Measurement-Invisible Quantum-Correlated Phase

Updated 20 November 2025

Measurement-invisible quantum-correlated phases are quantum states exhibiting hidden entanglement that remains undetected by standard local measurements due to information delocalization.
They arise in diverse platforms such as many-body unitary circuits, optomechanical systems, and fermionic chains, often revealed only by nonlinear, collective, or higher-order detection protocols.
Their discovery challenges conventional measurement methods, prompting the development of advanced detection schemes to harness hidden quantum resources for computation and metrology.

A measurement-invisible quantum-correlated phase is a phase of quantum matter or a dynamical regime in which quantum correlations—including entanglement and other nonclassical resources—are present but cannot be detected, harnessed, or even made manifest by standard forms of local measurement on subsystems, or by routine readout protocols such as homodyne detection, two-point correlators, or single-qubit tomography. Instead, such correlations are either revealed only via nonlinear, collective, or higher-order operations, or remain “hidden” in all outcome-conditioned ensembles accessible by local measurement. This concept spans diverse settings, from many-body unitary circuits, monitored fermionic systems, and spin chains, to engineered cavity optomechanics and quantum optical interferometry.

1. Conceptual Definition and Fundamental Mechanisms

Measurement-invisible quantum-correlated phases arise when nontrivial entanglement, nonlocality, or quantum correlations are present in a quantum system, but none of the conditional statistics on localizable subsystems reveal any dependence on the outcomes of projective (or more general) measurements on complementary subsystems. That is, although the global state is quantum correlated, for every measurement outcome $m$ on subsystem $B$ , the conditional state $\rho_{A|m}$ on $A$ is identical to $\rho_A$ , and thus all observable correlations in $A$ are independent of $m$ (Sherry et al., 31 Oct 2024). Operationally, all joint probabilities $P(a_A,a_B)$ of local observables $A\otimes B$ factorize.

The general mechanism for such phases includes:

Quantum information scrambling, where entanglement becomes highly delocalized;
Monitored quantum systems, where measurement unravels but does not extract certain global correlations;
Non-phase-sensitive readout protocols, which fundamentally discard or average out certain quadrature or phase information;
Hidden-variable or higher-order correlation models, in which phase or coherence dependencies exist but are not extractable by single-shot or first-order measurement.

2. Paradigmatic Theoretical and Experimental Realizations

2.1 Scrambling Dynamics and Conditional Ensemble Invisibility

In the model of collectively entangled qubit systems subjected to scrambling unitary evolution, projective measurement on a subsystem $B$ can partition entanglement phases into:

A disentangled phase (no entanglement between $A$ and $B$ ),
A measurement-invisible quantum-correlated (MIQC) phase (finite entanglement, but all conditional states $\rho_{A|m}$ equal to $\rho_A$ ), and
A measurement-visible phase (conditional states depend on measurement outcomes) (Sherry et al., 31 Oct 2024).

This structure is mathematically delineated via entanglement monotones (e.g., logarithmic negativity $\mathcal{N}_{AB}$ ) and measurement-invisibility metrics (relative-entropy spread $\mathcal{D}_{AB}$ , trace-distance $\Delta_{AB}$ ). In the MIQC phase, $\mathcal{N}_{AB}>0$ but $\mathcal{D}_{AB} = \Delta_{AB} = 0$ for all measurement bases on $B$ , establishing the strict criterion for invisibility.

2.2 Optomechanical Quantum Squeezing and Phase-Invisible Correlations

In cavity optomechanics under strong measurement, output field quadratures develop complex quantum correlations, including cross correlations $S_{XY}(\omega)$ whose imaginary part is inaccessible to standard homodyne detection (which only measures $\mathrm{Re}\,S_{XY}$ ). These “measurement-invisible” correlations can be made visible via a phase-sensitive detection protocol employing two local oscillators at separated frequencies and phases (Ockeloen-Korppi et al., 2018). This recovers the full complex-valued matrix of output correlations and reveals the system's minimal-squeezed quadrature, which had been invisible to single-phase measurement.

2.3 Long-Range Monitored Fermionic “Dark-State” Phases

In free-fermion chains with long-range hopping ( $t_{ij}\propto|i-j|^{-p}$ ) and local density monitoring, an unconventional phase arises for $1<p\lesssim1.5$ —the algebraic (dark-state) phase—characterized by algebraic growth of entanglement entropy $S(\ell)\sim \ell^{b}$ and power-law decay of density correlations. This phase remains invisible to local measurement, as local observables saturate, but nonlocal entanglement persists below the critical measurement rate (Müller et al., 2021).

2.4 Measurement-Induced Nonlocality in Integrable Spin Chains

The “measurement-induced nonlocality” (MIN) between subsystem pairs, as quantified via the trace norm $N_1(\rho)$ , can be strictly positive even in regions where the concurrence (entanglement) $C(\rho)$ vanishes. Thus, nonlocal quantum correlations exist that are not detectable by conventional entanglement or discord measures but only via global state perturbation (Bhuvaneswari et al., 2021).

2.5 MBQC Resource States Hidden in Disordered Phases

The ground state of the 2D AKLT model, in its gapped disordered (short-range-correlated) regime, appears featureless to two-point observables. However, local filtering followed by adaptive single-particle measurements “activates” a universal graph-state suitable for measurement-based quantum computation, revealing hidden long-range entanglement that is otherwise invisible to standard probes (Darmawan et al., 2011).

