Twirl - Revision history

Nathaniel: /* Argument descriptions */

2014-11-28T01:51:46Z

Argument descriptions

Nathaniel: Added Pauli twirling

2014-11-28T01:51:05Z

Added Pauli twirling

Nathaniel: Updated to v0.51 (since the BrauerStates function changed)

2014-11-12T17:25:59Z

Updated to v0.51 (since the BrauerStates function changed)

Nathaniel at 19:49, 6 November 2014

2014-11-06T19:49:00Z

Nathaniel: Now supports multipartite real orthogonal twirling

2014-11-06T19:48:46Z

Now supports multipartite real orthogonal twirling

Nathaniel at 15:36, 29 September 2014

2014-09-29T15:36:37Z

Nathaniel: Created page with "{{Function |name=Twirl |desc=Twirls a bipartite or multipartite operator |rel=IsotropicState
RandomUnitary
WernerState |upd=September 19, 2014 |v=1.00}} ..."

2014-09-19T12:49:10Z

Created page with "{{Function |name=Twirl |desc=Twirls a bipartite or multipartite operator |rel=IsotropicState<br />RandomUnitary<br />WernerState |upd=September 19, 2014 |v=1.00}} ..."

New page

{{Function
|name=Twirl
|desc=Twirls a bipartite or multipartite operator
|rel=[[IsotropicState]]<br />[[RandomUnitary]]<br />[[WernerState]]
|upd=September 19, 2014
|v=1.00}}
<tt>'''Twirl'''</tt> is a [[List of functions|function]] that twirls an operator. That is, it implements the following superoperator:

<math>X \mapsto \int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU,</math>

where integration is performed with respect to [http://en.wikipedia.org/wiki/Haar_measure Haar measure] on the unitary group. Multipartite twirling can also be performed, as can some other related twirls (in particular, twirling over the real orthogonal group and isotropic twirling). The output of this function is always sparse.

==Syntax==
* <tt>TX = Twirl(X)</tt>
* <tt>TX = Twirl(X,TYPE)</tt>
* <tt>TX = Twirl(X,TYPE,P)</tt>

==Argument descriptions==
* <tt>X</tt>: A square operator to have its twirl computed.
* <tt>TYPE</tt> (optional, by default <tt>'werner'</tt>): A string indicating what type of twirl should be performed. Can equal one of the following three values:
** If <tt>TYPE = 'werner'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices. Can also perform the natural multipartite generalization of the above integral (see the optional parameter <tt>P</tt>).
** If <tt>TYPE = 'isotropic'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes \overline{U})X(U \otimes \overline{U})^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices.
** If <tt>TYPE = 'real'</tt> then the twirl performed is:<math>X \mapsto \int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO,</math>where integration is performed with respect to Haar measure over the group of real orthogonal matrices.
* <tt>P</tt> (optional, by default <tt>2</tt>): The number of parties that <tt>X</tt> acts on. That is, $X \in M_n^{\otimes P}$ for some $n$. Must equal <tt>2</tt> unless <tt>TYPE = 'werner'</tt>.

==Examples==
===Werner twirling===
Werner twirling a quantum state always results in a [[WernerState|Werner state]]. More specifically, in the bipartite case we have the simple formula

<center><math>\displaystyle\int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU = \frac{\mathrm{Tr}\big( P_S X \big)}{\binom{d+1}{2}}P_S + \frac{\mathrm{Tr}\big(P_A X\big)}{\binom{d}{2}}P_A,</math></center>

where $P_S$ is the [[SymmetricProjection|projection onto the symmetric subspace]] and $P_A$ is the [[AntisymmetricProjection|projection onto the antisymmetric subspace]]. We can see this fact numerically as follows:
<pre<noinclude></noinclude>>
>> d = 2;
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
>> PS = [[SymmetricProjection|SymmetricProjection(2)]];
>> PA = [[AntisymmetricProjection|AntisymmetricProjection(2)]];
>> full(Twirl(rho))

ans =

0.2486 0 0 0
0 0.2514 -0.0029 0
0 -0.0029 0.2514 0
0 0 0 0.2486

>> full(trace(PS*rho)*PS/nchoosek(d+1,2) + trace(PA*rho)*PA/nchoosek(d,2))

