Measurement-Induced Entanglement (MIE)

• Measurement-Induced Entanglement (MIE) is the process of generating and controlling quantum entanglement via local or global measurements, enabling recovery, purification, and teleportation in quantum systems.
• Protocols using partial measurements with quantum erasure have demonstrated robust entanglement recovery and Bell-CHSH inequality violations, underscoring the impact of measurement basis and circuit design.
• MIE underpins studies of phase transitions and many-body dynamics, with applications ranging from cavity-QED and circuit experiments to quantum error correction and computational advantage.

Measurement-Induced Entanglement (MIE) refers to the generation, modification, or control of quantum entanglement in multipartite systems as a direct consequence of local or global measurements. MIE encompasses protocols that range from the recovery of entanglement lost to decoherence, to the explicit creation of complex, long-range, and even multipartite entangled states via projective or continuous (weak) measurement. The phenomenon is central to contemporary quantum information theory, non-equilibrium quantum dynamics, and underlies emergent critical phenomena such as measurement-induced phase transitions in monitored circuits.

1. Fundamentals of Measurement-Induced Entanglement

At its core, MIE exploits the wavefunction collapse and measurement back-action to redistribute quantum correlations, often converting local or short-range entanglement into nonlocal or long-range entanglement depending on the measurement protocol and system structure. Measurement-induced operations can overcome limitations associated with coherent unitaries, disentangle or reroute existing correlations, and, in some instances, teleport entanglement across spatially separated subsystems. The amount and nature of MIE depend crucially on the measurement basis, the premeasurement entanglement structure, and, for extended systems, the circuit architecture and presence of unitary scrambling dynamics.

Key entropic measures for MIE include:

• The average post-measurement entanglement entropy for a bipartition A/B, after measuring region M in basis $b$:

$$\mathrm{MIE}_b(A,B)[|\psi\rangle] = \sum_m p_m S(A)[|\psi_m\rangle],$$

where $|\psi_m\rangle$ is the post-measurement state after outcome $m$ and $p_m$ is its probability.

• For distinguishing the measurement effect from preexisting entanglement, the measurement-induced information (MII) subtracts the premeasurement mutual information:

$$\mathrm{MII}_b(A,B) = \sum_m p_m I(A,B)[|\psi_m\rangle] - I(A,B)[|\psi\rangle].$$

2. Single- and Few-Particle MIE: Recovery, Purification, and Nonlinear Dynamics

Recovery through Measurement: Partial Measurement and Quantum Erasure

A prototypical demonstration of measurement-induced entanglement recovery is the two-qubit photon experiment reported by Xu et al. (Xu et al., 2010). Two entangled photons traverse a dephasing channel, modeled with birefringent quartz plates. Without intervention, the off-diagonal density matrix components are damped, with concurrence (entanglement) decaying exponentially as:

$C(\rho_1) = \exp(-\alpha^2\sigma^2/8).$

Inserting a measurement apparatus, where a beam displacer partially measures (tags) the photon polarization before dephasing is completed, followed by a reversal/quantum erasure step (implemented via waveplates), enables recovery of the lost coherence. The restored state’s concurrence $C(\rho_2) = |k_b'|$ can approach its maximum value, even after apparent “sudden death” of entanglement. The explicit formula for $k_b'$ shows recovery is sensitive to phase matching between the dephasing intervals and the measurement reversal.

Significantly, Bell-CHSH inequality violation is observed with $S$ parameters well above the classical bound, confirming the nonlocality of recovered states. These protocols establish that tailored measurement interactions, followed by reversal/erasure, can robustly protect or even revive entanglement lost to environmental noise.

Measurement-Based Purification and Measurement-Induced Nonlinear Chaos

Iterative measurement-conditioned protocols can drive ensembles of two-qubit states toward highly entangled or separable regimes depending on initial conditions and noise (Kiss et al., 2011). Here, sequential measurement and conditional selection (squaring amplitudes, postselected on desired outcomes), combined with local unitary updates, induce a nonlinear evolution that can be cast as iterations of a rational map with strong sensitivity to the initial state. Pure-state dynamics lead to fractal basin boundaries (Julia sets) separating regions converging to Bell states from separable states. Incoherent noise further introduces new mixed-state attractors.

This protocol highlights that measurement-induced nonlinear dynamics can be highly sensitive to initial states and noise, with implications for entanglement purification, robust state stabilization, and error correction.

Cavity-QED, Heralded Photonics, and Bad-Cavity Schemes

Measurement-induced protocols in cavity and circuit-QED architectures provide practical tools for entanglement generation between distant static qubits. In the bad-cavity regime (Julsgaard et al., 2012), two qubits couple to a leaking cavity; continuous monitoring of the cavity output via homodyne detection probabilistically projects the system onto a maximally entangled singlet state. Here, selective post-processing (e.g., DC and AC lock-in analysis of the measurement record) discriminates between the singlet (dark) and triplet (bright) manifolds with infidelity scaling linearly with the qubit decoherence rate.

