The Observer Problem

Nidhin Sudarsanan Ragini, Mário Ziman

We identify and study a particular class of distinguishability problems for quantum observables (positive-operator-valued measures), in which observables with permuted effects are involved, which we call the labeling problem. Consequently, we identify binary observables, which can be labeled perfectly. In this work, we study these problems in the single-shot regime.

Massimiliano Sassoli de Bianchi

Seungbeom Choi

Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure.

Leslie Eastman

Hector Dehesa

M. Yoshida, Takayuki Miyadera, Hideki Imai

In 2001, Jefferey Bub proposed a quantum key distribution protocol using the Mean King problem. In the protocol, King chooses one of the observables σ x or σ z , while Alice is required to measure an observable that solves the Mean King problem for three observables σ x , σ y and σ z . The observable employed by Alice is thus rather complicated. In this paper, we give other simple solutions of the Mean King problem for two observables and examine the quantum key distribution protocols using them. Our result shows that not all the solutions of the Mean King problem are available to construct a secure protocol. In addition, we consider a protocol that is tolerant with three attack scenarios and derive a trade-off inequality between Eve's information gain and error probability observed by the legitimate users.