ans =

0.2486 0 0 0
0 0.2514 -0.0029 0
0 -0.0029 0.2514 0
0 0 0 0.2486
</pre>

Multipartite twirling does not have such a simple formula. Nonetheless, the following code demonstrates the effect of twirling a 3-qubit state <tt>X</tt> using the <tt>Twirl</tt> function versus approximately twirling it by generating many [[RandomUnitary|Haar-uniform random unitary matrices]].
<pre<noinclude></noinclude>>
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(8)]]; % random 3-qubit density matrix
>> TX = Twirl(rho,'werner',3);
>> s = 1000000; % we will now compute an approximate twirl using this many unitary matrices
approx_twirl = zeros(8);
for j = 1:s
U = [[RandomUnitary|RandomUnitary(2)]];
approx_twirl = approx_twirl + [[Tensor|Tensor(U,U,U)]]*rho*Tensor(U,U,U)';
end
>> norm(TX - approx_twirl/s) % TX is the twirl as computed by the Twirl function, approx_twirl/s is the approximate twirl computed by many random unitaries

ans =

1.6331e-04
</pre>

===Isotropic twirling===
Isotropic twirling a quantum state always results in an [[IsotropicState|isotropic state]]. More specifically, if $|\psi_+\rangle$ is the [[MaxEntangled|standard maximally-entangled pure state]] then we have

<center><math>\displaystyle\int_{U(d)} (U \otimes \overline{U})X(U \otimes \overline{U})^\dagger \, dU = \big( \langle\psi_+|X|\psi_+\rangle \big)|\psi_+\rangle\langle\psi_+| + \frac{\mathrm{Tr}\big((I - |\psi_+\rangle\langle\psi_+|) X\big)}{d^2-1}(I - |\psi_+\rangle\langle\psi_+|).</math></center>

We can see this fact numerically as follows:
<pre<noinclude></noinclude>>
>> d = 2;
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
>> psi = [[MaxEntangled|MaxEntangled(d)]];
>> full(Twirl(rho,'isotropic'))

ans =

0.2405 0 0 -0.0191
0 0.2595 0 0
0 0 0.2595 0
-0.0191 0 0 0.2405

>> (psi'*rho*psi)*(psi*psi') + trace((eye(d^2)-psi*psi')*rho)*(eye(d^2)-psi*psi')/(d^2-1)

ans =

0.2405 0 0 -0.0191
0 0.2595 0 0
0 0 0.2595 0
-0.0191 0 0 0.2405
</pre>

===Real orthogonal twirling===
Twirling a quantum state by real orthogonal matrices always results in a linear combination of an isotropic state and a Werner state (equivalently, a state in the 3-dimensional space spanned by the [[SymmetricProjection|projection onto the symmetric subspace]], the [[AntisymmetricProjection|projection onto the antisymmetric subspace]], and the [[MaxEntangled|standard maximally-entangled pure state]]). More specifically, the following formula holds:

<center><math>\displaystyle\int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO = \big( \langle\psi_+|X|\psi_+\rangle \big)|\psi_+\rangle\langle\psi_+| + \frac{\mathrm{Tr}\big(P_A X\big)}{\binom{d}{2}}P_A + \frac{\mathrm{Tr}\big( (P_S-|\psi_+\rangle\langle\psi_+|) X \big)}{\binom{d+1}{2}-1}(P_S - |\psi_+\rangle\langle\psi_+|).</math></center>

The following code generates the real orthogonal twirl of a random state in two different ways: first by using the Twirl function, and then by constructing 1000000 random (according to Haar measure) orthogonal matrices and averaging the value of $(O \otimes O)X(O \otimes O)^\dagger$ over them:
<pre<noinclude></noinclude>>
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(4)]];
>> full(Twirl(rho,'real'))

ans =

0.2219 0 0 0.0517
0 0.2781 -0.1079 0
0 -0.1079 0.2781 0
0.0517 0 0 0.2219

>> s = 1000000; % we will now compute an approximate twirl using this many unitary matrices
approx_twirl = zeros(4);
for j = 1:s
U = [[RandomUnitary|RandomUnitary(2,1)]]; % the second parameter being 1 ensures that this is an orthogonal matrix
approx_twirl = approx_twirl + kron(U,U)*rho*kron(U,U)';
end
>> approx_twirl/s

ans =

0.2220 + 0.0000i 0.0000 - 0.0001i 0.0002 + 0.0001i 0.0517 - 0.0000i
0.0000 + 0.0001i 0.2781 + 0.0000i -0.1079 - 0.0001i 0.0001 - 0.0000i
0.0002 - 0.0001i -0.1079 + 0.0001i 0.2780 + 0.0000i -0.0001 + 0.0000i
0.0517 + 0.0000i 0.0001 + 0.0000i -0.0001 - 0.0000i 0.2219 + 0.0000i
</pre>

