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Community-contributed installable additions to the Wolfram Language
Wolfram`QuantumFramework`
QuantumBasis  |
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A quantum basis may consist of distinct input and output qudits. For instance, a operates with both input and output qudits, while a typical only pertains to output. Therefore, it's important to recognize that the concept of a differs fundamentally from that of a basis in a Hilbert space. While users typically won't need to interact with this distinction directly, it is automatically handled behind the scenes for most practical purposes.
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The keys names can be any expression.
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Built-in names in the Hilbert space include:
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"Computational" or "I" | 2-dimensional computational basis | |
"Computational"[d] or "I"[d] | d-dimensional computational basis | |
"Bell" | Bell basis of 4-dimensional qudit | |
"Pauli" | basis of the Pauli matrices | |
"PauliX" or "X" | Pauli-X (Hadamard) basis | |
"PauliY" or "Y" | Pauli-Y basis | |
"PauliZ" or "Z" | Pauli-Z (computational) basis | |
"X"[d] or "Y"[d] or "Z"[d] | Generalized Pauli basis in d dimension (for qudits) | |
"JX" or "JY" or "JZ" | Generalized Spin basis with spin 1 2 | |
"JX"[j] or "JY"[j] or "JZ"[j] | Generalized Spin basis with spin j (or 2j+1 | |
"Fourier" | 2-dimensional basis of the quantum Fourier transform | |
"Fourier"[d] | d-dimensional basis of the quantum Fourier transform | |
"Schwinger" | 2-dimensional Schwinger basis | |
"Schwinger"[d] | d-dimensional Schwinger basis | |
"Dirac" | basis of the Dirac (gamma) matrices |
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Built-in names in the phase space that are informationally complete, include:
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"WignerMIC" | Phase space basis corresponding to the WignerMICPOVM measurement (Minimal Information-ally Complete) | |
"WignerMIC"[d] | WignerMIC basis in the d-dimension | |
"GellMannMIC" | Phase space basis corresponding to the GellMannMICPOVM (generalized Pauli) measurements | |
"GellMannMIC"[d] | GellMann basis in d-dimension | |
"Tetrahedron" | Phase space basis corresponding to the TetrahedronSICPOVM measurements (Symmetric Information-ally Complete), which eigenstates form vertices of a | |
"Tetrahedron"[a,b,c] | A tetrahedron rotated by "U"[a,b,c] | |
"HasseSIC" | ||
"HoggarSIC" | ||
"QBismSIC"[d] | ||
"RandomHaarMIC" | A random MIC basis based on Haar method | |
"RandomBlochMIC" | A MIC basis based on random Bloch sphere affine transformation |
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Built-in names in the phase space that are not informationally complete, include:
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"GellMann" | basis of the Gell-Mann matrices | |
"GellMann"[d] | basis of generalized Gell-Mann matrices for dimension d | |
"Wigner" | 2-dimensional basis of the Wigner phase space operators with respect to the computational basis | |
"Wigner"[d] | d-dimensional basis of the Wigner phase space operators with respect to the computational basis | |
"Wootters" | 2-dimensional phase space basis | |
"Wootters"[p] | p-dimensional phase space basis where p is a prime number | |
"Bloch" | Phase space basis of generalized Bloch coordinates |
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"Association" | An association with keys the formatted symbols of basis elements, and values the corresponding tensor | |
"Elements" | Corresponding elements of a basis | |
"OrthogonalElements" | Corresponding orthogonal elements of a basis |
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Besides its input and output qudits, a carries a picture, recovered with . The picture records where the time dependence of a system written in this basis is kept; it is part of the basis identity but says nothing about the basis elements themselves. Built-in pictures are:
qb["Picture"]
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"Schrodinger" | states carry the time dependence and operators are fixed; the default | |
"Heisenberg" | operators carry the time dependence and states are fixed | |
"Interaction" | the time dependence is divided between states and operators | |
"PhaseSpace" | the tag a phase-space basis carries; a basis whose elements form a phase-space frame sets it for itself, and a state expanded in such a basis is a quasi-probability distribution rather than an amplitude vector |
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A picture may be named positionally, on either side of the basis specification, or given as the option, and the three forms agree. So is the default basis in the Heisenberg picture, and is . A picture may also trail a two-sided specification, as in .
"Picture"
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When more than one picture is written, the one written first wins, positional and option alike: the arguments are folded in reverse, so the earliest is applied last. is in the Heisenberg picture.
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A trailing integer is read as a multiplicity rather than a dimension, so is three qubits in the Heisenberg picture.
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A phase-space basis name sets on its own, so a picture normally only needs naming for and .
"PhaseSpace"
"Heisenberg"
"Interaction"
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Such a basis will not be renamed to another picture positionally: its elements, not a label, are what put it in phase space, so is rejected with . The option is deliberately left open for code that has actually undone the transform.
QuantumBasis::phaseSpacePicture
"Picture"
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The picture travels with the basis into whatever is built on it, so and both land in the Heisenberg picture.
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Any picture outside the four above is rejected with . A picture followed by a tail that is not a basis specification at all, such as a bare symbol or a non-integer number, is rejected with naming that tail; a tail that fails on its own terms reports its own reason instead, so an unrecognized basis name gives and a negative integer gives .
QuantumBasis::picture
QuantumBasis::invalidSpec
QuditBasis::invalidName
QuditBasis::invalidArgs
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option | default | effect | |
"Picture" | "Schrodinger" | the picture the basis is written in, one of the four names above | |
"Label" | the label the basis displays under; a named basis, or one given a single dimension, fills this in for itself, so "PauliX" "I"[3] None | ||
"ParameterSpec" | {} | symbolic parameters the basis depends on, given as {p,min,max} "Parameter" "Parameters" |
Examples
(37)
Basic Examples
(11)
Create a 2-dimensional basis:
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QuantumBasis
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Note with no input, the basis is automatically set to 2D by default:
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| QuantumBasis |
Out[2]=
True
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Create a 3-dimensional basis:
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QuantumBasis
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Create a 2×2×2 dimensional basis (three qubits):
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Out[1]=
QuantumBasis
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Create a composite basis of 2- and 3-dimensional qudits (a qubit with a qutrit):
In[1]:=
| QuantumBasis |
Create a 2-dimensional basis using an explicit element representation with arbitrary names:
Construct a Pauli-Y basis:
Construct a Bell basis for a single 4-dimensional qudit (quqrit):
Return its association (basis names with corresponding representations):
A quantum basis with Pauli-X as the output and computational 2-dimensional basis as the input:
Show its elements:
Return the list of its orthogonalized basis elements:
Represent the 2-dimensional Schwinger basis of rank-2 (matrix) elements:
Every basis is also written in a picture, which defaults to the Schrodinger one:
Name a picture to get the default basis written in it:
The picture may sit on either side of the basis specification, or be given as an option, and the three agree: