Chapter 39
Complex Amplitudes, and Where the Coherence Is
This chapter has been rewritten more often than any other in this part of the book, and the reader should know what each rewriting changed. The first version presented the complex instance of the decoration as delivering quantum dynamics outright. It does not. The second version located two obstructions to its doing so and found one place — the equational theory — where coherence could live untouched by either. The present version reports that the graded where-clause of Chapter 14 takes the route the second version called “a strictly larger construction and not a repair of this one”, and reports what that route delivers: a construction on which a finite, contractive, staged fragment of graded rho admits a candidate quantum interpretation, and on which the extension to unrestricted reflective concurrency remains open. That is the claim, stated as the source states it [276], and it is deliberately modest. What follows is a construction, a boundary, and two views of where unitarity lives that this book does not choose between.
39.1 The question
Take \(\Semi = \CC\). The amplitude of a path is the product of its step amplitudes and the transition amplitude from \(P\) to \(Q\) is the sum over paths, \[\langle Q \mid P \rangle \;=\; \sum_{\gamma : P \to Q} W(\gamma),\] with \(|\langle Q \mid P\rangle|^2\) the candidate transition probability. This much is a construction and is not in doubt. It is also, so far, only bookkeeping: complex numbers have been attached to steps and multiplied along paths.
What would make it quantum is cancellation. Two paths \(\gamma_1, \gamma_2\) from \(P\) to \(Q\) with \(W(\gamma_1) = -W(\gamma_2)\) should annihilate, so that the transition has zero probability despite having two distinct causal histories. That is the phenomenon; everything else is notation. In the vocabulary of Chapter 14 it is Principle 14.2: a false disjunction with true disjuncts. The question is whether the construction can be made to have one.
39.2 The first construction, and its two obstructions
The first construction worked in the linear fragment — weight maps keyed by a partition — and it is worth keeping on the page, because several other chapters refer to it and because what it could not do is what the second construction is for.
Assume the set of reachable weight maps is finite. Let \(\Hilb = \ell^2(\Cfg)\) with orthonormal basis \(\{\,|\mathfrak{c}\rangle : \mathfrak{c} \in \Cfg\,\}\) indexed by configurations \((P,\dec)\), not by terms.
For a base rule \(r\) and key \(\varphi \in \Phi_r\) with \(\dec(\varphi) = z = |z|e^{i\theta}\), write \(\lambda(z) = |z|^2\) and put \[\Lop_{r,\varphi} \;=\; e^{i\theta}\!\!\!\sum_{\mathfrak{c},\,\mathfrak{c}''} \sqrt{\;\sum_{(k,\sigma)\,:\,\mathfrak{c} \to \mathfrak{c}''} \lambda(z)\,\gfac(k)\,\chi(r,k,\sigma)\;}\; |\mathfrak{c}''\rangle\langle\mathfrak{c}| ,\] the inner sum over redexes of \(r\) in class \(\varphi\) carrying \(\mathfrak{c}\) to \(\mathfrak{c}''\).
Let \(E_h\) be a distinguished subset of the equations of the theory, not quotiented into the basis, presented with real amplitudes \(h_e\). Put \[H \;=\; \sum_{e \in E_h} h_e \!\!\sum_{\mathfrak{c} \equiv_e \mathfrak{c}'}\!\! \bigl(|\mathfrak{c}'\rangle\langle\mathfrak{c}| + |\mathfrak{c}\rangle\langle\mathfrak{c}'|\bigr),\] and let the quantum continuous-time Markov chain of the weighted theory be \((\Hilb, \Lind)\) with \(\Lind(\rho) = -i[H,\rho] + \sum_{r,\varphi}\bigl( \Lop\rho\Lop^{\dagger} - \tfrac12\{\Lop^{\dagger}\Lop, \rho\}\bigr)\) [4].
The square root in Definition 39.2 is outside the inner sum, and the choice was forced by a principle the first version got right.
A complex weighting is conservative over the corresponding real one when, at zero Hamiltonian, the populations it evolves are exactly those the real weighting evolves.
Under Definition 39.2, distinct redexes of one class with \(\equiv\)-equal contracta do not interfere: their rates are summed under the square root, so \(|\langle\mathfrak{c}''|\Lop_{r,\varphi}|\mathfrak{c}\rangle|^2\) is exactly the classical rate. Distinct classes do not interfere either, their dissipators being summed as quantum operations. Consequently at \(H = 0\) the populations of \((\Hilb,\Lind)\) evolve by the classical generator of Theorem 14.1, and every interference effect in the model is attributable to the commutator \(-i[H,\rho]\).
Unraveling \((\Hilb,\Lind)\) into trajectories by the quantum-jump method reduces at \(H = 0\) to the sampling of Theorem 14.1, exactly.
