The ancilla transports across the padded state (lem:qld-pauli-selfcons, the exact steps) #
The paper's chain for the self-consistency of the exact Pauli observables --- the step
eq:qld-pulling-3a, and again eq:qld-pulling-4 --- moves a generalized Pauli from one party's
ancilla half to the other's, and calls the move approx_0: it is exact, because the two halves are
maximally entangled and the syndrome projectors are symmetric.
The register model has that (BipartiteModel.reg_mirror): a matrix P on the first player's
half acts on the state as Pᵀ on the second player's. The generalized Paulis over a field of
characteristic two are real symmetric matrices, and so are their spectral and syndrome projectors
(proj_weylOf_transpose, syn_weylOf_transpose), so the transpose costs nothing
(stateVec_hatVec_syn). What the chain needs is the same statement in the model K of a
SimulPair, which is not literally the register model --- the structure only provides an
embedding ι of the register model into it. That is enough: ι intertwines each player's
operators and carries the state to the state, so the vector identity is carried exactly
(SimulPair.stateVec_ancSyn), and so is the resolution of the identity by the matched pairs of
syndrome projectors (SimulPair.sum_ancSyn_mulVec).
Stated in a bipartite model #
Stated in a bipartite model (Phase 5 of planning/mipco-track.md). The matrix route read the
expanded state hatVec ψ = expVec epr ψ entry by entry (stateVec_expVec_kron_one) and carried
the transport to the padded state Φ as the vanishing of a squared norm, a quantity the padded
state reproduces. In the model the expanded state is the register model M.reg (Anc F m), the
ancilla operator 1 ⊗ S of the first player is smulKron 1 S, the transport on it is
BipartiteModel.reg_mirror, and the transport to K is the intertwining of the embedding, with no
detour through a norm.
The symmetric ancilla projectors #
A spectral projector of a generalized Pauli family is symmetric: the family is real symmetric, and the projector is a real Fourier average of it.
The transport, on the expanded state #
The ancilla's syndrome projector transports across the expanded state, exactly. The two halves of the register model's EPR pair are maximally entangled and the syndrome projectors are symmetric, so moving one from Alice's half to Bob's costs nothing.
The expanded state's transport, as the vanishing of a cross-party deviation.
The two ancilla halves resolve the identity on the expanded state (eq:qld-pulling-4):
by the transport each matched pair of syndrome projectors acts as the projector on one side alone,
and those sum to the identity.
The transport, in the model of a simultaneous pair measurement #
The ancilla's syndrome projector transports across the padded state, exactly.
This is the approx_0 step of the paper's pulling chain (eq:qld-pulling-3a, and again at
eq:qld-pulling-4), on the state the chain is actually run on. The model K is not the register
model, but the embedding ι intertwines each player's operators and carries the state to the
state, so the transport on the register model (stateVec_hatVec_syn) is carried exactly.
The two ancilla halves resolve the identity on the padded state (eq:qld-pulling-4).
Summing the matched pairs of syndrome projectors, one on each party, leaves the state unchanged:
this is the resolution on the register model (sum_hatVec_syn_mulVec), carried along the
embedding. This is the step that lets the chain insert Bob's ancilla resolution.