In the strong-coupling circuit-QED regime (Ohm et al., 2015), a Mach–Zehnder interferometer with a single microwave photon interacting resonantly with transmon qubits in both arms implements a parity measurement. Photon detection in an output port heralds projection onto a maximally entangled Bell state in a single shot, achieving deterministic high-fidelity entanglement robust against photon loss.

3. Measurement-Induced Entanglement Phase Transitions and Many-Body Dynamics

Hybrid Unitary-Measurement Circuits and Phase Transitions

When measurement is alternated with entangling unitary evolution in extended quantum circuits, the competition induces sharp entanglement phase transitions. The canonical setting involves random, monitored (projective) circuits on $N$ qubits, with single-qubit measurement applied with probability $p$ after units of unitary evolution (Lunt et al., 2020, Yu et al., 2022, Moghaddam et al., 2023, Yanay et al., 30 Jan 2024, Tang et al., 22 Jun 2025).

• Volume-law phase ($p < p_c$): Entanglement entropy $S(l)$ scales as $S(l)\sim l$ for large subsystems $l$, tracking the Page curve for random pure states.
• Area-law phase ($p > p_c$): Entanglement satisfies $S(l)\sim \mathrm{const}$ or $S(l)\sim \log l$ (at the critical point).

The critical measurement rate $p_c$ is model-dependent, reflecting the interplay of measurement and scrambling. In many-body localized systems (Lunt et al., 2020), $p_c$ depends on the measurement basis: measurements in the basis of local integrals of motion (l-bits) immediately destroy volume-law entanglement ($p_c=0$), while “scrambled” basis measurements retain volume-law up to $p_c>0$.

Critical exponents and scaling: Finite-size scaling analyses extract correlation length exponents (e.g., $\nu \approx 1.3(2)$) and dynamical exponents ($z \approx 1$), often linked to conformal field theory (CFT)-like universality at criticality. For example, at $p=p_c$, Rényi-$n$ entropies scale logarithmically $S_n^l \sim \alpha(n)\ln l + \beta(n)$, with $\alpha(n)=\alpha(\infty)(1+1/n)$ confirming emergent conformal symmetry (Lunt et al., 2020).

Analytical Frameworks and Classical Spin Mapping

Recent progress employs mappings to classical spin or rotor models for the averaged purity (second Rényi entropy) under random circuits (Tang et al., 22 Jun 2025, Yu et al., 2022). Here, the competition between unitary scrambling and measurement-induced purification is encoded in a Lindbladian acting on a permutation-symmetric (large spin $S$) sector:

$$L_{\mathrm{U}} \sim -2^{1-q} \bigl[(1+z)^q + (1-z)^q\bigr], \quad L_{\mathrm{M}} \sim -2^{-q'} \bigl[(1+z)^{q'} + (1-z)^{q'} + (x+iy)^{q'} + (x-iy)^{q'}\bigr].$$

Phase diagrams are governed by symmetry-breaking in the effective spin model, e.g., spontaneous $Z_2$ symmetry breaking corresponds to the emergence of a volume-law entangled “scrambled” phase with Page-like entanglement. Emergent continuous $U(1)$ or $SU(2)$ symmetries can arise for certain measurement types (e.g., two-body or (1,1)-body measurements) in the large-$d$ limit, altering phase boundaries and entanglement dynamics.

A key result is that nonlocal (frustrated, high-order) measurements can themselves induce a globally scrambled phase, with the entanglement scaling governed by both the measurement range $q'$ and local Hilbert space dimension $d$:

$d_c = \frac{q'}{q' - 2}$

sets the critical threshold for volume-law MIE under measurement-only protocols.

4. Universality, Symmetry, and Scaling of MIE in Many-Body Ground States

Universal Structure in 1D and 2D Models

Entropic measures of MIE in many-body ground states reveal universal structures controlled by phase and criticality (Cheng et al., 2023, Khanna et al., 4 Aug 2025):

• In 1D critical systems (e.g., Luttinger liquids, XX chain, CFTs), MIE between spatially separated regions $A$ and $B$ decays as a power-law in the conformally invariant cross-ratio $\eta$, $\mathrm{MIE}\sim \eta^\alpha$ with universal $\alpha$ depending on operator content. Analytical approaches (using the replica trick and boundary CFT techniques) show that physical MIE, involving Born-averaging over outcomes, is sensitive to winding sectors in the CFT and reflects conformal invariance (Khanna et al., 4 Aug 2025).
• In 2D systems, generic finite-depth circuits can display a “teleportation transition,” so even trivial states may show extensive long-range MIE. Here, universal information is extracted from subleading corrections via “scaled” or “partially-traced” MIE measures:

$$\mathcal{T}\mathrm{MIE}(A,B) = 2\,\mathrm{MIE}_{1/2}(A/2,B/2) - \mathrm{MIE}(A,B)$$

which cancels nonuniversal teleportation contributions, isolating the universal phase-sensitive components (Cheng et al., 2023).

Topological orders (toric code, string-net models) display MIE corresponding to the quantum dimension ($S=\log D$), and criticality manifests in the algebraic decay exponents.

