Realignment
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| Realignment | |
| Computes the realignment of a bipartite operator | |
| Other toolboxes required | none |
|---|---|
| Related functions | PartialTranspose PermuteSystems Swap |
| Function category | Superoperators |
Realignment is a function that computes the realignment of a bipartite operator.
Syntax
- RX = Realignment(X)
- RX = Realignment(X,DIM)
Argument descriptions
- X: A bipartite operator to have its realignment computed.
- DIM (optional, by default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that X lives on. DIM can be provided in one of three ways:
- If DIM is a scalar, it is assumed that the first subsystem has dimension DIM and the second subsystem has dimension length(X)/DIM.
- If $X \in M_{n_1} \otimes M_{n_2}$ then DIM should be a row vector containing the dimensions (i.e., DIM = [n_1, n_2]).
- If the subsystems aren't square (i.e., $X \in M_{m_1, n_1} \otimes M_{m_2, n_2}$) then DIM should be a matrix with two rows. The first row of DIM should contain the row dimensions of the subsystems (i.e., the mi's) and its second row should contain the column dimensions (i.e., the ni's). In other words, you should set DIM = [m_1, m_2; n_1, n_2].
Examples
A two-qubit example
When viewed as a map on block matrices, the realignment map takes each block of the original matrix and makes its vectorization the rows of the realignment matrix. This is illustrated by the following small example:
>> X = reshape(1:16,4,4)'
X =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
>> Realignment(X)
ans =
1 2 5 6
3 4 7 8
9 10 13 14
11 12 15 16A non-square example
The realignment map sends $|i\rangle\langle j| \otimes |k\rangle\langle \ell|$ to $|i\rangle\langle k| \otimes |j\rangle\langle \ell|$. Thus it changes the dimensions of matrices if the subsystems aren't square and of the same size. The following code computes the realignment of an operator $X \in M_{5,2} \otimes M_{3,7}$:
>> X = reshape(1:210,14,15)'
X =
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23 24 25 26 27 28
29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98
99 100 101 102 103 104 105 106 107 108 109 110 111 112
113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150 151 152 153 154
155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182
183 184 185 186 187 188 189 190 191 192 193 194 195 196
197 198 199 200 201 202 203 204 205 206 207 208 209 210
>> Realignment(X,[5,3;2,7])
ans =
1 2 3 4 5 6 7 15 16 17 18 19 20 21 29 30 31 32 33 34 35
8 9 10 11 12 13 14 22 23 24 25 26 27 28 36 37 38 39 40 41 42
43 44 45 46 47 48 49 57 58 59 60 61 62 63 71 72 73 74 75 76 77
50 51 52 53 54 55 56 64 65 66 67 68 69 70 78 79 80 81 82 83 84
85 86 87 88 89 90 91 99 100 101 102 103 104 105 113 114 115 116 117 118 119
92 93 94 95 96 97 98 106 107 108 109 110 111 112 120 121 122 123 124 125 126
127 128 129 130 131 132 133 141 142 143 144 145 146 147 155 156 157 158 159 160 161
134 135 136 137 138 139 140 148 149 150 151 152 153 154 162 163 164 165 166 167 168
169 170 171 172 173 174 175 183 184 185 186 187 188 189 197 198 199 200 201 202 203
176 177 178 179 180 181 182 190 191 192 193 194 195 196 204 205 206 207 208 209 210Source code
Click here to view this function's source code on github.