Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumEntanglementMonotone  |
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The following measures or metrics are supported:
"Concurrence" | the generalized concurrence for two-qudits | |
"EntanglementEntropy" | the von-Neumann entropy of reduced subsystems for pure states | |
"LogNegativity" | the logarithmic negativity (Peres-Horodecki criterion) | |
"Negativity" | the negativity (Peres-Horodecki criterion) | |
{"RenyiEntanglementEntropy",α}or{"RenyiEntropy",α} | the Renyiα-entropy | |
"Realignment" | the realignment metric, which is total of the list of nonzero singular values of realiged density matrix | |
"ConcurrenceVector" | the generalized concurrence vector |
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The bipartition list has the form . For example, divides the state into subsystem and subsystem . If no bipartition list is specified, , will be performed first, which divides the system into first subsystem, and others (i.e. ).
bipart
{{,,…},{,,…}}
i
1
i
2
j
1
j
2
{{1,2},{3}}
{1,2}
{3}
bipart
qs["Bipartition"]
{{1},{2,3,...}}
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The measure or metric is the concurrence, by default, unless specified otherwise.
t
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For concurrence, we use the Rungta-Buzek-Caves-Hillery-Milburn generalization of the Hill-Wootters concurrence, applicable to any dimensions. It is exact for two qubits and for pure states of any dimension, and a lower bound on the convex-roof concurrence for a higher-dimensional mixed state, where (unlike the two-qubit case) it is frame-dependent and not invariant under local unitaries. This generalized formula is the norm of concurrence vector, where each element of the concurrence vector is given by with the singular values of where are the set of positive roots of Lie algebra for each bipartite. Note that the concurrence vector is (-1)×(-1)dimensional vector, with the dimension of first bipartite, and of the second one. When the state is pure, the norm of concurrence vector is the same as with =[] is the reduced density matrix across the bipartitioned state .
Max[0,-]
σ
1
∑
i=2
σ
i
σ
i
ω=
ρ
λ
α
λ
β
*
ρ
λ
α
λ
β
λ
j
d
1
d
1
2
d
2
d
2
2
d
1
d
2
2(1-Tr[]
2
ρ
rbp
ρ
rbp
Tr
A
ρ
A,B
ρ
A,B
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If no α is given for the RenyiEntanglementEntropy, it is automatically set as .
α=1/2
Examples
(13)
Basic Examples
(5)
Compute the concurrence of a quantum state:
In[1]:=
 | QuantumEntanglementMonotone |
 | QuantumState |
Out[1]=
0.96
Compute entanglement entropy:
In[1]:=
| QuantumEntanglementMonotone |
| QuantumState |
Out[1]=
0.942683
b
Compute negativity:
In[1]:=
| QuantumEntanglementMonotone |
| QuantumState |
Out[1]=
0.48
Compute logarithmic negativity:
In[1]:=
| QuantumEntanglementMonotone |
| QuantumState |
Out[1]=
0.970854
Compute Renyi α-entropy with :
α=0.25
In[1]:=
| QuantumEntanglementMonotone |
| QuantumState |
Out[1]=
0.985339
Scope
(3)
Generalizations & Extensions
(1)
Applications
(3)
Properties & Relations
(1)
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