2.6 Higher-Order Intensity Products and Hidden Phases

In single-photon interferometry with polarization-tagged arms and relative phase $\phi$ , first-order (intensity) measurement yields no information about $\phi$ . Only second- or higher-order (coincidence) measurements, i.e., products $I_1(\phi)I_2(\phi)$ , reveal a quantized dependence on $2\phi$ , manifesting photonic de Broglie-wave fringes and Bell inequality violation (Ham, 2023). Here, the phase is operationally “measurement-invisible” at first order.

3. Phase Diagrams, Scaling Laws, and Analytical Criteria

Measurement-invisible phases are typically bounded by:

An entanglement transition (vanishing of a monotone such as negativity $\mathcal{N}_{AB}$ at, e.g., $p_c=\frac{1-\gamma}{2}$ ),
A measurement-invisibility transition (onset of $\mathcal{D}_{AB},\Delta_{AB}>0$ at $p_c=1-\gamma$ in scrambling circuits) (Sherry et al., 31 Oct 2024).

Scaling relations and critical exponents, where computed, include algebraic scaling of entropy with block size in monitored fermion chains $S(\ell)\sim \ell^{3/2-p}$ if $1<p\leq1.5$ ; $S(\ell)\sim \log\ell$ at $p_c=1.5$ ; $S(\ell)\sim\text{const}$ above the critical point. Analytical criteria often exploit Haar-random or replica-bosonic field theory to bound the spread of conditional state ensembles and establish the invisibility regime.

4. Operational and Experimental Implications

Measurement-invisible phases profoundly impact:

The operational ability to harness quantum resources for measurement-feedback or quantum computation. In the MIQC regime, although entanglement exists, measurement-feedback protocols on observed subsystems yield no information gain or steerability (Sherry et al., 31 Oct 2024).
Quantum-enhanced metrology, where undetected squeezing or correlations limit optimal force sensing or phase estimation. Phase-sensitive detection protocols explicitly designed to reveal measurement-invisible components can extend sensitivity below standard quantum limits (Ockeloen-Korppi et al., 2018, Ham, 2023).
The practical visibility of entanglement in noisy, monitored or open systems—certain high-entanglement phases are experimentally invisible unless nonlocal observables or specific detection schemes are implemented (Khindanov et al., 12 Oct 2024).
Resource activation, wherein a system's computational utility is masked in its “natural” observable ensemble but is revealed via nonlocal, adaptive, or filtering measurements (Darmawan et al., 2011).

5. Measurement Protocols and Detection Strategies

Strategies for revealing measurement-invisible phases include:

Bi-chromatic or multi-tone local oscillator schemes to access full quadrature correlations (recovering both real and imaginary parts) (Ockeloen-Korppi et al., 2018).
Conditional state ensemble measurements, utilizing full tomography across measurement outcomes—practically challenging due to exponential sample complexity.
Reversal or time-inversion protocols, allowing direct observation of a measurement-induced phase transition via the polynomially measurable return probability in repeated blocks (Khindanov et al., 12 Oct 2024).
Nonlinear (higher-order) or coincident detection protocols in photonic settings (Ham, 2023).
Local filtering and adaptive post-processing, as in activatable computational phases of many-body ground states (Darmawan et al., 2011).
Use of robust nonlocality measures (e.g., MIN) over conventional entanglement monotones (Bhuvaneswari et al., 2021).

Measurement-invisible quantum-correlated phases are sharply distinct from:

Classical or disentangled phases (zero monotone, trivial conditional ensembles),
Measurement-visible entangled phases (conditional ensembles are nontrivial, enabling steering, feedback, and operational correlation),
Quantum phases detectable by local correlation functions or linear response,
Measurement-induced phase transitions as identified by abrupt changes in entanglement scaling, but which may remain physically inaccessible due to measurement limitations (Sherry et al., 31 Oct 2024, Khindanov et al., 12 Oct 2024, Müller et al., 2021).

A comparative schema is illustrated below:

Phase Type	Entanglement ( $\mathcal{N}$ )	Conditional Ensemble Spread ( $\mathcal{D}$ )	Measurement-Feedback Utility
Disentangled	0	0	None
Measurement-Invisible (MIQC)	$>$ 0	0	None
Measurement-Visible Entangled	$>$ 0	$>$ 0	Possible

7. Broader Theoretical and Practical Significance

Measurement-invisible quantum-correlated phases provide new insight into the nonlocal structure of quantum many-body states and circuits, revealing that entanglement and nonlocality may exist independently of any operational local extractability. This has direct implications for fault-tolerant quantum computing, error correction, quantum metrology, and the understanding of information scrambling and non-Hermitian quantum dynamics. The identification and activation of such “hidden” resource phases represent an active frontier in both theoretical quantum science and experimental design (Ockeloen-Korppi et al., 2018, Sherry et al., 31 Oct 2024, Darmawan et al., 2011, Bhuvaneswari et al., 2021, Khindanov et al., 12 Oct 2024).

A plausible implication is that the design of readout protocols—whether in optomechanical sensors, monitored quantum processors, or photonic interferometers—must be matched to the target resource: failure to select measurement-adaptive schemes may overlook fundamentally useful quantum correlations that are operationally “invisible” to standard probes.