DiamondNorm - Revision history

Rmmateri at 16:40, 13 June 2018

2018-06-13T16:40:10Z

← Older revision		Revision as of 16:40, 13 June 2018
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	\|name=DiamondNorm		\|name=DiamondNorm
	\|desc=Computes the [[diamond norm]] of a superoperator		\|desc=Computes the [[diamond norm]] of a superoperator
	\|req=[http://cvxr.com/cvx/ ~~cvx~~]		\|req=[http://cvxr.com/cvx/ CVX]
	\|rel=[[CBNorm]]<br />[[InducedSchattenNorm]]<br />[[MaximumOutputFidelity]]		\|rel=[[CBNorm]]<br />[[InducedSchattenNorm]]<br />[[MaximumOutputFidelity]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]

Nathaniel at 00:22, 12 January 2016

2016-01-12T00:22:36Z

← Older revision		Revision as of 00:22, 12 January 2016
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	\|desc=Computes the [[diamond norm]] of a superoperator		\|desc=Computes the [[diamond norm]] of a superoperator
	\|req=[http://cvxr.com/cvx/ cvx]		\|req=[http://cvxr.com/cvx/ cvx]
	\|rel=[[CBNorm]]<br />[[MaximumOutputFidelity]]		\|rel=[[CBNorm]]<br />[[InducedSchattenNorm]]<br />[[MaximumOutputFidelity]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=November 21, 2014		\|upd=November 21, 2014

Nathaniel: added CVX parameter

2014-12-24T16:35:01Z

added CVX parameter

← Older revision		Revision as of 16:35, 24 December 2014
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	\|rel=[[CBNorm]]<br />[[MaximumOutputFidelity]]		\|rel=[[CBNorm]]<br />[[MaximumOutputFidelity]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=November 21, 2014}}		\|upd=November 21, 2014
			\|cvx=yes (convex)}}
	<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the diamond norm $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.		<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the diamond norm $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.

Nathaniel: Simplified example

2014-12-16T18:04:32Z

Simplified example

← Older revision		Revision as of 18:04, 16 December 2014
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	variable Phi(4^num_qubits,4^num_qubits) hermitian;		variable Phi(4^num_qubits,4^num_qubits) hermitian;
	variable p(4^num_qubits);		variable p(4^num_qubits);

	~~% Let's construct a general Pauli channel.~~
	~~PauliLinComb = zeros(4^num_qubits,4^num_qubits);~~
	~~for j = 1:4^num_qubits~~
	~~bitind = bitget(j-1,1:2*num_qubits);~~
	~~ind = bitind(1:2:2num_qubits) + 2bitind(2:2:2*num_qubits);~~
	~~PauliLinComb = PauliLinComb + p(j)*ChoiMatrix({Pauli(ind)});~~
	~~end~~

	minimize DiamondNorm(Phi - Psi)		minimize DiamondNorm(Phi - Psi)

	subject to		subject to
	% these ~~first two~~ constraints force Phi to be a ~~quantum~~ channel		% these constraints force Phi to be a Pauli channel
	~~PartialTrace(~~Phi) == ~~eye~~(~~2^num_qubits~~)~~; % trace-preserving~~		Phi == PauliChannel(p);
	~~Phi >= 0; % completely positive~~

	~~% the following constraints force Phi to have Pauli Kraus operators~~
	~~Phi == PauliLinComb~~;
	sum(p) == 1;		sum(p) == 1;
	p >= 0;		p >= 0;

Nathaniel: /* Can be used within CVX */ corrected Pauli example

2014-11-28T03:22:48Z

Can be used within CVX: corrected Pauli example

← Older revision		Revision as of 03:22, 28 November 2014
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	Phi >= 0; % completely positive		Phi >= 0; % completely positive

	% the following ~~constraint forces~~ Phi to have Pauli Kraus operators		% the following constraints force Phi to have Pauli Kraus operators
	Phi == PauliLinComb;		Phi == PauliLinComb;
	sum(p) == 1;		sum(p) == 1;
			p >= 0;
	cvx_end		cvx_end
	cvx_optval		cvx_optval

Nathaniel: removed red links

2014-11-27T21:32:43Z

removed red links

← Older revision		Revision as of 21:32, 27 November 2014
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	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=November 21, 2014}}		\|upd=November 21, 2014}}
	<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.		<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the diamond norm $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.

	==Syntax==		==Syntax==
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	==Argument descriptions==		==Argument descriptions==
	* <tt>PHI</tt>: A superoperator. Should be provided as either a [[Choi matrix]], or as a cell with either 1 or 2 columns (see the [[tutorial]] page for more details about specifying superoperators within QETLAB). <tt>PHIC</tt> will be a cell of Kraus operators if <tt>PHI</tt> is a cell of Kraus operators, and similarly <tt>PHIC</tt> will be a Choi matrix if <tt>PHI</tt> is a Choi matrix.		* <tt>PHI</tt>: A superoperator. Should be provided as either a Choi matrix, or as a cell with either 1 or 2 columns (see the [[tutorial]] page for more details about specifying superoperators within QETLAB). <tt>PHIC</tt> will be a cell of Kraus operators if <tt>PHI</tt> is a cell of Kraus operators, and similarly <tt>PHIC</tt> will be a Choi matrix if <tt>PHI</tt> is a Choi matrix.
	* <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix.		* <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix.