Sign Structure and MIE Bounds

The sign structure of a many-body wavefunction fundamentally affects MIE. For sign-free (real, nonnegative) stabilizer (CSS) states, the MIE after projective measurement in the sign-free basis is bounded by the mutual information (Lin et al., 2022):

$\mathrm{MIE}(A:B) \leq \mathrm{MI}(A:B)$

with stronger bounds in simple cases, e.g., for single qubits:

$$\mathrm{MIE} \leq \langle \psi|X_A X_B|\psi\rangle.$$

States with nontrivial sign structure can violate these bounds, allowing MIE to be more long-ranged than mutual information—a diagnostic for quantum criticality, SPT phases, and computational complexity.

5. Multipartite, Long-Range, and Operational Aspects

Multipartite Entanglement, Power Laws, and Graphical Methods

Hybrid unitary-measurement circuits can generate genuine multipartite entanglement (GME) over long distances (Avakian et al., 24 Apr 2024). Criteria based on multipartite entanglement witnesses detect GME even for distant, nonadjacent groups of qubits. At criticality, multipartite entanglement measures (e.g., specialized witness functions $W$ for three or four parties) decay as power laws with exponents larger than their bipartite counterparts, reflecting the increased difficulty of establishing multipartite coherence.

A graphical "spanning graph" representation tracks GME growth—minimal spanning roots connect the relevant subsystems, offering a practical diagnostic for GME in monitored dynamics.

Measurement-Induced Teleportation and Conditional Mutual Information

Protocols utilizing local measurements and postselection can transform short-range entanglement into long-range entanglement suitable for teleportation and secret sharing [(Zhang et al., 5 Feb 2024), Phys. Rev. A 86, 052335]. The conversion is characterized by growth in quantum conditional mutual information (CMI), whose universal bound and maximization structure depend on the measurement protocol and system structure.

In non-Markovian settings, MIE dynamics can be explicitly engineered or bounded by optimizing measurement orderings and intermediate state factorization.

6. Experimental, Computational, and Quantum Information Implications

Efficient Detection and Measurement-Device Independence

Direct tomography of entanglement under monitored dynamics is exponentially costly. Recent advancements (Moghaddam et al., 2023, Yanay et al., 30 Jan 2024) leverage polynomially efficient proxies, such as bipartite or multipartite fluctuations (e.g., conserved spin variance), and hybrid quantum–classical “mirror” circuit protocols using matrix product states (MPS) to bound entanglement entropy from above and below. These approaches enable scalable demonstration of MIE phase transitions and critical points even on near-term quantum hardware.

Measurement-device-independent (MDI) schemes based on semiquantum nonlocal games provide universal, operationally accessible entanglement measures, facilitating certification of MIE in scenarios with untrusted or imperfect measurement implementations (Shahandeh et al., 2016).

MIE and Quantum Advantage

Long-range, measurement-induced entanglement is a definitive resource for quantum computational advantage. In random shallow 2D Clifford circuits, the presence of long-range (GHZ-type) MIE is proven to preclude efficient classical simulation by any shallow circuit with bounded fan-in, establishing unconditional quantum advantage in random circuit sampling (Watts et al., 30 Jul 2024). Conversely, when only short-range MIE is present (as in circuits with efficient local-with-shielding simulation), classical parallelization remains possible.

Probing Quantum Gravity

Measurement-induced entanglement provides a theoretically viable route for seeking non-classical signatures in contexts where direct detection is implausible. For example, in gravitational wave detectors, computation of bipartite measurement-induced entanglement entropy of the output state for a fixed number of gravitons demonstrates a detectable, normalized quantum signature—achievable with practical coincident detection protocols and sidestepping the infeasibility of single-graviton projective detection (Jones et al., 23 Nov 2024).

7. Open Problems, Universality, and Future Directions

The robust universality of MIE phase transitions—critical exponents, scaling relations, and conformal field theory links—remains a highly active research area. Open questions include:

• The evolution of universality classes as a function of disorder, interaction, symmetry, or circuit architecture (Lunt et al., 2020, Tang et al., 22 Jun 2025).
• Optimization of measurement protocol locality and basis to control MIE and enhance computational resourcefulness or error correction (Tang et al., 22 Jun 2025, Lin et al., 2022).
• Extension to higher-dimensional, nontrivial topological, or fracton phases where teleportation transitions can obscure or enhance universal MIE signatures (Cheng et al., 2023).
• Experimental realization of multipartite, long-range MIE in noisy hardware, and construction of scalable classical-quantum hybrid detection protocols (Hoke et al., 2023, Moghaddam et al., 2023).
• Application of MIE bounds and diagnostics for detecting and classifying sign problems in quantum simulation and tensor network representations (Cheng et al., 2023, Lin et al., 2022).

Consequently, MIE constitutes both a central theoretical arena for entanglement dynamics and a critical operational resource underpinning the new frontier of quantum information processing, simulation, and foundational tests of quantum reality.