We consider a restricted two-qubit model in which an unknown local unitary acts on one subsystem of a maximally entangled qubit pair and only local Pauli-product correlations are measured. We show that the three same-axis correlators XX, YY, and ZZ determine the magnitudes of the four components of the associated unit quaternion but, at generic interior points, leave an eightfold ambiguity in the underlying local rotation. We express this ambiguity as a sign fiber isomorphic to F2^3. The six cross-axis Pauli correlators separate into three rank-two sign-character subspaces. This structure yields a complete classification of the fifteen possible pairs of cross-axis correlators: the three reciprocal pairs XY/YX, XZ/ZX, and YZ/ZY remain rank deficient and generically leave a twofold ambiguity, whereas the remaining twelve pairs generically resolve the eightfold alias under explicit amplitude-nondegeneracy conditions. We then formulate a finite-shot adaptive reconstruction procedure that selects well-separated correlator groups and abstains near degeneracy, and we distinguish conditional reconstruction from held-out model auditing. The results concern identifiability within a restricted local-unitary model and do not constitute a new general tomography bound or evidence for additional physical states.
We consider a restricted two-qubit model in which an unknown local unitary acts on one subsystem of a maximally entangled qubit pair and only local Pauli-product correlations are measured. We show that the three same-axis correlators XX, YY, and ZZ determine the magnitudes of the four components of the associated unit quaternion but, at generic interior points, leave an eightfold ambiguity in the underlying local rotation. We express this ambiguity as a sign fiber isomorphic to F2^3. The six cross-axis Pauli correlators separate into three rank-two sign-character subspaces. This structure yields a complete classification of the fifteen possible pairs of cross-axis correlators: the three reciprocal pairs XY/YX, XZ/ZX, and YZ/ZY remain rank deficient and generically leave a twofold ambiguity, whereas the remaining twelve pairs generically resolve the eightfold alias under explicit amplitude-nondegeneracy conditions. We then formulate a finite-shot adaptive reconstruction procedure that selects well-separated correlator groups and abstains near degeneracy, and we distinguish conditional reconstruction from held-out model auditing. The results concern identifiability within a restricted local-unitary model and do not constitute a new general tomography bound or evidence for additional physical states.
We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.
Operational data can be complete inside a chosen subsystem frame while leaving the frame itself non-unique. A previous finite certification construction exhibits this phenomenon explicitly: an unpartitioned three-qubit Pauli presentation supports three inequivalent finest full tensor frames even though its generated observable algebra is all of M_8(\mathbb C). We ask what additional information can select one frame without treating that information as merely another unlabelled control direction. We introduce relational anchors: gauge-equivariant constraints on a finite refinement poset of direction-generated frames. A general finite-poset theorem characterizes exact anchor collapse and turns target certification into a hitting-set problem, while global frame identification becomes a test-cover problem. In the three-qubit presentation we enumerate the complete nontrivial full witness poset, prove that one two-direction locality anchor separates all three finest frames, and determine exactly twelve single anchors of this type that pass the full-poset test. For robustness, exact support is replaced by conditional-expectation signatures, yielding a sharp \gamma/2 separation criterion. We prove a zero-radius result for the exact witness poset under unrestricted raw perturbations and derive a sharp projective Pauli decoding radius. Six Hamiltonian-derivation squares reconstruct the projective Pauli geometry. A residual synchronization analysis separates ordinary graph cycles from gauge-invariant parity constraints, and four atomic Jordan–Lie anchors are necessary and sufficient, within the declared anchor class, to reduce the remaining scale and sign freedom to Clifford gauge. An explicit relative-defect rigidity modulus and a protocol-conditioned end-to-end error certificate are obtained. The result does not claim that probabilities alone determine a physical tensor-product structure. Rather, unpartitioned process data together with a finite relational anchor package robustly identify a subsystem frame up to the declared gauge.
The three-qubit CCZ state is the smallest rank-three hypergraph state and an elementary entangled magic resource. Its cubic phase is governed by generalized stabilizers that are not Pauli strings, so standard graph-state self-testing arguments do not apply directly. We show that twenty correlators, all obtainable from five of the eight global input triples in the tripartite two-input, two-output scenario, determine this state and the action of the Pauli X/Z measurements up to local isometries. The proof fixes eight equally weighted computational branches and propagates conditional X-flip relations across the branch cube, recovering the minus sign of the 111 amplitude. These five-context correlations are nonlocal, but the canonical Pauli measurements cannot attain the largest quantum value of any Bell inequality that they violate: whenever they maximize a Bell expression, its local bound has the same value. Introducing an independent third measurement makes self-testing from maximal Bell violation possible. We construct an explicit Bell inequality whose maximal quantum violation self-tests the CCZ state and all three local measurements. An exact sum-of-squares decomposition proves the quantum bound, and its equality conditions yield an analytic SWAP extraction. Together, these results give two explicit device-independent self-tests of the CCZ state and demonstrate that determining a state and its measurements from several correlator equalities is distinct from identifying them through the maximal violation of a single Bell inequality.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with N spatial orbitals encoded via Jordan-Wigner, the X-rank, the binary rank of the X parts of its strings, of any commuting Pauli string family is no more than 2(N−1). Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH4, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on n qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at 2n+1.
This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.