A convergent hierarchy probes classical explanations of quantum states

A set of quantum states can encode information in relationships that have no classical counterpart. Yet noncommuting states—states that cannot all be diagonalized in the same basis—do not by themselves rule out every classical account of how those states were prepared. Shared randomness can mix different preparation procedures, each of which is classical on its own. The question in this paper is whether an entire set of states can be reproduced through such a mixture.

The authors call a set classically simulable if it can be produced using shared randomness and preparation devices that, when conditioned on that randomness, generate states that commute with one another. The mixture as a whole may be noncommuting. This makes the question more subtle than simply checking whether the supplied states commute.

Connecting states to measurements

The authors’ main contribution is a complete hierarchy of semidefinite programs (SDPs). These are optimization problems with constraints on positive semidefinite matrices. Although SDPs can be approached with numerical optimization tools, their size and resource demands can limit which instances are practical to solve.

To build the hierarchy, the paper reformulates classical simulability in terms of deterministic response functions—rules assigning an outcome to each preparation label—and auxiliary positive-operator-valued measures, or POVMs. A POVM describes a general quantum measurement. The reformulation connects the original question to whether those auxiliary measurements can be simulated by mixtures of rank-one projective measurements, a more restricted kind of measurement.

Each finite hierarchy level supplies an outer approximation to the set of classically simulable state sets. “Outer” means it may include candidates that are not truly simulable. Thus, if a state set falls outside a level’s approximation, it cannot be classically simulable; if it falls inside, that finite-level result alone does not establish that a classical simulation exists. The authors prove that the approximations converge to the exact set as the hierarchy continues through all levels. This is a guarantee about the full sequence, not a promise that a modest or practical calculation settles every case.

The dual form of an SDP can also produce an affine witness: a mathematical expression built from expectation values of the states. If the values violate its threshold, the witness rules out classical simulability. The authors describe these witnesses as potentially experimentally accessible without full state-set tomography. That is a theoretical possibility, not a reported experimental demonstration or evidence that fewer measurements have been achieved in practice.

Noise bounds and examples

The researchers apply the hierarchy to state sets under depolarizing noise, which mixes each state with the maximally mixed state. They calculate upper bounds on critical visibility: the largest signal strength for which the noisy set could still be classically simulable. Because finite hierarchy levels use outer approximations, their bounds are upper bounds on the true critical visibility, not generally exact thresholds.

For four symmetric qubit examples—BB84, six-state, trine and tetrahedral state sets—the level-two upper bounds are matched by explicit classical simulations constructed with guidance from the numerical solutions. In these cases, the matching bounds and simulations establish the reported thresholds. The paper also reports a matched threshold for one qutrit example built from states in two mutually unbiased bases. Other higher-dimensional examples have level-two upper bounds above the available lower bounds from explicit simulations, leaving a gap rather than determining the exact threshold.

The authors also compare a detailed hierarchy with an “aggregated” version. Aggregation is computationally cheaper, but gives a weaker outer approximation. The detailed version retains more of the hidden-variable structure and can be tighter, at greater computational cost. For the higher-dimensional examples where both versions could be computed, their level-two bounds agreed within numerical precision. This is a result for those examples, not a general equivalence. The paper notes that at least one calculation could not be completed with the available memory.

Limits and context

The convergence proof establishes what the hierarchy captures in the limit; it does not show that finite levels will be efficient or decisive for arbitrary state sets. Evidence that low levels can be informative comes from selected examples, particularly symmetric ones. Numerical agreement should also be distinguished from an exact proof: the matched cases are supported by explicit simulations, while other reported bounds remain unresolved.

This is a theoretical and computational study, not a laboratory experiment. The source is an arXiv preprint, so its claims are reported by the authors and should not be treated as independent confirmation. The work offers a systematic way to bound or rule out classical explanations, while leaving open how well its computational approach will scale across broader classes of quantum state sets.