New Localizable Entanglement (NLE)

• New Localizable Entanglement (NLE) is a refined measure that optimizes the minimum concurrence over all measurement outcomes, ensuring a guaranteed entanglement in a selected two-qubit subsystem.
• It addresses an ambiguity in standard localizable entanglement by shifting focus from average performance to worst-case performance in post-measurement states.
• Numerical analyses indicate that the gap between LE and NLE increases with system size, highlighting its operational significance in four- and five-qubit systems.

Searching arXiv for the core NLE paper and closely related LE literature to ground the article in the cited record. New localizable entanglement (NLE) is a variant of localizable entanglement introduced to remove what its authors describe as an ambiguity in the standard definition of localizable entanglement (LE): LE optimizes an average over post-measurement outcomes, whereas a single experimental run produces only one collapsed state on the target subsystem. In the formulation of "New Localizable Entanglement" (Sabour et al., 15 Jul 2025), NLE replaces the outcome-averaged objective by a worst-case objective, optimizing the minimum post-measurement entanglement over all local measurement settings. The resulting quantity is intended to quantify the entanglement that can be guaranteed on a chosen target pair after local measurements on the remaining parties.

1. Origin of the concept and the ambiguity it addresses

The point of departure for NLE is the standard LE framework associated with Verstraete, Popp, and Cirac: given a multipartite pure state, one performs local measurements on all parties except two targets, and LE is defined as the maximum average entanglement obtainable on that target pair after optimization over measurement settings (Sabour et al., 15 Jul 2025). In that setting, the averaged nature of LE is operationally important but also, according to the NLE proposal, conceptually incomplete.

The ambiguity identified in the NLE paper is that an optimized average does not determine the entanglement of the actual post-measurement state produced in a single run. Even when LE is large, the realized two-party outcome may have entanglement anywhere between $0$ and $1$. The paper supports this claim by numerically generating many random three-qubit pure states, grouping states by approximately equal LE, and examining the minimum and maximum entanglement among the measurement-induced collapsed states. As the sample size increases, those extrema approach $0$ and $1$, respectively, even when the LE is fixed in a small interval (Sabour et al., 15 Jul 2025).

Within this perspective, NLE is introduced not as a replacement for all uses of LE, but as a sharpened operational quantity. LE characterizes what can be achieved on average across all outcomes; NLE characterizes what can be guaranteed regardless of which outcome occurs. This suggests a distinction between an ensemble-level resource and a one-shot resource.

2. Formal definition and measurement framework

The NLE construction preserves the basic architecture of entanglement localization. One starts from a multipartite pure state $\lvert \psi\rangle$, chooses two target subsystems $A$ and $B$, and performs local projective measurements on all remaining parties. In the paper’s concrete four-qubit setting, the target pair is $(1,4)$, so qubits $2$ and $3$ are measured after local unitary rotations $1$0 are applied to tune the measurement basis (Sabour et al., 15 Jul 2025).

For that four-qubit case, the standard LE is written as

$1$1

where $1$2 is the post-measurement state on qubits $1$3 and $1$4, $1$5 is the corresponding probability, and $1$6 is the concurrence for pure two-qubit states,

$1$7

NLE changes only the optimization target. Instead of maximizing the average concurrence over outcomes, it maximizes the minimum concurrence over outcomes: $1$8 The conceptual shift is therefore from mean performance to guaranteed performance. For any fixed measurement setting, one inspects all possible collapsed states on the target pair and takes the least entangled one; one then optimizes the measurement setting so that this least favorable outcome is as entangled as possible (Sabour et al., 15 Jul 2025).

The paper performs its concrete analysis for projective von Neumann measurements. It notes that more general local measurement classes such as LOCC or POVMs could be considered in principle, but does not develop that generalization.

3. Relation to standard localizable entanglement

A central theorem of the NLE proposal is the inequality

$1$9

The argument is elementary but structurally important. For any fixed measurement setting, the average of nonnegative numbers is always at least as large as their minimum. Applied to the set of outcome-dependent concurrences,

$0$0

Maximizing both sides over all allowed measurement settings yields $0$1 (Sabour et al., 15 Jul 2025).

The paper does not claim that LE and NLE always coincide, nor does it provide a complete analytic classification of when they do. Instead, it reports a numerical comparison over about $0$2 random states for each of three-, four-, and five-qubit systems. The reported summary statistics are:

• for $0$3: $0$4, $0$5, $0$6;
• for $0$7: $0$8, $0$9, $1$0;
• for $1$1: $1$2, $1$3, $1$4 (Sabour et al., 15 Jul 2025).

Here $1$5 is the maximum relative difference, $1$6 is the maximum absolute difference, and $1$7 is the NLE value of the state for which the maximum absolute difference was observed. The reported trend is that for three-qubit systems the discrepancy between LE and NLE is very small, whereas for four- and five-qubit systems the difference becomes clearly noticeable and then somewhat larger. The paper’s interpretation is that the LE–NLE gap is negligible in small systems but can become significant as the number of parties increases.

4. Examples, lower bounds, and worked illustrations

The NLE paper uses several examples to clarify the distinction between measurement-based localization and simpler reduced-state reasoning. A standard motivating example is the three-qubit GHZ state,

$1$8

If one traces out the third qubit, the remaining two-qubit state is mixed and unentangled. If, instead, one measures the third qubit in an appropriate local projective basis, the other two qubits can be projected onto a Bell state. In the NLE context, this example underscores why entanglement localization is fundamentally a measurement task rather than a reduced-state task (Sabour et al., 15 Jul 2025).