{{SourceCode|name=Twirl}}

@@ Line 21: / Line 21: @@
 ** If <tt>TYPE = 'werner'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices. Can also perform the natural multipartite generalization of the above integral (see the optional parameter <tt>P</tt>).
 ** If <tt>TYPE = 'isotropic'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes \overline{U})X(U \otimes \overline{U})^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices.
-** If <tt>TYPE = 'real'</tt> then the twirl performed is:<math>X \mapsto \int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO,</math>where integration is performed with respect to Haar measure over the group of real orthogonal matrices.
+** If <tt>TYPE = 'real'</tt> then the twirl performed is:<math>X \mapsto \int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO,</math>where integration is performed with respect to Haar measure over the group of real orthogonal matrices. Can also perform the natural multipartite generalization of the above integral (see the optional parameter <tt>P</tt>).
 ** If <tt>TYPE = 'pauli'</tt> then the twirl performed is:<math>X \mapsto \frac{1}{4^q}\sum_{\text{Paulis } Q}(Q \otimes Q)X(Q \otimes Q)^\dagger,</math>where $X \in M_{2^q} \otimes M_{2^q} \cong M_{4^q}$ for some $q \geq 1$.
 * <tt>P</tt> (optional, by default <tt>2</tt>): The number of parties that <tt>X</tt> acts on. That is, $X \in M_n^{\otimes P}$ for some $n$. Must equal <tt>2</tt> if <tt>TYPE = 'isotropic'</tt> or if <tt>TYPE = 'pauli'</tt>.

@@ Line 2: / Line 2: @@
 |name=Twirl
 |desc=Twirls a bipartite or multipartite operator
-|rel=[[IsotropicState]]<br />[[RandomUnitary]]<br />[[WernerState]]
+|rel=[[IsotropicState]]<br />[[Pauli]]<br />[[RandomUnitary]]<br />[[WernerState]]
 |cat=[[List of functions#Superoperators|Superoperators]]
-|upd=November 12, 2014
+|upd=November 27, 2014}}
-|v=0.51}}
+<tt>'''Twirl'''</tt> is a [[List of functions|function]] that twirls an operator. That is, it implements a superoperator like the following one:
 <math>X \mapsto \int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU,</math>
-where integration is performed with respect to [http://en.wikipedia.org/wiki/Haar_measure Haar measure] on the unitary group. Multipartite twirling can also be performed, as can some other related twirls (in particular, twirling over the real orthogonal group and isotropic twirling). The output of this function is always sparse.
+where integration is performed with respect to [http://en.wikipedia.org/wiki/Haar_measure Haar measure] on the unitary group. Multipartite twirling can also be performed, as can some other related twirls (in particular, twirling over the real orthogonal group, isotropic twirling, and Pauli twirling). The output of this function is always sparse.
 ==Syntax==
@@ Line 19: / Line 18: @@
 ==Argument descriptions==
 * <tt>X</tt>: A square operator to have its twirl computed.
-* <tt>TYPE</tt> (optional, by default <tt>'werner'</tt>): A string indicating what type of twirl should be performed. Can equal one of the following three values:
+* <tt>TYPE</tt> (optional, by default <tt>'werner'</tt>): A string indicating what type of twirl should be performed. Can equal one of the following four values:
 ** If <tt>TYPE = 'werner'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes U)X(U \otimes U)^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices. Can also perform the natural multipartite generalization of the above integral (see the optional parameter <tt>P</tt>).
 ** If <tt>TYPE = 'isotropic'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes \overline{U})X(U \otimes \overline{U})^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices.
 ** If <tt>TYPE = 'real'</tt> then the twirl performed is:<math>X \mapsto \int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO,</math>where integration is performed with respect to Haar measure over the group of real orthogonal matrices.
-* <tt>P</tt> (optional, by default <tt>2</tt>): The number of parties that <tt>X</tt> acts on. That is, $X \in M_n^{\otimes P}$ for some $n$. Must equal <tt>2</tt> if <tt>TYPE = 'isotropic'</tt>.
+** If <tt>TYPE = 'pauli'</tt> then the twirl performed is:<math>X \mapsto \frac{1}{4^q}\sum_{\text{Paulis } Q}(Q \otimes Q)X(Q \otimes Q)^\dagger,</math>where $X \in M_{2^q} \otimes M_{2^q} \cong M_{4^q}$ for some $q \geq 1$.
 ==Examples==
@@ Line 163: / Line 163: @@
    -0.0091 - 0.0094i   0.1525 - 0.0000i   0.0000 + 0.0000i
 .0000 + 0.0000i   0.0000 + 0.0000i   0.1194 + 0.0000i
 </syntaxhighlight>
 {{SourceCode|name=Twirl}}

