The same system can be observed through different questions, sensors or classification rules. Each gives some information about its state, but the observations may not correspond to one shared set of latent categories. In a sequence of assessments, an earlier step may also change the information or state available to the next one.
E.g. an options portfolio can have low net delta but substantial exposure to volatility or limited liquidity. A delta-based assessment and a liquidity-based assessment describe different aspects of the same positions. Simply calculating both on a fixed portfolio does not create an order effect. If an assessment also triggers hedging or liquidation, however, it can change the portfolio presented to the next assessment.
Consider a hypothetical workflow that applies a delta hedge and reduces positions to meet a liquidity constraint. Reversing these steps can leave different holdings, depending on how the hedge and position reduction are defined. This motivates specifying the sequence of operations. It does not, by itself, establish that a quantum-inspired model is needed: a classical execution model can also represent such behaviour.
A useful starting question is what a quantum-inspired representation adds to a classical latent model, and how that addition can be connected to ordinary data. Commutativity helps make the distinction more precise.
The data do not have to be quantum
Measurements, responses and category labels remain classical data. A quantum-inspired model changes the latent representation used to explain their probabilities. It does not change the physical nature of the observations or require a quantum computer.
Instead of describing a latent state only by a probability vector, use a density matrix ρ. It is positive semidefinite and has trace one. A lens has outcome effects Ea, each positive semidefinite, summing to the identity. Its outcome probabilities are
These probabilities can enter an ordinary likelihood for observed categories. Classical computation can estimate the parameters. Whether this is a useful model depends on the restrictions it imposes across lenses and whether those restrictions describe the data reasonably well.
A matrix encoding is also possible. For example, a real-valued feature vector z could be mapped to ρ(z) = zzT/(zTz), for z ≠ 0. This is a valid rank-one density matrix, but it discards magnitude and maps z and −z to the same state. If z contains signed portfolio exposures, reversing all positions would leave this encoding unchanged. Scaling the exposures up would also leave it unchanged, despite changing their absolute size. A valid matrix construction is therefore not automatically a suitable representation of the data; direction and scale may need to be preserved separately.
There is also a distinction between encoding an observed feature vector and estimating a latent state from observed outcomes. The first specifies ρ as a function of the input. The second treats ρ as something to infer. Both can use the same probability rule, but they make different assumptions about what the data determine.
When the lenses commute
For two operators A and B, their commutator is [A, B] = AB − BA. Finite-dimensional Hermitian operators that commute can be diagonalised in a common orthonormal basis.
If all outcome effects share that basis, their probabilities depend only on the diagonal entries of ρ in that basis. Those entries form a classical probability vector. Each lens reads this vector through its own outcome weights. Off-diagonal terms do not contribute to these particular measurements.
The time evolution needs attention too. Classical-looking probabilities at one instant do not establish that the full dynamic model is classical. For example, dynamics might allow off-diagonal terms to affect the populations later. A reduction to a classical continuous-time Markov chain also needs closed population dynamics of the appropriate form. Commuting observations alone do not establish this.
Commutativity is therefore not a flaw to remove. It identifies a setting where a simpler common-state interpretation may be sufficient. A model that contains this case provides a useful reference for asking whether incompatible lenses add anything.
Different bases are not just different correlations
Non-commuting projectors cannot all be interpreted as reading the same fixed basis of latent categories. Applying one projective assessment can change the state used for a subsequent assessment. To discuss that change, the outcome probabilities alone are not enough; a state-update rule is needed.
For illustration, use a projective outcome P with the Lüders update ρ ↦ PρP/tr(Pρ). This is a particular modelling choice, not a rule that follows automatically whenever a density matrix is introduced. Other measurement instruments can give different sequential behaviour.
For outcome P followed by outcome Q, the joint probability under this update is
p(Q then P) = tr(PQρQP).
If P and Q commute, the two expressions agree. If they do not, disagreement is possible, but not guaranteed for every state and outcome. In particular, choosing a state can make an otherwise non-commuting pair give equal sequential probabilities.