	==Examples==		==Examples==
	===A completely positive map===		===A completely positive map===
	If $\Phi$ is completely positive then $\\|\Phi\\|_{\diamond} = \\|\Phi^\dagger(I)\\|$, where $I$ is the identity matrix, $\Phi^\dagger$ is the [[dual map]] of $\Phi$, and $\\|\cdot\\|$ is the usual operator norm, which we can verify in a special case via the following code:		If $\Phi$ is completely positive then $\\|\Phi\\|_{\diamond} = \\|\Phi^\dagger(I)\\|$, where $I$ is the identity matrix, $\Phi^\dagger$ is the dual map of $\Phi$, and $\\|\cdot\\|$ is the usual operator norm, which we can verify in a special case via the following code:
	<syntaxhighlight>		<syntaxhighlight>
	>> Phi = {[1 2;3 4] ; [0 1;2 0] ; [1 1;-1 3]};		>> Phi = {[1 2;3 4] ; [0 1;2 0] ; [1 1;-1 3]};

Nathaniel: Updated since it can now be used within CVX optimization problems

2014-11-27T21:31:57Z

Updated since it can now be used within CVX optimization problems

@@ Line 3: / Line 3: @@
 |desc=Computes the [[diamond norm]] of a superoperator
 |req=[http://cvxr.com/cvx/ cvx]
-|rel=[[CBNorm]]
+|rel=[[CBNorm]]<br />[[MaximumOutputFidelity]]
 |cat=[[List of functions#Norms|Norms]]
-|upd=October 6, 2014
+|upd=November 21, 2014}}
 <tt>'''DiamondNorm'''</tt> is a [[List of functions|function]] that computes the [[diamond norm]] $\|\Phi\|_\diamond$ of a superoperator $\Phi$.
@@ Line 58: / Line 57: @@
 .4142
 </syntaxhighlight>
 {{SourceCode|name=DiamondNorm}}

Nathaniel at 17:55, 6 October 2014

2014-10-06T17:55:54Z

← Older revision		Revision as of 17:55, 6 October 2014
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	\|rel=[[CBNorm]]		\|rel=[[CBNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=~~January 22~~, ~~2013~~		\|upd=October 6, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.		<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.

Nathaniel at 16:29, 22 September 2014

2014-09-22T16:29:27Z

← Older revision		Revision as of 16:29, 22 September 2014
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	\|name=DiamondNorm		\|name=DiamondNorm
	\|desc=Computes the [[diamond norm]] of a superoperator		\|desc=Computes the [[diamond norm]] of a superoperator
			\|req=[http://cvxr.com/cvx/ cvx]
	\|rel=[[CBNorm]]		\|rel=[[CBNorm]]
			\|cat=[[List of functions#Norms\|Norms]]
	\|upd=January 22, 2013		\|upd=January 22, 2013
	\|v=1.00}}		\|v=0.50}}
	<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.		<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.

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	===A completely positive map===		===A completely positive map===
	If $\Phi$ is completely positive then $\\|\Phi\\|_{\diamond} = \\|\Phi^\dagger(I)\\|$, where $I$ is the identity matrix, $\Phi^\dagger$ is the [[dual map]] of $\Phi$, and $\\|\cdot\\|$ is the usual operator norm, which we can verify in a special case via the following code:		If $\Phi$ is completely positive then $\\|\Phi\\|_{\diamond} = \\|\Phi^\dagger(I)\\|$, where $I$ is the identity matrix, $\Phi^\dagger$ is the [[dual map]] of $\Phi$, and $\\|\cdot\\|$ is the usual operator norm, which we can verify in a special case via the following code:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> Phi = {[1 2;3 4] ; [0 1;2 0] ; [1 1;-1 3]};		>> Phi = {[1 2;3 4] ; [0 1;2 0] ; [1 1;-1 3]};
	>> DiamondNorm(Phi)		>> DiamondNorm(Phi)
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	37.6510		37.6510

	>> norm([[ApplyMap]](eye(2),~~[[DualMap\|~~DualMap(Phi)]]))		>> norm(ApplyMap(eye(2),DualMap(Phi)))

	ans =		ans =

	37.6510		37.6510
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	===A difference of unitaries channel===		===A difference of unitaries channel===
	If $\Phi(X) = X - UXU^\dagger$, then the diamond norm of $\Phi$ is the diameter of the smallest circle that contains the eigenvalues of $U$, which we can verify in a special case via the following code:		If $\Phi(X) = X - UXU^\dagger$, then the diamond norm of $\Phi$ is the diameter of the smallest circle that contains the eigenvalues of $U$, which we can verify in a special case via the following code:
	<~~pre~~>		<syntaxhighlight>
	>> U = [1 1;-1 1]/sqrt(2);		>> U = [1 1;-1 1]/sqrt(2);
	>> Phi = {eye(2),eye(2); U,-U};		>> Phi = {eye(2),eye(2); U,-U};
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	1.4142		1.4142
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=DiamondNorm}}

Nathaniel at 20:54, 22 January 2013

2013-01-22T20:54:23Z

← Older revision		Revision as of 20:54, 22 January 2013
Line 5:		Line 5:
	\|upd=January 22, 2013		\|upd=January 22, 2013
	\|v=1.00}}		\|v=1.00}}
	<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] of a superoperator.		<tt>'''DiamondNorm'''</tt> is a [[List of functions\|function]] that computes the [[diamond norm]] $\\|\Phi\\|_\diamond$ of a superoperator $\Phi$.

	==Syntax==		==Syntax==