Both results stand, and the theorem is worth having: “Gillespie-inspired” is a statement about a limit and not a resemblance. But the construction cannot do what the question asks, for two reasons that should be kept apart.
The equivalence on which this book rests is bisimulation, and bisimulation is a branching-time notion: it distinguishes systems by the shape of their tree of possibilities, not merely by the set of runs they admit. Two distinct paths to the same term are, for bisimulation, two distinct pieces of structure, and no assignment of weights to steps can make a branching-time equivalence identify them — still less make them annul one another. Interference adds and cancels contributions to a shared outcome, which is a statement about runs; bisimulation refuses to pool runs in the first place [62, §13].
Independently of the equivalence, the linear fragment attaches one amplitude per class. If \(m\) redexes of one class carry a configuration to one successor, summing amplitudes and squaring would give \(m^2\lambda\) where the real instance gives \(m\lambda\); conservativity forces the rates under the root, and under the root nothing cancels. The two paths do not cancel — not because branching time declines to pool them, but because the only normalization under which the complex reading lifts the classical one adds them where cancellation cannot occur.
They agree in their verdict and disagree in everything else. Remark 39.2 is a claim about observation: a cancellation could be computed but would not be seen. Remark 39.3 is a claim about arithmetic: under the linear fragment’s normalization there is no cancellation to compute. The second is specific to the fragment. The first is not, and it is the one the rest of this chapter has to answer.
The second version of this chapter drew one conclusion from this pair that survives, and one that must now be restated. The surviving conclusion is that a presentation has a second slot for coherence — its equations — and that equations withheld from the quotient supply a Hermitian hopping term untouched by either obstruction; that is Definition 39.3, and Section 39.5 returns to it as one of two views. The conclusion to restate is that cancellation between rewrites would need “a decoration indexed by derivations rather than by refinement classes”, a strictly larger construction, and that whether it could be given without destroying locality was open. It has been given. It is the graded where-clause, and locality is exactly what it costs, in a way the next section makes precise.
39.3 The second construction: amplitudes on candidates
Chapter 14 evaluates a clause once per candidate (Definition 14.12) — per occurrence-indexed match, not per class. Two candidates with the same contractum are two entries of the vector \(\sem{\psi}_r\), and with \(\Semi = \CC\) they may carry different phases. Equation (14.2) then sums them as amplitudes, before anything is squared. That is the derivation-indexed decoration the previous version asked for, obtained not by enlarging the weight map but by discarding the partition that made it a lookup. Conservativity is not violated: on a clause in normal form (Theorem 14.4) every candidate of a class carries the same value and the sum over them is the multiplicity, which is the classical count. Cancellation becomes available exactly when the clause leaves normal form — when it distinguishes candidates the partition would have identified — and that is the right place for it to become available, because it is the place where the programmer has said that two ways of firing are not the same way.
39.3.1 The semantic obstruction, answered
Remark 39.2 said that a branching-time equivalence will not pool runs. It still will not, and it does not have to. Principle 14.1 is the reason: the resolver is a parameter that no context can read. Bisimulation is a relation on terms under the unweighted transition relation; the Born rule is a way of resolving the races that relation leaves open; and the two never meet, because the first is a statement about which races exist and the second about how they come out. Two paths to one normal form remain two paths for bisimulation. What the complex resolver adds is a distribution over normal forms in which that normal form may have probability zero, and Corollary 14.1 says precisely what has happened: a reachable outcome whose coefficient is a sum of nonzero terms summing to zero. Support adequacy holds. Its converse fails, and the failure is the interference.
This is the right time to say what becomes of the concession this part of the book once carried — that recovering interference would cost the ontology in which bisimulation classes are the real things. It does not, and the reason is cleaner than the one the previous version gave. The ontology is a claim about what distinctions exist. The resolver is a claim about how the distinctions that exist get decided. A theory may have a Born resolver and keep every bisimulation class it had, because nothing in the resolver identifies two terms; it only declines, with some probability, to arrive at one of them. Part Part VI survives, and it survives for a reason that does not depend on where the coherence is sited.
39.3.2 What is normalized
The route costs locality, and the cost arrives through Definition 14.15. A single clause cannot be made norm-preserving by a condition on itself, because the one-step operator sums over sites as well as over candidates, and candidates at different sites may be alternatives or may be concurrent. Only alternatives are weighed against each other; concurrent candidates both fire. The unit of normalization is therefore the contention component, and the alternatives within it are its maximal matchings — the RSpace zip.
Let \(T_P\) be the one-step operator at \(P\) with \(\Semi = \CC\). \(T_P\) is an isometry at \(P\) if, for each contention component \(C_i\), \[\sum_{o \,\in\, \Out(C_i)} \Bigl| \sum_{\substack{m \,\text{a maximal matching of } C_i\\ \Out(m) = o}} w(m) \Bigr|^{2} \;=\; 1 ,\] together with the column-orthogonality condition of Section 39.4. Under staging and unit occupancy (Definition 39.5) the condition is also necessary, because the components then act on disjoint factors of the state space and \(T_P\) is their tensor product.
