The maximally entangled state of the Weyl system #
|EPR_q⟩^{⊗ n} on two copies of (C^q)^{⊗ n}, normalized, and the one property the Pauli basis
test uses it for: the two parties' Weyl operators act identically on it.
That property is the reason the expansion stage of the appendix works at all. Its expanded state
adjoins |EPR_q⟩^{⊗ M} to each player, and the sign (-1)^γ that the strategy's own observables
carry is cancelled against one the ancilla observables carry; for that the ancilla halves must
be tied together, and stateVec_epr_wX/stateVec_epr_wZ is the tie.
The general fact, and why the Weyl case is clean #
For any operator, (M ⊗ Id)|EPR⟩ = (Id ⊗ Mᵗ)|EPR⟩ (stateVec_epr) --- the maximally entangled
state transports an operator to the other factor at the cost of a transpose. Generalized Paulis
over a field of characteristic two are real symmetric matrices, so the transpose is the
identity map on them and the cost vanishes. Over F_p for odd p it would not: τ^Z(b) has
entries ω^{tr(bj)}, whose transpose is τ^Z(b) but whose conjugate is τ^Z(-b), and the
bookkeeping reappears. This is the second dividend of the characteristic-two restriction, after
the operators being involutions.
The state #
The normalizing scalar 1/√N of the maximally entangled state, N = q^n.
Equations
- MIPRE.Weyl.eprScale = (√↑(Fintype.card (n → F)))⁻¹
Instances For
The normalized maximally entangled state on two copies of (C^q)^{⊗ n}.
Equations
- MIPRE.Weyl.epr p = if p.1 = p.2 then ↑MIPRE.Weyl.eprScale else 0
Instances For
The maximally entangled state is symmetric under exchanging the two halves.
Transport to the other factor #
The maximally entangled state transports an operator to the other factor, at the cost of
a transpose: (M ⊗ Id)|EPR⟩ = (Id ⊗ Mᵗ)|EPR⟩.
The Weyl operators are symmetric #
So the transpose in stateVec_epr costs nothing: the two parties' operators act identically on
the state.
The same for the spectral projectors of either family, which are built from the operators by a real Fourier average and are therefore symmetric too.
The syndrome projectors #
syn w v a collects the spectral projectors along a level set of the pairing. Two facts the
expansion stage needs: it is a projective measurement, and it transports across the maximally
entangled state like each projector does.
A coarse-grained eigenbasis measurement is a projective measurement, indexed by the values of the label. Orthogonality of the spectral projectors makes each element idempotent, and the level sets partition the index group.
The syndrome projectors form a projective measurement, indexed by the value of the pairing.
Coarse-grained eigenbasis projectors transport across the maximally entangled state,
exactly: each spectral projector does (stateVec_epr_proj) and the level-set projector is a sum of
them.
The syndrome projectors transport across the maximally entangled state, exactly.