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arXiv:2606.09765

Complexity-driven transitions in quantum observation

Zhenyu Du, Siyuan Cheng, Han Ye, Junjie Chen, Xiao Yuan, Xiongfeng Ma

quant-phSubmitted Jun 8, 2026• Announced Jun 9, 2026• 0 citations
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Abstract

Observing the physical world is a foundational pursuit in science. In the quantum realm, however, observation necessitates a fundamental quantum-to-classical conversion: destructive measurements irreversibly project quantum states into classical data, inevitably incurring a loss of information. What physical principles govern this information loss, and how can we construct optimal measurements to maximize the readout? Here, we address these questions by establishing an intrinsic relationship between readout capability--quantified by the ratio of accessible classical Fisher information to the total quantum Fisher information (QFI), and measurement complexity--defined as the quantum circuit depth required prior to projection. Remarkably, we uncover a sudden emergence of observability: a sharp hidden-to-visible transition driven entirely by measurement complexity. We rigorously prove that below critical depth thresholds--$Θ((\log n)^{1/δ})$ for $δ$-dimensional architectures and $Θ(\log\log n)$ for all-to-all connectivity--readout capability decays exponentially with system size $n$, rendering the quantum information fundamentally inaccessible. Surprisingly, immediately above this threshold, the system enters a visible regime: we demonstrate that randomized measurements universally recover a constant fraction of the QFI using approximate unitary 3-designs, for which we explicitly develop optimal-depth circuit constructions tailored to finite-dimensional architectures. By unveiling the fundamental scaling laws and transitions that govern quantum observation, our results delineate definitive resource boundaries for quantum learning, state certification, and quantum metrology.

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39 pages, 3 figures, comments are welcome!

