Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumDistance  |
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returns the distance between two input states, a number for numeric input, a quantity in bits for , and a object for beyond a single qubit. In the following distance measures t are supported:
"RelativeEntropy"
"Bloch"
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"Fidelity" | Distance between two states in terms of fidelity | |
"Trace" | Trace distance between two states | |
"Bures" | Bures distance between two states | |
"BuresAngle" | Bures angle between two states | |
"HilbertSchmidt" | Hilbert-Schmidt distance | |
"Bloch" | Half the Euclidean distance between two Bloch vectors (single qubit only) | |
"RelativeEntropy" | Relative von Neumann entropy between two states | |
"RelativePurity" | 1-Tr[ ρ 1 ρ 2 |
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If no measure t is given, is used.
"Fidelity"
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The distance based on fidelity is calculated as with the corresponding density matrices and the fidelity. The trace distance is Tr[]. The Bures distance is , and Bures angle is . The Hilbert-Schmidt distance is the of the difference between density matrices . The Bloch distance is half the between two Bloch vectors. And the relative entropy is , returned as a quantity in bits.
1-Tr[..]
1/2
ρ
1
ρ
2
1/2
ρ
1
ρ
1,2
Tr[..]
1/2
ρ
1
ρ
2
1/2
ρ
1
1
2
2
(-)
ρ
1
ρ
2
2(1-fidelity)
Tr[]
2
(-)
ρ
1
ρ
2
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The measures are defined for density matrices, which are Hermitian and positive semidefinite with unit trace. An input whose trace is a positive number other than 1, such as a state vector whose squared norm is not 1, is divided by its trace first. An exact positive trace other than 1 issues a message; a machine-precision trace is divided in any case, and issues the message when it is more than from 1. A symbolic input is rescaled only when its trace evaluates to a number, and is otherwise used as given; gives the state it stands for. An input whose trace is 0, negative, or not real, or a numeric input that, once divided by its trace, is not Hermitian (tested exactly when its entries, once divided, are exact rational or complex rational numbers, and otherwise by with a tolerance of ) or has a machine-precision eigenvalue below , issues a message. The entries of an exact input multiplied by an irrational number, as for a matrix of rational entries, need not be rational once divided by the trace, and such an input is then tested with the tolerance. A machine-precision trace counts as 0 when its absolute value is at most times the Frobenius norm of the numeric entries of the matrix, as positive when its real part exceeds, and its imaginary part is at most, times the larger of that norm and the absolute value of the trace, and otherwise as negative or not real. A machine-precision state vector whose squared norm underflows to 0 counts as the zero vector. An exact trace whose sign cannot be decided, numerically or, for an algebraic trace, with , is used as given, and a matrix with a symbolic entry is checked only through its trace.
QuantumDistance::notnormalized
-8
10
qs["Normalized"]
-8
10
-
-8
10
QuantumDistance::notphysical
ρ/
2
ρ
-8
10
-8
10
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The distance is defined for a single qubit, whose Bloch vectors give the trace distance; for any other dimension it returns a object.
"Bloch"
Examples
(14)
Basic Examples
(5)
Find the distance between two pure states in terms of fidelity:
In[1]:=
 | QuantumDistance |
 | QuantumState |
| QuantumState |
Out[1]=
1
Find the distance between two mixed states in terms of fidelity:
In[1]:=
| QuantumDistance |
| QuantumState |
| QuantumState |
Out[1]=
1--
3
2
2
1
2
2
Find the distance between a pure state and a mixed state:
In[1]:=
| QuantumDistance |
| QuantumState |
| QuantumState |
Out[1]=
1
2
Find the distance between two quantum states using the trace metric:
In[1]:=
| QuantumDistance |
| QuantumState |
| QuantumState |
Out[1]=
1
2
2
Find the distance between a pure and mixed state in terms of the Bures angle:
In[1]:=
| QuantumDistance |
| QuantumState |
| QuantumState |
Out[1]=
ArcCos
1
3
Scope
(3)
Applications
(1)
Properties & Relations
(2)
Possible Issues
(3)
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