@@ Line 4: / Line 4: @@
 |rel=[[IsotropicState]]<br />[[RandomUnitary]]<br />[[WernerState]]
 |cat=[[List of functions#Superoperators|Superoperators]]
-|upd=November 6, 2014
+|upd=November 12, 2014
-|v=0.50}}
+|v=0.51}}
 <tt>'''Twirl'''</tt> is a [[List of functions|function]] that twirls an operator. That is, it implements the following superoperator:

@@ Line 4: / Line 4: @@
 |rel=[[IsotropicState]]<br />[[RandomUnitary]]<br />[[WernerState]]
 |cat=[[List of functions#Superoperators|Superoperators]]
-|upd=September 19, 2014
+|upd=November 6, 2014
 |v=0.50}}
 <tt>'''Twirl'''</tt> is a [[List of functions|function]] that twirls an operator. That is, it implements the following superoperator:

@@ Line 23: / Line 23: @@
 ** If <tt>TYPE = 'isotropic'</tt> then the twirl performed is:<math>X \mapsto \int_{U(d)} (U \otimes \overline{U})X(U \otimes \overline{U})^\dagger \, dU,</math>where integration is performed with respect to Haar measure over the group of unitary matrices.
 ** If <tt>TYPE = 'real'</tt> then the twirl performed is:<math>X \mapsto \int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO,</math>where integration is performed with respect to Haar measure over the group of real orthogonal matrices.
-* <tt>P</tt> (optional, by default <tt>2</tt>): The number of parties that <tt>X</tt> acts on. That is, $X \in M_n^{\otimes P}$ for some $n$. Must equal <tt>2</tt> unless <tt>TYPE = 'werner'</tt>.
+* <tt>P</tt> (optional, by default <tt>2</tt>): The number of parties that <tt>X</tt> acts on. That is, $X \in M_n^{\otimes P}$ for some $n$. Must equal <tt>2</tt> if <tt>TYPE = 'isotropic'</tt>.
 ==Examples==
@@ Line 56: / Line 56: @@
 </syntaxhighlight>
-Multipartite twirling does not have such a simple formula. Nonetheless, the following code demonstrates the effect of twirling a 3-qubit state <tt>X</tt> using the <tt>Twirl</tt> function versus approximately twirling it by generating many [[RandomUnitary|Haar-uniform random unitary matrices]].
+Multipartite Werner twirling does not have such a simple formula. Nonetheless, the following code demonstrates the effect of twirling a 3-qubit state <tt>X</tt> using the <tt>Twirl</tt> function versus approximately twirling it by generating many [[RandomUnitary|Haar-uniform random unitary matrices]].
 <syntaxhighlight>
 >> rho = RandomDensityMatrix(8); % random 3-qubit density matrix
@@ Line 103: / Line 103: @@
 ===Real orthogonal twirling===
-Twirling a quantum state by real orthogonal matrices always results in a linear combination of an isotropic state and a Werner state (equivalently, a state in the 3-dimensional space spanned by the [[SymmetricProjection|projection onto the symmetric subspace]], the [[AntisymmetricProjection|projection onto the antisymmetric subspace]], and the [[MaxEntangled|standard maximally-entangled pure state]]). More specifically, the following formula holds:
+Twirling a bipartite quantum state by real orthogonal matrices always results in a linear combination of an isotropic state and a Werner state (equivalently, a state in the 3-dimensional space spanned by the [[SymmetricProjection|projection onto the symmetric subspace]], the [[AntisymmetricProjection|projection onto the antisymmetric subspace]], and the [[MaxEntangled|standard maximally-entangled pure state]]). More specifically, the following formula holds:
 <center><math>\displaystyle\int_{O(d)} (O \otimes O)X(O \otimes O)^\dagger \, dO = \big( \langle\psi_+|X|\psi_+\rangle \big)|\psi_+\rangle\langle\psi_+| + \frac{\mathrm{Tr}\big(P_A X\big)}{\binom{d}{2}}P_A + \frac{\mathrm{Tr}\big( (P_S-|\psi_+\rangle\langle\psi_+|) X \big)}{\binom{d+1}{2}-1}(P_S - |\psi_+\rangle\langle\psi_+|).</math></center>
@@ Line 133: / Line 133: @@
 .0002 - 0.0001i  -0.1079 + 0.0001i   0.2780 + 0.0000i  -0.0001 + 0.0000i
 .0517 + 0.0000i   0.0001 + 0.0000i  -0.0001 - 0.0000i   0.2219 + 0.0000i
 </syntaxhighlight>
 {{SourceCode|name=Twirl}}