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Observing the physical world is a foundational pursuit in science. In the quantum realm, however, observation necessitates a fundamental quantum-to-classical conversion: destructive measurements irreversibly project quantum states into classical data, inevitably incurring a loss of information. What physical principles govern this information loss, and how can we construct optimal measurements to maximize the readout? Here, we address these questions by establishing an intrinsic relationship between readout capability--quantified by the ratio of accessible classical Fisher information to the total quantum Fisher information (QFI), and measurement complexity--defined as the quantum circuit depth required prior to projection. Remarkably, we uncover a sudden emergence of observability: a sharp hidden-to-visible transition driven entirely by measurement complexity. We rigorously prove that below critical depth thresholds--Θ((log⁡n)1/δ)Θ((\log n)^{1/δ})Θ((logn)1/δ) for δδδ-dimensional architectures and Θ(log⁡log⁡n)Θ(\log\log n)Θ(loglogn) for all-to-all connectivity--readout capability decays exponentially with system size nnn, rendering the quantum information fundamentally inaccessible. Surprisingly, immediately above this threshold, the system enters a visible regime: we demonstrate that randomized measurements universally recover a constant fraction of the QFI using approximate unitary 3-designs, for which we explicitly develop optimal-depth circuit constructions tailored to finite-dimensional architectures. By unveiling the fundamental scaling laws and transitions that govern quantum observation, our results delineate definitive resource boundaries for quantum learning, state certification, and quantum metrology.
Quantum Fisher information (QFI) determines the ultimate precision of a quantum sensor after optimization over all allowed measurements, but it does not quantify how difficult the optimal readout is to implement. We introduce a moment Fisher hierarchy FM(k)F_M^{(k)}FM(k)​, which measures the fraction of QFI accessible to collective spin moments up to degree kkk, together with a readout-complexity scale k90k_{90}k90​ defined by the minimum degree required to recover 90%90\%90% of the QFI. Applying this framework to a nonlinear collective-spin sensor, we show that many-body dynamics can preserve large QFI while transferring the optimal readout into high-rank SU(2)SU(2)SU(2) tensor sectors inaccessible to low-order collective observables. For system sizes up to N=100N=100N=100, first moments access only ∼10−2\sim 10^{-2}∼10−2 of the available QFI, while the degree required to recover 90%90\%90% of it grows to k90≃28k_{90}\simeq 28k90​≃28. A rank-resolved analysis of the symmetric logarithmic derivative (SLD) under weak damping shows that the nonlinear protocol pushes the optimal readout toward the highest available tensor ranks while strongly depleting its low-rank weight. These results identify nonlinear tangent-space shear as the mechanism behind a finite-size readout-complexity bottleneck in many-body quantum metrology.
We present a unitary and operational account of quantum measurement in which outcome irreversibility arises from computational complexity rather than fundamental nonunitarity. Measurement is modeled as a two-stage physical process. First, local interactions dynamically select a pointer basis in which stable classical records are formed via decoherence. Second, subsequent local scrambling dynamics distribute these records globally across a finite quantum memory register. Using explicit circuit constructions based on local random circuits that form approximate unitary k-designs, we show that measurement records are encoded in high-order nonlocal correlations. We prove that reversing such records requires quantum circuits of exponential depth, even allowing ε-approximate reconstruction. This establishes a quantitative complexity barrier rendering outcome reversal physically unrealizable despite global unitarity. We apply the framework to multi-register (Wigner-type) scenarios and show that no physically realizable operation can expose inconsistency between distinct register-relative descriptions. Finally, we prove a conditional no-go theorem excluding any global collapse mechanism compatible with unitary dynamics, locality, and computational feasibility. Measurement irreversibility is therefore shown to be an emergent, complexity-protected phenomenon within standard quantum mechanics
Quantum-enhanced sensing is commonly benchmarked using the quantum Fisher information (QFI), often interpreted as a direct indicator of achievable precision. However, such an interpreta- tion is only justified within a fully specified inference framework that consistently incorporates state preparation, measurement design, resource accounting, estimator construction, prior information, and finite data effects. Here we develop an operational, end-to-end framework for quantum sensing under finite resources and identify general principles required for meaningful performance assess- ment. A central conceptual point is that the relevant unit of estimation is not a single detection event, but the complete data set required to construct a consistent estimator. Rather than relying on abstract statistical formalism, we analyze a set of paradigmatic and experimentally relevant ex- amples that expose common pitfalls in widely discussed quantum sensing strategies. These include quantum interferometry, sensing with squeezed states, and criticality-inspired protocols. Our results clarify when large Fisher information or nonclassical resources translate into genuine metrological advantage, and when they do not. They further provide a practical methodology for designing and evaluating quantum sensing protocols under realistic experimental constraints.
We present a unitary and operational account of quantum measurement in which outcome irreversibility arises from computational complexity rather than fundamental nonunitarity. Measurement is modeled as a two-stage physical process. First, local interactions dynamically select a pointer basis in which stable classical records are formed via decoherence. Second, subsequent local scrambling dynamics distribute these records globally across a finite quantum memory register. Using explicit circuit constructions based on local random circuits that form approximate unitary k-designs, we show that measurement records are encoded in high-order nonlocal correlations. We prove that reversing such records requires quantum circuits of exponential depth, even allowing ε-approximate reconstruction. This establishes a quantitative complexity barrier rendering outcome reversal physically unrealizable despite global unitarity. We apply the framework to multi-register (Wigner-type) scenarios and show that no physically realizable operation can expose inconsistency between distinct register-relative descriptions. Finally, we prove a conditional no-go theorem excluding any global collapse mechanism compatible with unitary dynamics, locality, and computational feasibility. Measurement irreversibility is therefore shown to be an emergent, complexity-protected phenomenon within standard quantum mechanics.
The quantum-to-classical transition is often operationally identified through one preferred indicator, such as coherence loss, classical record formation or measurement stability. This paper develops a different reconstruction test: classicality is treated as a witness-indexed, observer-explicit and recoverability-sensitive cross-scale transition rather than as a single hard boundary inserted by assumption. Starting from a fine description X, a candidate effective state Γ and a future observable family Y, we define a predictive-retention ratio η_pred=I(Γ;Y)/I(X;Y) and residual ε_cl=I(X;Y|Γ). We distinguish raw relation count from record-forming capacity K_rec^(δ), introduce relation-to-record efficiency κ_R, define a transition spectrum T_C={t_W} across a declared witness family and its width ΔT_C=max_W t_W−min_W t_W, and require observer counterfactuals so that conditioning or readout cannot manufacture a physical transition. An observer-conditioned recoverability diagnostic separates information destroyed, information dispersed into wider system–environment relations and information inaccessible to a restricted observer. The resulting classicality vector is deliberately non-compensatory unless a physical model proves otherwise. The construction is a falsifiable physical-regression architecture, not a derivation of quantum mechanics from Ω₀/A2, and it makes explicit the calculations and counterexamples required before any stronger claim is admissible.
Random unitaries are fundamental to quantum information and many-body physics, with widespread applications ranging from quantum learning and metrology to device benchmarking. A central pursuit is to minimize the space and circuit depth required to generate them. However, existing methods for generating low-depth random unitaries rely heavily on an extensive number of ancillary qubits, imposing severe spatial overhead. In this work, we prove that random unitaries can be generated in optimal depth without ancillae. For multiplicative-error approximate kkk-designs on nnn qubits, our circuits achieve a depth of O~(k)(log⁡n)1/δ\widetilde{O} (k) (\log n)^{1/δ}O(k)(logn)1/δ on δδδ-dimensional architectures and O~(k)log⁡log⁡n\widetilde{O}(k) \log \log nO(k)loglogn with all-to-all connectivity. Furthermore, by introducing a general exactification lemma, we lift our construction to optimal-depth exact kkk-designs, yielding an exponential resource reduction over state-of-the-art exact constructions. Our results minimize the space-time costs for a wide range of quantum protocols.