Assumptions, results and research status
Exact finite-unitary identities; declared coherent test family
Assumptions
- Directed links use U_yx:V_x→V_y, with endpoint law U′_yx=g_yU_yxg_x⁻¹ and right-to-left products.
- Controlled alternative histories and their relative phases are available and calibrated.
- Claims about SU(3) reconstruction assume unitarity and determinant one; these constraints must be checked or supplied.
Results
- Two calibrated probe phases recover the real and imaginary parts of a normalized trace.
- For one SU(3) loop, the complex fundamental trace determines the characteristic polynomial and conjugacy class.
- Separate traces of fundamental cycles do not determine their simultaneous-conjugation class.
- Forced singlet-return probability loses central phase information that coherent comparison can retain.
Open questions
- Which coherent references and representation-sensitive probes does the source actually provide?
- Which joint loop words can be measured on the intended graph?
- How are calibration errors and unitarity/determinant assumptions independently checked?
Specify the object that is to be reconstructed
For a based closed path, forward transport W transforms by W′=gWg−1. Its eigenvalues and traces are invariant under a change of the base frame. A matrix tomography performed in a chosen laboratory frame can recover entries, but those entries are not themselves frame-independent conclusions. The intrinsic target for one loop is its conjugacy class.
There is more than one meaning of “the loop measurement.” A fundamental complex trace, its real part, its absolute square and a trace in an adjoint representation are different functions of W. They can agree on some loops while disagreeing on others. A predictive record is sufficient only relative to the admitted tests, so choosing among those functions changes which distinctions must survive completion. Completed records, v21, Theorem 2.3 and §5.2, pp.4 and 10–11.
Our orientation remains W=U03U32U21U10 for the cycle 0→1→2→3→0. The earliest operation acts on the right. Reversing the path replaces W by W†. Its real trace remains unchanged, while its imaginary trace changes sign. A phase-sensitive measurement must therefore report its path orientation.
Recover a complex trace with two calibrated phases
Prepare a carrier in Im/m and coherently compare the loop with a reference history. Put q=TrW/m. The plus-port probability at relative phase φ is
Two phases give Re q=2p+(0)−1 and Im q=1−2p+(π/2). The minus sign in the second relation follows from Re(iq)=−Imq. An opposite phase convention would reverse that sign consistently; it cannot be changed halfway through reconstructing the loop. The isotropic input removes a preferred carrier polarization from this specific protocol. Temporal completion, v20, Proposition 17.3, p.53; coherent histories are specified in v21, §2.2, p.4.
For the central SU(3) loop W=e2πi/3I, q=−1/2+i√3/2. Then p+(0)=1/4 and p+(π/2)=(2−√3)/4≈0.0669873. Reversal leaves the first probability unchanged but changes the second to (2+√3)/4. Both results require the calibrated alternative history; an arbitrary redefinition of an absent phase reference cannot perform this experiment.
Recover eigenvalues from trace data and group constraints
For a general m-dimensional unitary, sufficiently many power traces pk=TrWk determine the elementary symmetric functions through Newton's identities. For example e₁=p₁, e₂=(p₁²−p₂)/2 and e₃=(p₁³−3p₁p₂+2p₃)/6. They are coefficients of the characteristic polynomial. Traces of powers can be obtained by coherent repeated traversal, provided that repeated history is part of the available protocol.
SU(3) has a useful simplification. Let z=TrW and let its unit-modulus eigenvalues be λ₁,λ₂,λ₃ with product one. Then λ₁λ₂+λ₁λ₃+λ₂λ₃=λ̄₁+λ̄₂+λ̄₃=z̄. Therefore
The complex trace determines the unordered spectrum and hence the unitary conjugacy class. For z=0 the polynomial is λ³−1, giving the three cube roots of unity. This reconstruction is an algebraic extension of the source's loop-spectrum description, not an assumption that every complex z is admissible. V21, Proposition 5.3, p.11. The SU(3) domain calculation supplies the missing domain constraint.
The real Wilson term cannot reconstruct the full SU(3) loop
The Wilson-type trace deficit uses m−ReTrW. It contains only one real number. A generic SU(3) conjugacy class needs two real parameters, so a single real trace cannot determine it. W and W† already provide a simple ambiguity whenever ImTrW≠0: they share the trace deficit while producing opposite phase quadratures.
