Sekatski, Yadin, Renou, Dür, Gisin, and Fröwis (New J. Phys. 20, 013025 (2018)) quantified the macroscopic distinctness of a quantum superposition by the information a limited-resolution ('classical') detector can extract about its branches, defining the macroscopicness of information content MICb\mathrm{MIC}_b as the largest detector resolution scale at which bb bits of information about the branch label remain accessible. The framework covers pure states, a convex-roof extension MIC^b(ρ)\widehat{\mathrm{MIC}}_b(\rho) to mixed states, and a discord-based alternative, but it explicitly left open the three ingredients needed to make the measures usable in practice: how to measure or rigorously bound them from experimental data, how to compute the convex-roof extension for general high-rank mixed states, and how to optimize over detector designs with more than a single resolution parameter. This problem asks for these ingredients: certification protocols (including lower-bound witnesses) valid under realistic detector imperfections and finite statistics without full state tomography, tractable algorithms or certified bounds for the mixed-state extensions, and the general multi-parameter detector optimization.

Background

A recurring goal in the foundations of quantum mechanics is to quantify how 'macroscopic' a quantum superposition is — the size of a Schrödinger-cat state, i.e. a coherent superposition of two or more states that differ at a large, classical scale. Early proposals measured the size of two-branch superpositions by the number of microscopic constituents that must be traced or disturbed to erase the interference: Korsbakken, Whaley, Dubois, and Cirac (Phys. Rev. A 75, 042106 (2007)) defined a measurement-based effective size, and Sekatski, Sangouard, and Gisin (Phys. Rev. A 89, 012116 (2014)) defined the distinctness of the two branches by how well they can be told apart with a coarse-grained, 'classical' detector of limited resolution. Other general measures choose a reference observable and quantify the spread of the state in its spectrum: the quantum-Fisher-information (QFI) based effective size of Fröwis and Dür (New J. Phys. 14, 093039 (2012)) — the QFI F(ρ,A)F(\rho,A) is a metric sensitivity of the state under unitaries generated by the observable AA — or the phase-space measure of Lee and Jeong (Phys. Rev. Lett. 106, 220401 (2011)). These either apply only to a pre-selected two-branch structure or lack the operational intuition of detection with a realistic, low-resolution device.

Sekatski, Yadin, Renou, Dür, Gisin, and Fröwis (New J. Phys. 20, 013025 (2018)) closed this gap for the idealized theory, and the present problem is the practical completion they explicitly left open. Their construction: fix an observable A=∑ℓaℓ∣Aℓ⟩⟨Aℓ∣A=\sum_\ell a_\ell|A_\ell\rangle\langle A_\ell| whose eigenstates define the 'branches' (e.g. a collective spin or photon-number operator) and model the classical detector by von Neumann's pointer model — an auxiliary system (the pointer) coupled by U=e−iA⊗p^U=e^{-iA\otimes\hat p} and then read out, with the pointer's initial wave packet of width Δ\Delta playing the role of the detector's finite resolution. A projective readout of the pointer is then a general measurement described by a POVM (positive-operator-valued measure), i.e. a set of positive operators summing to the identity, with elements ξΔ2(x−A)\xi^2_\Delta(x-A). For a superposition ∣Ψ⟩=∑ℓpℓ∣Aℓ⟩|\Psi\rangle=\sum_\ell\sqrt{p_\ell}|A_\ell\rangle the branch label ℓ\ell and the pointer outcome xx are jointly distributed, and their mutual information (Shannon information shared by two random variables) IΔ(A:ℓ)I_\Delta(A:\ell) quantifies what the detector learns about the branches at resolution Δ\Delta. The measure

MICb=max⁡{Δ∣IΔ(A:ℓ)≥b}\mathrm{MIC}_b=\max\{\Delta \mid I_\Delta(A:\ell)\ge b\}

is the largest detector width at which bb bits of branch information remain extractable; the parameter bb acts as a 'rank' counting the effective number of superposed components. The measure is calibrated on reference states of k+1k+1 equally spaced peaks in the spectrum of AA with b=log⁡2(k+1)b=\log_2(k+1), so any state can be reported as 'as macroscopic as k+1k+1 peaks spread over a width NN'.