@@ Line 3: / Line 3: @@
 |desc=Twirls a bipartite or multipartite operator
 |rel=[[IsotropicState]]<br />[[RandomUnitary]]<br />[[WernerState]]
 |upd=September 19, 2014
-|v=1.00}}
+|v=0.50}}
 <tt>'''Twirl'''</tt> is a [[List of functions|function]] that twirls an operator. That is, it implements the following superoperator:
@@ Line 31: / Line 32: @@
 where $P_S$ is the [[SymmetricProjection|projection onto the symmetric subspace]] and $P_A$ is the [[AntisymmetricProjection|projection onto the antisymmetric subspace]]. We can see this fact numerically as follows:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> d = 2;
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
+>> rho = RandomDensityMatrix(d^2);
->> PS = [[SymmetricProjection|SymmetricProjection(2)]];
+>> PS = SymmetricProjection(2);
->> PA = [[AntisymmetricProjection|AntisymmetricProjection(2)]];
+>> PA = AntisymmetricProjection(2);
 >> full(Twirl(rho))
@@ Line 53: / Line 54: @@
    -0.0029    0.2514         0
          0         0    0.2486
-</pre>
+</syntaxhighlight>
 Multipartite twirling does not have such a simple formula. Nonetheless, the following code demonstrates the effect of twirling a 3-qubit state <tt>X</tt> using the <tt>Twirl</tt> function versus approximately twirling it by generating many [[RandomUnitary|Haar-uniform random unitary matrices]].
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(8)]]; % random 3-qubit density matrix
+>> rho = RandomDensityMatrix(8); % random 3-qubit density matrix
 >> TX = Twirl(rho,'werner',3);
 >> s = 1000000; % we will now compute an approximate twirl using this many unitary matrices
     approx_twirl = zeros(8);
     for j = 1:s
-        U = [[RandomUnitary|RandomUnitary(2)]];
+        U = RandomUnitary(2);
-        approx_twirl = approx_twirl + [[Tensor|Tensor(U,U,U)]]*rho*Tensor(U,U,U)';
+        approx_twirl = approx_twirl + Tensor(U,U,U)*rho*Tensor(U,U,U)';
     end
 >> norm(TX - approx_twirl/s) % TX is the twirl as computed by the Twirl function, approx_twirl/s is the approximate twirl computed by many random unitaries
@@ Line 70: / Line 71: @@
 .6331e-04
-</pre>
+</syntaxhighlight>
 ===Isotropic twirling===
@@ Line 78: / Line 79: @@
 We can see this fact numerically as follows:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> d = 2;
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
+>> rho = RandomDensityMatrix(d^2);
->> psi = [[MaxEntangled|MaxEntangled(d)]];
+>> psi = MaxEntangled(d);
 >> full(Twirl(rho,'isotropic'))
@@ Line 99: / Line 100: @@
          0    0.2595         0
     -0.0191         0         0    0.2405
-</pre>
+</syntaxhighlight>
 ===Real orthogonal twirling===
@@ Line 107: / Line 108: @@
 The following code generates the real orthogonal twirl of a random state in two different ways: first by using the Twirl function, and then by constructing 1000000 random (according to Haar measure) orthogonal matrices and averaging the value of $(O \otimes O)X(O \otimes O)^\dagger$ over them:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(4)]];
+>> rho = RandomDensityMatrix(4);
 >> full(Twirl(rho,'real'))
@@ Line 121: / Line 122: @@
     approx_twirl = zeros(4);
     for j = 1:s
-        U = [[RandomUnitary|RandomUnitary(2,1)]]; % the second parameter being 1 ensures that this is an orthogonal matrix
+        U = RandomUnitary(2,1); % the second parameter being 1 ensures that this is an orthogonal matrix
         approx_twirl = approx_twirl + kron(U,U)*rho*kron(U,U)';
     end
@@ Line 132: / Line 133: @@
 .0002 - 0.0001i  -0.1079 + 0.0001i   0.2780 + 0.0000i  -0.0001 + 0.0000i
 .0517 + 0.0000i   0.0001 + 0.0000i  -0.0001 - 0.0000i   0.2219 + 0.0000i
-</pre>
+</syntaxhighlight>
 {{SourceCode|name=Twirl}}