Does Real Quantum Theory Survive Commuting Measurements and Superselection Rules?

Two ways to encode Alice and Bob's separation

A bipartite quantum theory has to say what makes Alice's and Bob's measurements separate. The tensor-product formulation assigns each party its own Hilbert-space factor, so measurements take the form E^a_x ⊗ 1_B and 1_A ⊗ F^b_y, and the observed behaviour is p(a,b|x,y) = Tr[ρ (E^a_x ⊗ F^b_y)]. The commuting formulation instead puts both parties on one common Hilbert space and requires their measurement operators to commute, [E^a_x, F^b_y] = 0, with probabilities p(a,b|x,y) = Tr[ρ E^a_x F^b_y].

Tsirelson's problem asks whether these two meanings of separation yield the same bipartite correlations. In finite-dimensional complex quantum theory (QT) they do: every finite-dimensional commuting-operator strategy can be rewritten exactly as a finite-dimensional tensor-product strategy. The paper's first result extends that equivalence to real quantum theory, where states and effects are represented by real symmetric matrices.

Why the complex proof does not carry over to the reals

The equivalence is not automatic in RQT. Using the usual proof methods, there exist pairs of commuting real POVMs {E^a_x} and {F^b_y} that would need to be represented by complex POVMs {Ẽ^a_x ⊗ 1} and {1 ⊗ F̃^b_y} to fit the tensor-product form. That possibility is reinforced when the set of valid effects is linearly restricted, for example by a superselection rule.

If the commuting definition produced a strictly larger set of behaviours, it would amount to a different and potentially stronger variant of RQT, one in which local measurements were not exhausted by tensor-product local measurements. That is the gap the paper sets out to close before extending the framework to superselected theories.

Real commuting and tensor-product correlations coincide

Theorem 8 proves that C^fin_tensor,R = C^fin_commuting,R. The left-hand side is the set of behaviours obtained from real symmetric states and tensor-product local real POVMs; the right-hand side is the set obtained from real symmetric states and commuting real POVMs on one finite-dimensional real Hilbert space. Combining this with the earlier complex-to-real embedding, Theorem 12 states C^fin_tensor,R = C^fin_commuting,R = C^fin_tensor,C = C^fin_commuting,C.

The proof has two ingredients. First, the finite-dimensional C*-algebra structure theorem known as Artin-Wedderburn decomposes the Hilbert space into a finite direct sum of blocks H ≅ ⊕_k (H_A,k ⊗ H_B,k), so commuting finite-dimensional measurements are already tensor-product measurements up to a direct-sum label k that acts as classical correlation. Second, the locality-preserving realification converts the blockwise complex tensor-product realisations into real symmetric ones without changing any probabilities, and the block label k is absorbed as shared classical randomness in a larger real tensor-product model.

One caveat is stated explicitly: the total matrix dimension of the real tensor-product representation is larger than that of the commuting representation. The authors call this minor because, when arbitrary local measurements are allowed, the optimisation runs over arbitrarily large finite dimensions, so the caveat evaporates.

The equivalence is deliberately finite-dimensional

The result is stated only for finite dimensions, and the paper is explicit about why that boundary matters. In complex QT the corresponding infinite-dimensional assertion is false: the result MIP* = RE of Ji, Natarajan, Vidick, Wright, and Yuen implies a strict separation between tensor-product and commuting-operator correlation models.

The authors therefore expect that an infinite-dimensional commuting-measurement version of RQT would also be more powerful than the corresponding tensor-product version, but they do not prove such an infinite-dimensional real separation here. Readers should treat the equivalence as a finite-dimensional statement, not as a claim about all RQT models.

The paper also notes that there is no equally direct dual theorem obtained by swapping the roles of states and measurements. If both states and measurements were forced to be literal tensor products, all correlations would factorise. More importantly, RQT is not locally tomographic: correlations present in a global real state need not be detectable by local real measurements, so a state-side analogue would require a different operational formulation rather than a naive replacement of commuting measurements by commuting states.

Superselection as a restriction, not a replacement

The paper treats a superselection rule as a restriction on which preparations, measurements, and transformations are physically available, not as a replacement of the underlying system. Fix a group G, a finite-dimensional complex Hilbert space H_A per system, and a unitary representation U_A: G → U(H_A). The associated superselected, or invariant, theory permits exactly those states and effects fixed by the action, U_A(g)(ρ) = ρ and U_A(g)(E) = E, and those transformations equivariant under it, Φ ∘ U_A(g) = U_B(g) ∘ Φ for every g in G.

This makes superselected theories sub-theories of quantum theory: if QT^inv_G is the invariant theory and QT the unrestricted one, then QT^inv_G ⊆ QT. Passing from the invariant theory to unrestricted QT changes no Hilbert space, matrix, channel, outcome, or classical control label; it simply forgets the rule that prohibited the non-invariant alternatives. Every Born probability and every joint or conditional probability in a finite adaptive protocol is therefore identical before and after the restriction is forgotten.

