Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumState  |
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The following native properties (those distinguishing QuantumState from QuantumBasis) are the most commonly accessed via state["PropertyName"].
"Amplitudes" | association of basis-name amplitude, including zeros | |
"Amplitude" | sparse association of basis-name non-zero amplitude | |
"StateVector" | the state as a 1D SparseArray of complex amplitudes | |
"DensityMatrix" | the state as a density matrix ρ | |
"Formula" | typeset Dirac-notation expression of the state | |
"Norm" | norm of the state vector or trace of the density matrix | |
"TraceNorm" | Schatten 1-norm, sum of singular values of ρ | |
"VonNeumannEntropy" | Von Neumann entropy −Tr[ρlogρ] | |
"LogicalEntropy" | 1−Tr[ 2 ρ | |
"Purity" | Tr[ 2 ρ | |
"PureStateQ" | True if the state is pure ( Type==="Pure" | |
"MixedStateQ" | True if the state is mixed ( Type==="Mixed" | |
"Probabilities" | association of basis-name probability, including zeros | |
"Probability" | sparse association of basis-name non-zero probability | |
"BlochCartesianCoordinates" | {x,y,z} | |
"BlochSphericalCoordinates" | {r,θ,ϕ} | |
"BlochPlot" | Graphics3D of the Bloch vector on the unit sphere (qubit-only; alias: BlochSpherePlot) | |
"Eigenvalues" | eigenvalues of the density matrix | |
"Eigenvectors" | eigenvectors of the density matrix | |
"Eigenstates" | list of eigenstates as QuantumState objects | |
"SchmidtBasis" | QuantumState in the Schmidt basis ∑ λ k u k v k | |
"SchmidtDecompose" | symbolic ∑ p i U i V i | |
"Bend" | flatten ρ | |
"Unbend" | inverse of Bend; recover the density-matrix state | |
"Purify" | qs ["Bend"] | |
"Unpurify" | 2 (qs["Unbend"]) | |
"Projector" | projector |ψ〉〈ψ| | |
"Operator" | QuantumOperator corresponding to the projector |
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See Properties & Relations for the complete native-property ladder with worked examples.
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The following named states are supported:
"Plus" | the normalized eigenstate of Pauli-X with +1 eigenvalue | |
"Minus" | the normalized eigenstate of Pauli-X with −1 eigenvalue | |
"Left" | the normalized eigenstate of Pauli-Y with −1 eigenvalue | |
"Right" | the normalized eigenstate of Pauli-Y with +1 eigenvalue | |
"PhiPlus" | the Bell state |Φ+〉=(|00〉+|11〉)/ 2 | |
"PhiMinus" | the Bell state |Φ-〉=(|00〉−|11〉)/ 2 | |
"PsiPlus" | the Bell state |ψ+〉=(|01〉+|10〉)/ 2 | |
"PsiMinus" | the Bell state |ψ-〉=(|01〉−|10〉)/ 2 | |
"BlochVector"[{ r x r y r z | a one-qubit state from Bloch vector { r x r y r z | |
"Register"[s] | a quantum register with s | |
"Register"[s,i] | a quantum register with s i | |
"UniformSuperposition" | a uniform superposition of 1 qubit | |
"UniformSuperposition"[s] | a uniform superposition of s | |
"UniformMixture" | a maximally mixed state of 1 qubit | |
"UniformMixture"[s] | a maximally mixed state of s | |
"RandomPure" | a random pure state of 1 qubit | |
"RandomPure"[s] | a random pure state of s | |
"RandomMixed" | a random mixed state of 1 qubit | |
"RandomMixed"[s] | a random mixed state of s | |
"GHZ" | Greenberger–Horne–Zeilinger (GHZ) state of 3 qubits | |
"GHZ"[s] | Greenberger–Horne–Zeilinger state of s | |
"Dicke"[n,k] | Dicke's state of n k | |
"Dicke"[{ k 1 k 2 | Dicke's state of n n=∑ k i k i i | |
"W" | W state of 3 qubits | |
"W"[s] | W state of s | |
"Werner"[p,d] | a Werner state of d×d p | |
"Graph"[g] | the graph state corresponding to the graph g |
Examples
(35)
Basic Examples
(6)
Pure states can be defined by inputting state vectors, given a basis:
In[1]:=
ψ=
[{α,β},2]
 | QuantumState |
Out[1]=
QuantumState
 |  |
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The first argument is the state vector and the second argument specifies the dimension of basis (which is computational). The default basis is the computational basis, unless specified otherwise. With no basis info, the basis is set by default as a -dimensional computational basis.
n
2
In[2]:=
| QuantumState |
| QuantumState |
Out[2]=
True
Return amplitudes:
In[3]:=
ψ["Amplitudes"]
Out[3]=
|0〉α,|1〉β
Return formula of the state:
In[4]:=
ψ["Formula"]
Out[4]=
α|0〉+β|1〉
Define a 2-qubit state ( Hilbert space):
2D⊗2D
In[1]:=
ψ=
[{3,2,5,1}]
| QuantumState |
Out[1]=
QuantumState
|  |
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Return qudits dimensions:
In[2]:=
ψ["Dimensions"]
Out[2]=
{2,2}
Again, note that the basis info can be given explicitly too.
In[3]:=
| QuantumState |
| QuantumState |
Out[3]=
True
Specify the dimension of qudit as 3D:
In[1]:=
ψ=
[{1,2+1,3},3]
| QuantumState |
Out[1]=
QuantumState
|  |
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Return qudits dimensions:
In[2]:=
ψ["Dimensions"]
Out[2]=
{3}
Note if no basis info is provided, the state vector will be padded to right by zeroes to reach the first -dimensional space
n
2
A built-in state:
Return amplitudes:
Show the tradition form (on Mac, click on the output cell and then Cmd+Shift+T)
One can define a state in a given basis. A state in 4D Schwinger basis:
Return amplitudes:
States (pure or mixed) can be also defined by matrices.
A mixed state:
Test if it is mixed:
Calculate Von Neumann Entropy:
Purity:
A pure state:
Purity:
Calculate Bloch Spherical Coordinates:
Note the Bloch vector can be given directly, too.