Three things in that statement do work, and each corrects a plausible alternative that was measured and found wrong [276]. Candidates in different components are concurrent and both fire, so treating them as alternatives double-counts. The alternatives within a component are its maximal matchings, not its individual candidates. And the norm is taken after summing over the fiber of \(\Out\), since matchings with coinciding outcomes interfere before they are measured. Normalizing over sites, over candidates, or over matchings without the fiber sum each give the wrong squared norm on one of five test configurations; only the fiber convention gives \(1\) on all five.
Independent redexes must be counted once. Summing over interleavings instead inflates the squared norm of a layer of \(n\) independent sites by \(n!\) — measured, exactly \(2\), \(6\) and \(24\) for \(n = 2,3,4\). Under Proposition 14.4 the quotient is not imposed on histories: concurrent firings are bundled into one matching and independent components tensor, so the inflation cannot arise. This is the same point Chapter 34 made about the history monad — the free history is a tree of causal histories, not of interleavings — arriving from the other side.
Completion is nonlocal: a clause can be normalized only in the light of what else is contending, so isometry is a property of a configuration and not of a clause. And the sum over maximal matchings is permanent-shaped, hence \(\#P\)-hard in general — which is the physics [280] but bites an implementation, so what a compiler checks must be a syntactic sufficient condition such as a bound on component size. These are Remark 14.12 with the stakes raised: in \(\Rnn\) a nonlocal normalization is an inconvenience; in \(\CC\) it is the difference between a state and a vector.
Proposition 39.2 holds at a configuration. That the induced operator preserves norm along arbitrary concurrent execution, with components merging and splitting as data are produced and consumed, is conjectured and not proved.
39.4 When is the denotation a quantum state?
Proposition 39.2 makes each column of \(T\) a unit vector. That is half of \(T^{\dagger}T = \mathrm{Id}\). The other half is that distinct columns be orthogonal, and it fails for reasons that have nothing to do with weights. Both failures are syntactic, and both are visible in the term.
A configuration with no enabled component is a fixed point of \(T\) under the absorbing convention of Definition 14.14. If another configuration in the same superposition reduces to it, the two columns are nested and no choice of weights makes them orthogonal. Termination is not a unitary phenomenon.
The repair is halt tagging: a configuration with no enabled component does not idle; it is tagged with the step at which it halted and thereafter frozen. The tag records when, never which, so branches halting together share it and it writes no which-path record.
A token not consumed by a step is a spectator; a spectator resting on a channel that a firing emits to is indistinguishable from the emitted quantum, since parallel composition is a multiset. A term is staged when every receipt emits only to channels that no other receipt emits to and that no spectator occupies — fresh wires per stage, which is what a circuit is.
Halt tagged, staged, dual-rail and token-conserving implies pairwise orthogonal columns. Consequently, on such terms orthogonality is automatic and isometry reduces to Proposition 39.2, which is local to a component.
The evidence is an exhaustive finite search over six small systems at one to three tokens and depth three [276]. Of the severe collisions found in unrestricted terms, almost all have a normal form on one side — absorption — and every non-severe one involves a spectator; under halt tagging and staging with the dual-rail encoding, none of either kind remains at any token count or depth searched. That is evidence and not a proof, and what the search does establish is the diagnosis: the two ways a term leaves the fragment are absorption and spectators. That no third way exists is exactly what is not established.
Circuit-shaped terms on fresh channels have one receipt per contention component, so a per-site normalization coincides there with Proposition 39.2, and they are staged, so they collide nowhere. A worked example drawn from the circuit model — Shor’s algorithm is carried out in full in the source, dual-rail and linearly, with no discard — cannot exercise this part of the theory. The fragment is where the construction is becoming credible; the boundary is more informative than a restriction imposed at the outset would have been.
A step that uses every bound name exactly once is block-diagonal in the input basis; this is true and its hypothesis is a property of the encoding. Under the dual-rail encoding every one-qubit gate and every entangling two-qubit gate is realized with no discard, so linearity does not cost universality. What it buys is Proposition 39.2’s necessity direction. What it does not buy is recursion: recursion by reflection runs into a known incompatibility between quantum alternation and the standard fixed-point semantics, and the source files this as an obligation rather than a result. The reflective calculus is not covered. Said plainly: the quantum reading is credible for the fragment of rho that looks like a circuit, and the part of rho that makes it rho is the part that is open.
39.5 Two views of where unitarity lives
Two accounts of the same calculus are now on the table, and this book does not choose between them. It states both, says what each commits to, and notes where they can be told apart.