For mixed states ρ\rho the pure-state measure is extended by the convex roof (the standard device for extending a pure-state quantity to mixed states by minimizing the average over all pure-state ensemble decompositions of ρ\rho), MIC^b(ρ)=min⁡{qk,∣Ψk⟩}∑kqk MICb(∣Ψk⟩)\widehat{\mathrm{MIC}}_b(\rho)=\min_{\{q_k,|\Psi_k\rangle\}}\sum_k q_k\,\mathrm{MIC}_b(|\Psi_k\rangle), and two more tractable variants were introduced: MIC^b′(ρ)\widehat{\mathrm{MIC}}'_b(\rho), based on the convex roof of the mutual information itself, and MIC~b(ρ)\widetilde{\mathrm{MIC}}_b(\rho), based on the measurement-induced entropy increase CΔ(ρ,A)=S(ρM′)−S(ρ)C_\Delta(\rho,A)=S(\rho'_{M})-S(\rho) (with SS the von Neumann entropy), which equals the fixed-basis quantum discord (the non-classical part of the correlations with respect to a fixed measurement) of the post-interaction state. The paper demands four criteria of a macroscopic-coherence measure: (C1) vanishing exactly on incoherent mixtures, (C2) non-increase under operations covariant with AA, (C3) convexity, and (C4) increase with the branch-separation scale; the underlying information quantities satisfy all four (for a Gaussian pointer), while the resulting size measures satisfy the monotonicity and scale criteria together with the vanishing criterion in the threshold-modified form (zero exactly when no detector resolution extracts bb bits; Eqs. (32) and (45) of the source). In the limit of small bb, MIC^b′(ρ)≤F(ρ,A)/(8ln⁡2 b)\widehat{\mathrm{MIC}}'_b(\rho)\le\sqrt{F(\rho,A)/(8\ln 2\, b)} with asymptotic equality, linking the family to QFI-based sizes. The paper also proved that the macroscopicity of the branches governs the fragility of the entanglement in micro-macro states under noise. A companion review (Fröwis, Sekatski, Dür, Gisin, and Sangouard, Rev. Mod. Phys. 90, 025004 (2018)) surveys the whole landscape of such measures.

The framework, however, stops at idealized theory. (i) It provides no protocol by which MICb\mathrm{MIC}_b, MIC^b\widehat{\mathrm{MIC}}_b, or the discord-based variants could be measured or bounded in an actual experiment — no witness inequalities, no estimator from accessible data, no treatment of finite statistics or detector imperfections; only the asymptotic small-bb QFI relation is noted as a possible route to upper bounds. (ii) The convex-roof extension is solved analytically only for one two-branch family of states; the paper states that the convex roof of the inverted function is 'generally difficult to handle in proofs or in calculations of specific examples', and no algorithm or certified bounds exist for general high-rank mixed states or many-body systems. (iii) The detector is modeled by a single parameter, the pointer width Δ\Delta (plus one over-complete phase-space POVM example); the authors explicitly exclude the case of detectors 'with additional knobs', for which an additional optimization would be necessary. Closing these three gaps is what makes the measure family directly applicable to the experiments it was designed to judge.

Problem Statement

Make the macroscopicness-of-information-content (MIC) family of measures of Sekatski, Yadin, Renou, Dür, Gisin, and Fröwis directly applicable to realistic experimental systems, by closing the three gaps the source paper explicitly left open. (a) Certification: construct protocols by which MICb\mathrm{MIC}_b, its convex-roof extension MIC^b(ρ)\widehat{\mathrm{MIC}}_b(\rho), or the discord-based variant MIC~b(ρ)\widetilde{\mathrm{MIC}}_b(\rho) can be measured or rigorously bounded — in particular lower-bounded via witness-type inequalities, which is what certifying macroscopic quantumness requires — from data accessible in an experiment (finite statistics, imperfect and calibratable detectors, no full state tomography). The existing asymptotic small-bb QFI relation yields only upper bounds in a limiting regime and does not answer this. (b) Computation: provide practical algorithms, or certified upper and lower bounds, for the convex-roof extension on high-rank mixed states and many-body regimes, beyond the single two-branch family solved analytically in the source paper. (c) Detector generality: extend the definition to detector families with multiple tunable parameters ('additional knobs' beyond the single pointer width Δ\Delta), including the associated optimization of the extracted mutual information over the detector design, and establish that the extended measure retains the properties the source requires of a macroscopic-coherence measure (monotonicity under operations covariant with AA, increase with the branch-separation scale, and vanishing on states without coherence — in the threshold form the source establishes for its own variants, or the exact form, vanishing precisely on incoherent mixtures, if achieved). These three strands share one objective — rendering the MIC measures experimentally certifiable and computationally tractable — and an answer must make explicit the state, detector, and noise-model assumptions under which its protocol, algorithm, or optimization is valid, and demonstrate it on a concrete experimental or realistic numerical case. Progress on only some strands, or only in restricted regimes, is partial progress on the problem as stated.

References

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