The two central loops e±2πi/3I illustrate the loss particularly clearly. Both give the unnormalized deficit 9/2 and forced circuit success one. Their calibrated complex traces differ. The exact static singlet energy also agrees for this pair because its secular function depends on cosθa. That energy is a definite response of its Hamiltonian, but it is not a complete tomographic statistic for oriented holonomy. V20, Theorem 2.1 and equation (5), pp.9–10; equation (80), p.62.
There are also equal-real-trace classes that are not simply reversals and have different exact singlet energies. They demonstrate a second limitation: the leading trace cost need not determine the full finite-hopping energy. The distinction between a useful leading invariant and a complete response function matters when fitting model data.
Separate loop spectra leave relative orientation undetermined
On a connected finite graph, set spanning-tree links to identity. The remaining E−V+1 fundamental cycle matrices share one base frame and transform by simultaneous conjugation. Individual conjugacy classes permit separate diagonalizations, but those separate diagonalizations need not be compatible with one common frame. V21, Proposition 5.3, p.11.
Take two SU(2) cycles A=eiaZ and either B∥=eibZ or B⊥=eibX. In both configurations TrA=2cosa and TrB=2cosb. Their ordered products differ:
At a=b=π/3, both separate cycle traces equal one. The mixed word has trace −1 in the parallel case and 1/2 in the perpendicular case. No common conjugation can identify the two pairs because it would preserve this mixed trace. Embedding the two-by-two matrices as diag(A,1) in SU(3) gives the same lesson in that group.
A graph record that retains only the separate fundamental-cycle spectra therefore loses some observable products. Which mixed words must be retained depends on which composed histories the future test family permits.
A mixed trace measures a concrete relational parameter
Write A=cos a I+i sin a n·σ and B=cos b I+i sin b r·σ, where n and r are unit vectors. The Pauli product identity gives
When sin a sin b≠0, the measured traces of A, B and AB determine the relative axis dot product after choosing consistent eigenangle branches. A common SU(2) conjugation rotates both axes together and preserves n·r. The mixed trace therefore extracts an explicitly relational quantity, rather than introducing an absolute color direction.
When either loop is central, its axis is unobservable and the displayed division is inappropriate. That is a genuine loss of identifiable information, not a numerical bug to fix by choosing an arbitrary axis. With more loops, additional mixed words can constrain further relative data; a list of single-loop eigenvalues is not a general substitute.
This worked calculation develops the simultaneous-conjugation statement in v21, p.11. It does not establish that an actual source supplies the coherent compositions or calibrated reference needed to measure every desired word. Those resources remain part of the experimental specification.
Keep the three common readouts separate
| Protocol or functional | Quantity | Central loop eⁱθI |
|---|---|---|
| Normalized trace deficit | 1−Re q | 1−cosθ |
| Interferometer minus port | (1−Re q)/2 | sin²(θ/2) |
| Forced single-circuit singlet return | |q|² | 1 |
The first is a nonnegative functional, the second a detector probability, and the third the success probability of a different preparation/transport/projection sequence. Multiplying a rejection probability by a selected penalty can give an energy-valued cost, but the penalty is additional input. A square of a trace loses information that a calibrated complex trace retains. V20, Proposition 17.3, p.53, and §20.4, pp.61–62.
Representation choice matters as well. Scalar center elements act trivially by conjugation on an observable matrix algebra, so adjoint-only tests cannot reproduce every defining-carrier phase test. The source explicitly distinguishes the full carrier and its reference tensors from the adjoint observable algebra. V21, §5.2, pp.10–11. Tomographic completeness must always name the representation and available reference family.
Report enough data to assess a tomographic claim
A reproducible record includes the ordered loop word, carrier representation, initial carrier state, probe phase convention and detector counts. For N independent shots at one phase, the estimator 2k/N−1 for a trace quadrature has variance 4p(1−p)/N≤1/N. Thus its standard error is at most 1/√N; 10,000 shots give an upper bound 0.01 for the normalized quadrature. Multiplying by m gives the corresponding unnormalized trace uncertainty.
This elementary counting estimate does not cover phase-calibration drift, preparation error or correlated shots. Those can rotate or shrink the reconstructed complex trace systematically. A point slightly outside the allowed SU(3) trace domain may therefore signal uncertainty or a failed model assumption; silently projecting it onto the domain discards useful diagnostic information.
Finally, state the claimed target: a probability, one loop's conjugacy class, a tuple under simultaneous conjugation, or a Hamiltonian parameter. None follows automatically from all the others. The source's sufficiency principle is valuable precisely because it asks which future tests a record reproduces. V21, Proposition 2.1 and Theorem 2.3, pp.3–4.
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Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.