View one: reductions carry coherent amplitude.
This is the construction of Sections 39.3 and 39.4. The graded clause at a receipt site is a matrix over candidates and outcomes; a firing is a coherent step; interference is between matchings of a contention component with a common outcome; measurement is whatever reads the support. On this view the equational theory is quotiented into the basis as usual and contributes nothing, and the entire quantum content is in the where-clauses. Its strength is that it is built, with a worked example and a measured boundary. Its cost is the one just stated: unrestricted reflective concurrency is open, and what is credible is the circuit-shaped fragment.
View two: unitary in the equations, measurement in the reductions.
This is the organizing intuition of a later design note [278], and it is the second version of this chapter’s observation taken as a principle rather than a slot. A GSLT presents two kinds of relation on terms: symmetric, cost-free equations and directed, costed rewrites. A weighted symmetric relation can be Hermitian; a directed one cannot. A communication in rho quotes: firing substitutes \(@Q\) into a continuation, the payload becomes a name, and names are compared by equality — so a communication writes a record, and a step that writes a record is the kind of step Chapter 34 charges for. And cost lands where information is erased, which is Landauer’s principle read operationally. On this view the coherent dynamics is the equational theory, suitably augmented — messages and names become pairs of a process and a phase, superposition enters through a binder over phase-valued variables, couplings are inert atoms governed by path-sum equations — and reductions are where coherence ends: resolving a value, or running received code, is a measurement; only binding a receipt is coherent. Its strength is that it puts unitarity where the structural facts of the calculus already point. Its cost is that it changes the calculus — paired messages, a binary drop, a value binder — and the source records it as a design session with its decisions and open questions, not as a semantics with a boundary measured.
Where they differ, and where they agree.
They agree that Principle 39.1 — a weighted theory is quantum exactly to the extent that its presentation withholds equations from the quotient — is true of view two by construction and true of view one only vacuously, since view one withholds nothing and puts the coherence elsewhere. They agree that bisimulation survives, on view one by Principle 14.1 and on view two because the equations that carry phase are structural congruence enlarged, which bisimulation already respects. They differ on the status of a firing: coherent step or measurement. That difference is testable inside the calculus. View one predicts that a which-path record written by a reduction destroys interference only if the record is read; view two predicts that the reduction is the reading. The source for view two works a single example — \(HZH\lvert 0\rangle = \lvert 1\rangle\) computed by equations alone, against a which-rail detector that destroys the interference by reductions alone — and the two views agree on it, because in that example the reduction does read the record. The example that separates them has not been written.
A weighted GSLT is quantum exactly to the extent that its presentation withholds equations from the quotient.
The principle is kept, restated, and now carries a qualification: it names the quantum content of view two and is silent about view one, whose quantum content is in the clauses. A reader who wants one sentence should take this one: coherence lives either in the where-clauses or in the equations, the two views place it differently, and nothing in this book decides which placement is the world’s.
Three things, and they should not be run together. First, on view one, Conjectures 39.1 and 39.2 and the recursion obligation are open; the fragment is finite, contractive and staged, and rho is none of those. Second, on view two, the calculus has been redesigned and not yet given a semantics with a measured boundary. Third, on either view, the notion of equivalence appropriate to a theory with coherence is not known: bisimulation is untouched but is also uninformative about phase, and the correctness frame of Chapter 14 — separations between resolvers at a fixed term — is a frame and not yet a theorem. That is the precise form in which the branching-time question returns, and it is a better form than the one this chapter started with.
39.6 Simulation, in the meantime
The stochastic instance is unobstructed, and it is what the implementation actually runs. The classical Gillespie algorithm [3] simulates a continuous-time Markov chain by computing exit rates from the current state, sampling a waiting time, and selecting the next rule proportionally to rate. With \(\Semi = \Rnn\) and clauses supplying the rates, this is a graded theory stepped forward: the simulator advances the term, re-evaluates the clauses against it, debits the account and records the trace as one step of one process. By Remark 14.10 the chain is over terms, the state being the term.
The complex instance, on view one, is simulated by the same loop with the contention-set completion in place of the exit rate and the Born rule in place of the exponential clock, on the fragment where Conjecture 39.2 is believed; outside it the loop runs and the numbers it produces are not guaranteed to be those of a quantum state, which is the sense in which the boundary of Section 39.4 is the boundary of what may be claimed. Theorem 39.1 remains true of the linear fragment and says that the two simulators are one simulator at \(H = 0\).
What remains withheld is the interpretation and not the computation. One may run either loop, and the numbers coming out will be the numbers of a quantum process on the fragment. Whether the system they describe is a quantum system depends on whether the amplitudes answer to anything, and that is a question about which of the two views is the world’s — rather than about the algorithm. The framework has made the question askable, has sharpened it to a single example that has not been written, and has not answered it.