A two-state calculation
Let |0⟩ and |1⟩ be an orthonormal basis and |+⟩ = (|0⟩ + |1⟩)/√2. Set P = |0⟩⟨0|, Q = |+⟩⟨+| and initially ρ = P. The two projectors do not commute.
Assess P first. Its selected outcome has probability one and leaves the state unchanged. The selected Q outcome then has probability one half. Therefore p(P then Q) = 1/2.
In the reverse order, the selected Q outcome has probability one half and leaves state Q. The selected P outcome then has probability one half. Therefore p(Q then P) = 1/4.
This is an exact calculation for the stated construction, not a fitted result. P and Q could be proposed as latent representations of passing a delta-exposure assessment and passing a liquidity assessment. But those labels do not derive the projectors or justify the Lüders update. A fitted observation model and evidence about the assessment procedure would still be needed. The probabilities 1/2 and 1/4 are illustrative, not estimates for an actual portfolio.
Changing assessment order is not reversing a conditional probability
In a classical joint distribution, P(A | B) and P(B | A) can differ. That alone is not an order effect. They have different denominators. The joint event still satisfies P(A ∩ B) = P(B ∩ A).
Sequential protocols are different. An assessment or intervention may change the situation before the next observation. In a portfolio workflow, a hedge changes positions, while an execution can change available cash or the subsequent trading cost. Classical state-space models with history-dependent transitions can also describe order effects. Observing a difference between two sequences is not, on its own, evidence that no classical explanation exists.
The protocol therefore needs to be explicit. Does applying a lens mean calculating a Greek on unchanged positions, updating a volatility estimate, or executing a trade? These are different operations. Reordering delta and liquidity columns in a dataset is not equivalent to performing portfolio interventions in different orders. Even a sequential update of beliefs requires a specified model; reversing the order of conditioning on the same information in exact Bayesian inference does not create an order effect.
What an empirical comparison would need
A static dataset with several labels or indicators may support a model of context-specific probabilities. Without suitable sequential observations or additional identification assumptions, it does not directly establish that assessment order changes outcomes. Operator non-commutativity in a fitted model remains a property of that representation until connected to observable restrictions.
The choice of classical comparator is equally important. A model with one shared latent mechanism is a different baseline from a flexible model that fits each context separately. An improvement over the former does not establish superiority over unrestricted classical probability. The possible benefit may instead be a compact representation, useful constraints across contexts, or better prediction with limited data. Each needs to be evaluated.
A practical comparison can separate static prediction from sequential prediction. For static data, evaluate held-out outcome probabilities in each context. For sequential data, evaluate the probabilities of complete observed sequences under a specified update rule. In the portfolio example, comparing two operation orders also requires accounting for initial holdings, prices, liquidity conditions and elapsed time. Otherwise an apparent order effect may just reflect a different starting portfolio or a market move between operations.
There is also a representation issue: jointly rotating ρ and every effect by the same unitary leaves tr(ρE) unchanged. A particular matrix entry or basis direction therefore cannot automatically be given a unique application-specific meaning. Interpretations should concern identifiable probabilities, constraints or relationships, unless extra structure fixes the basis.
The useful question is whether a compact state and a specified collection of lenses explain several observation contexts consistently. Non-commutativity gives that model additional structure. It is not a substitute for specifying what is observed, what changes between observations, and which classical alternatives have actually been tested.
A related application
These questions also arise in our preprint on systemic risk, where solvency, liquidity and market assessments provide different lenses. That is one application of the modelling ideas discussed here, rather than a requirement that the framework be financial.
- Sedai, Medda, Schroeder & Gong. A Quantum Latent Framework for Multi-Lens Systemic Risk. SSRN preprint. The connection here follows the public abstract. The two-state calculation and feature encoding in this note are independent explanatory examples, not reconstructions of the paper's experiments.
Comments or corrections: a.sedai@ucl.ac.uk.