A second worked setting is a four-qubit Ising-chain example with Hamiltonian

$1$9

with Pauli operators on neighboring sites. The paper selects the initial state that exhibits the largest LE–NLE discrepancy at $\lvert \psi\rangle$0, evolves it, and compares three quantities over time: LE, NLE, and a classical correlation quantity $\lvert \psi\rangle$1. In the plotted interval, the ordering

$\lvert \psi\rangle$2

is observed throughout (Sabour et al., 15 Jul 2025).

This observation is used to argue that the classical correlation function $\lvert \psi\rangle$3, already known to lower-bound standard LE in related literature, also appears to lower-bound NLE in this example. The paper is explicit that this is evidence from a concrete example rather than a general theorem. Accordingly, the status of $\lvert \psi\rangle$4 is illustrative rather than universally established.

These examples also clarify the operational distinction between LE and NLE. LE can be high because favorable outcomes dominate the average, even if unfavorable outcomes are weakly entangled or separable. NLE suppresses such averaging effects by construction.

5. Position within the broader literature on entanglement localization

NLE belongs to a broader family of measurement-induced entanglement-localization concepts, but it is distinct from several neighboring notions that are sometimes conflated with it.

A useful contrast is "Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions" (Manna et al., 20 Jan 2026), which does not redefine LE but uses standard LE as an order parameter for monitored-circuit measurement-induced phase transitions. In that work, LE is operationalized as a criterion for whether entanglement can be localized across macroscopic distances; the associated order parameter $\lvert \psi\rangle$5 exhibits a transition near $\lvert \psi\rangle$6, the intrinsic entanglement length $\lvert \psi\rangle$7 diverges as $\lvert \psi\rangle$8, and the extracted exponent is $\lvert \psi\rangle$9. That program is conceptually close to NLE in its operational emphasis, but its object remains standard LE rather than a worst-case variant.

Another distinct construction is the localized entanglement by canonical measurement (LECM) of "Entanglement Localization and Optimal Measurement" (Sahoo, 2012). There the issue is not outcome-averaged versus worst-case localization, but the choice of a canonical measurement basis: the eigenbasis of the environment’s reduced density matrix. LECM is presented as system-determined and easy to calculate in practice, and in symmetry-restricted spin-$A$0 settings it is shown to be optimal.

The term “new localized-entanglement framework” can also refer to regionally localized entanglement in "Block entanglement bounds distribution of regionally localized entanglement" (Krishnan et al., 7 Aug 2025), where entanglement is localized not onto a single pair but onto all two-qubit regions sharing a common hub. That work introduces total RLE as a sum of hub-centered localized contributions and derives bounds relating it to block entanglement and block localizable entanglement.

Similarly, "Localizing multipartite entanglement with local and global measurements" (Vairogs et al., 2024) defines multipartite entanglement of assistance (MEA) and localizable multipartite entanglement (LME), extending localization ideas to seed measures such as the $A$1-tangle, GME concurrence, and concentratable entanglement. The optimization architecture resembles LE, but the target is multipartite entanglement on the unmeasured subsystem rather than bipartite entanglement on a fixed pair.

By contrast, "Uniform Decoherence Effect on Localizable Entanglement in Random Multi-qubit Pure States" explicitly states that it does not define a new quantity called NLE; it studies standard LE, its distribution over random states, and its robustness under noise (Banerjee et al., 2019). This point matters because “NLE” is sometimes used informally for multiple distinct developments, whereas the specific term new localizable entanglement is tied to the worst-case definition of (Sabour et al., 15 Jul 2025).

6. Significance, limitations, and interpretive issues

The principal significance claimed for NLE is operational: it is intended to match the entanglement of the state that can actually be prepared in a single run more closely than LE does. In this sense, NLE is a guaranteed-entanglement quantity rather than an average-yield quantity (Sabour et al., 15 Jul 2025). A plausible implication is that NLE may be the more relevant benchmark when protocols require reliability across outcomes rather than favorable ensemble averages.

At the same time, the construction has explicit limitations. The evidence for differences between LE and NLE is primarily numerical, based on random-state sampling rather than a full analytic characterization. The study is largely restricted to projective von Neumann measurements. The claim that classical correlation lower-bounds NLE is supported by a specific Ising-chain example, not by a general proof. And the paper does not provide a complete criterion for when LE and NLE coincide (Sabour et al., 15 Jul 2025).

A further source of confusion is terminological. In the surrounding literature, “new” developments in entanglement localization include restricted-measurement lower bounds for LE in noisy graph states (Amaro et al., 2018), gain–loss relations linking localizable entanglement to entanglement destroyed during measurement (Krishnan et al., 2022), and criteria for long-range localizable entanglement in matrix product states (Wahl et al., 2012). None of these works defines NLE in the specific sense of maximizing the minimum post-measurement entanglement. Encyclopedic usage therefore benefits from reserving “new localizable entanglement” for the worst-case quantity of (Sabour et al., 15 Jul 2025) and treating other proposals under their own established names.

In that restricted and precise sense, NLE is a one-shot refinement of localizable entanglement. LE asks how much entanglement can be localized on average; NLE asks how much can be guaranteed. The distinction is negligible for some small systems and measurement scenarios, but the available numerical evidence indicates that it can become substantial already at four and five parties.