Embedding superselected theories in real quantum theory

The realification map Γ_d(A) = ((Re(A), -Im(A)), (Im(A), Re(A))) sends Hermitian positive matrices to real symmetric positive matrices and doubles the trace, so the encoded RQT states ρ̃ = ½ Γ_d(ρ) and effects Ẽ = Γ_d(E) preserve the Born pairing: Tr_R[ρ̃ Ẽ] = Tr_C[ρE]. The same map sends each unitary U_A(g) to an orthogonal matrix O_A(g) = Γ(U_A(g)), carrying the invariance and equivariance equations into orthogonal-conjugation equations on the encoded real operators.

Theorem 17 identifies an embedded RQT subtheory with exactly the same Born probabilities. Writing X for an encoded real state or effect and J_d := J ⊗ 1_d = Γ(i 1_d) for the real operator representing multiplication by i, these states and effects satisfy both the original symmetry restriction and compatibility with the encoded complex structure: [X, O(g)] = 0 for every g in G, and [X, J_d] = 0. The second condition can itself be expressed as invariance under the cyclic group C4 generated by J_d, so the embedded states and effects are selected by a combined C4 × G symmetry. Invariance under O(G) alone would generally allow a larger RQT theory.

For finite multipartite protocols, assuming the symmetry acts on-site relative to the chosen tensor factors, Theorem 20 shows that every such superselected QT protocol admits an RQT realisation with the same joint and conditional probabilities and the same prescribed locality of operations. The real representative of each operation depends only on that operation and its declared input and output blocks, so the construction applies consistently across protocols rather than assigning a separate simulation to each experiment. As in the unrestricted case, preserving operational locality does not require product initial states to remain product states under the mapping.

Fermionic information theory gets a real image

Fermionic information theory (FIT) is the concrete example, with G = Z2 implementing parity superselection. The paper explicitly realifies the Fock-space description, the parity operator, and the allowed states, effects, and channels, obtaining the same probabilities while retaining the graded commutation relations that specify fermionic locality. The realified Fock space decomposes as F̃_m = ⊕_{k=0}^m F̃_{m,k} ≅ ⊕_{k=0}^m ∧^k C^m, with sectors given by eigenspaces of the realified number operator Ñ.

The realified operator space is B̂(F_m) = Γ(B(F_m)) = {X ∈ B(F̃_m) : [J_{F̃_m}, X] = 0} ≅ {X ∈ M_{2d_m}(R) : [J_{2d_m}, X] = 0}. The parity-superselected real theory is then characterised by three conditions on states, effects, and channels: positivity and unit trace, commutation with J_{d_m}, and commutation with the realified parity operator P̃_m. The J_{2d_m}-commutation condition selects precisely the real matrices in the image of the quantum operator algebra, while the P̃_m-commutation condition is the parity-invariance superselection rule.

The paper stresses that the J_{2d_m} condition is essential. If it is omitted, one obtains all parity-invariant RQT states and effects, a strictly larger theory than the image of fermionic quantum theory. Proposition 22 states the resulting operational equivalence: defined as an operational probabilistic theory, fermionic information theory has a real image that preserves all probabilities and locality assignments, with the sets mapped isometrically via the triple (½ Γ, Γ, Ad Γ).

The authors also flag what they did not do here. They did not attempt to show that the realified FIT can itself be understood as a C4-superselected real FIT. Defining what real FIT is would require choices about what should constitute the real CAR algebras, and the dimensional mismatch between ∧^k C^m and ∧^k R^{2m} would require reformulating the construction using a factor-wise realification such as Γ^(k).

What the equivalence does and does not settle

Taken together, the two results say that in finite dimensions neither the choice of real or complex scalars nor the choice between the commuting and tensor-product formulations of measurement locality changes the observable statistics, and that finite-dimensional superselected QTs admit RQT representations preserving their symmetry restrictions, all probabilities in finite adaptive protocols, and the declared tensor-product locality of operations. Parity-superselected fermionic QT is the concrete case where this is worked out in full.

The most decision-relevant distinction for readers is therefore not real versus complex, but finite versus infinite dimensional, and unrestricted versus superselection-restricted. The equivalence is proved for finite-dimensional behaviours and finite adaptive protocols; the infinite-dimensional commuting-measurement version of RQT is expected to be strictly more powerful, but that separation is not proved here. The paper also leaves open the link between representations of complex-superselected theories in RQT and genuinely superselected subtheories of RQT. Within the finite-dimensional, symmetry-restricted setting the paper actually covers, the operational predictions of RQT and QT remain indistinguishable.

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