From qudits to qubits: the generalized Pauli system is a qubit Pauli system #
Blueprint lem:pauli-binary. A self-dual basis of F_q over F_2 identifies F_q^n with
F_2^{n x k}, and under that identification every piece of the Weyl data over F_q becomes the
corresponding piece over F_2: the form trDot, the two operator families, their spectral
projectors, and the maximally entangled state.
The relabelling #
The coordinate relabelling F_q^n ≃ F_2^{n x k} attached to a basis b of F_q over
F_2: send a to its matrix of coordinates.
Equations
- MIPRE.Weyl.binEquiv b = (Equiv.piCongrRight fun (x : n) => b.equivFun.toEquiv).trans (Equiv.curry n ι (ZMod 2)).symm
Instances For
The form #
Self-duality is exactly what makes the form transport.
The operator families #
Both are carried over by Matrix.submatrix along the relabelling --- which is conjugation by the
permutation unitary the paper calls the isometry phi, written as a reindexing of rows and
columns so that no unitary has to be constructed.
The X-type family transports, and needs no self-duality: the relabelling is additive,
which is all a shift knows about.
The Z-type family transports, by self-duality of the basis.
The spectral projectors #
The Fourier average commutes with the relabelling: the index group is reindexed by it, the character is the transported form, and the normalization is the same cardinality counted on either side.
The Fourier average commutes with the relabelling, for an arbitrary family: no group law and no self-duality beyond the form's.
The Z-side projectors transport.
The X-side projectors transport.
The maximally entangled state #
The maximally entangled state transports: phi ⊗ phi |EPR_q⟩^{⊗ n} = |EPR_2⟩^{⊗ nk},
in the relabelled form. Both sides are the normalized diagonal, and a relabelling of the index set
preserves the diagonal; the two normalizations agree because the two index sets have the same
cardinality.
The five items together #
lem:pauli-binary. Given a self-dual basis of F_q over F_2, the coordinate
relabelling carries the whole Weyl system over F_q on n registers to the Weyl system over
F_2 on n x k registers --- the maximally entangled state, both operator families, and both
families of spectral projectors.
The paper's phi is a genuine isometry (C^q)^{⊗ L} → (C^2)^{⊗ Lk} and the transported operator
is phi^† (...) phi; here the isometry is a bijection of basis index sets, so conjugating by it is
Matrix.submatrix, and no unitary has to be built.
The relabelling exists for every field of characteristic two and odd degree over F_2:
lem:pauli-binary composed with the existence half of lem:self-dual-basis. The oddness is the
hypothesis of exists_isSelfDualBasis_isNormalBasis, not of anything here; the paper's admissible
field sizes satisfy it.
The target system is qubits, register by register #
The binary Weyl system is Weyl at F = F_2 and index set n x k, whose matrices are indexed by
the functions n x k → F_2 --- that is, by the computational basis of (C^2)^{⊗ nk}. These three
identities say that the operators are the tensor products of the single-qubit ones, in the only
sense the one-matrix presentation can state it: entry by entry, as a product over the qubits. They
are what makes "these are qubit Paulis" a checked statement rather than a reading of the encoding.
Over F_2 the form is the plain dot product: the field trace of a one-dimensional extension
is the identity.
The binary Z-type operator is the tensor product of single-qubit Zs: it is diagonal
(wZ_apply) and its diagonal entries are products over the qubits.