:48, 2 October 2014
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	==Syntax==		==Syntax==
	* <tt>DIST = Distinguishability(X)</tt>		* <tt>DIST = Distinguishability(X)</tt>
			* <tt>DIST = Distinguishability(X,P)</tt>

	==Argument descriptions==		==Argument descriptions==
	* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.		* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.
			* <tt>P</tt> (optional, default <tt>[1/k, 1/k, ..., 1/k]</tt>, where <tt>k</tt> is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above.

	==Examples==		==Examples==

Nathaniel: Created page with "{{Function |name=Distinguishability |desc=Computes the maximum probability of distinguishing quantum states |req=[http://cvxr.com/cvx/ cvx] |rel=ChannelDistinguishability<..."

2014-10-02T17:13:03Z

Created page with "{{Function |name=Distinguishability |desc=Computes the maximum probability of distinguishing quantum states |req=[http://cvxr.com/cvx/ cvx] |rel=ChannelDistinguishability<..."

New page

{{Function
|name=Distinguishability
|desc=Computes the maximum probability of distinguishing quantum states
|req=[http://cvxr.com/cvx/ cvx]
|rel=[[ChannelDistinguishability]]<br />[[LocalDistinguishability]]
|cat=[[List of functions#Distinguishing_objects|Distinguishing objects]]
|upd=October 2, 2014
|v=0.50}}
<tt>'''Distinguishability'''</tt> is a [[List of functions|function]] that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.

==Syntax==
* <tt>DIST = Distinguishability(X)</tt>

==Argument descriptions==
* <tt>X</tt>: The quantum states to be distinguished. <tt>X</tt> can either be a cell containing 2 or more density matrices, or <tt>X</tt> can be a matrix whose columns are pure vector states.

==Examples==
===Orthogonal states can be perfectly distinguished===
Any number of quantum states can be perfectly distinguished (i.e., distinguished with probability 1) if they are mutually orthogonal. The following code general a random $6\times 6$ unitary matrix (i.e., a matrix with orthogonal pure states as columns) and verifies that those pure states are perfectly distinguishable:
<syntaxhighlight>
>> Distinguishability(RandomUnitary(6))

ans =

1
</syntaxhighlight>

===Two states===
The maximum probability of distinguishing two quantum states $\rho$ and $\sigma$ is exactly $\frac{1}{2} + \frac{1}{4}\|\rho - \sigma\|_1$<ref>John Watrous. [https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/all.pdf Theory of Quantum Information lecture notes], Fall 2011.</ref>, where $\|\cdot\|_1$ is the [[TraceNorm|trace norm]]. We can verify this in a special case as follows:
<syntaxhighlight>
>> rho = RandomDensityMatrix(4);
>> sigma = RandomDensityMatrix(4);
>> Distinguishability({rho, sigma})

ans =

0.7762

>> 1/2 + TraceNorm(rho - sigma)/4

ans =

0.7762
</syntaxhighlight>

===Three or more states===
We can also compute the maximum probability of distinguishing three or more states, but no simple formula is known in this case.
<syntaxhighlight>
>> for j = 1:6
rho{j} = RandomDensityMatrix(4);
end
>> Distinguishability(rho)

ans =

0.4156
</syntaxhighlight>
{{SourceCode|name=Distinguishability}}

==References==
<references />