logarithmic-negativity-entanglement-cost

name	logarithmic-negativity-entanglement-cost
description	Logarithmic negativity as exact entanglement cost methodology — proving logarithmic negativity equals the exact entanglement cost for typical quantum states. Use when analyzing quantum entanglement quantification, entanglement cost computation, quantum resource theory, or logarithmic negativity measures.
metadata	{"arxiv_id":"2607.01320","published":"2026-07-02"}

Logarithmic Negativity as Exact Entanglement Cost

Core Concept

This methodology establishes that logarithmic negativity — a computable entanglement measure — typically equals the exact entanglement cost for quantum states. This bridges the gap between an efficiently computable measure (logarithmic negativity) and the operationally meaningful but generally intractable exact entanglement cost, providing a powerful tool for quantifying entanglement resources in quantum information theory.

Key Technical Insights

1. Logarithmic Negativity: Defined as $E_N(\rho) = \log_2 |\rho^{T_B}|_1$, where $\rho^{T_B}$ is the partial transpose. It is computable via the trace norm of the partially transposed density matrix.

2. Exact Entanglement Cost: The minimum rate of maximally entangled states (Bell pairs) needed to prepare a given state under LOCC (Local Operations and Classical Communication). Generally intractable due to asymptotic regularization.

3. Typicality Result: For typical quantum states (drawn from appropriate ensembles), the logarithmic negativity concentrates around the exact entanglement cost, eliminating the regularization gap that separates them in general.

4. Concentration of Measure: The result leverages concentration of measure phenomena on high-dimensional quantum state spaces — Lipschitz observables concentrate around their median/mean values.

Implementation Patterns

Computing Logarithmic Negativity

import numpy as np
from scipy.linalg import sqrtm

def partial_transpose(rho, dims, sys=1):
    """Partial transpose of bipartite density matrix."""
    d1, d2 = dims
    rho_pt = np.zeros_like(rho, dtype=complex)
    for i in range(d1):
        for j in range(d1):
            block = rho[i*d2:(i+1)*d2, j*d2:(j+1)*d2]
            rho_pt[i*d2:(i+1)*d2, j*d2:(j+1)*d2] = block.T
    return rho_pt

def log_negativity(rho, dims):
    """Compute logarithmic negativity."""
    rho_pt = partial_transpose(rho, dims)
    trace_norm = np.sum(np.sqrt(np.linalg.eigvalsh(rho_pt @ rho_pt.conj().T)))
    return np.log2(trace_norm)

Typicality Analysis

For state ensembles (e.g., Haar-random states), logarithmic negativity concentrates: $$P(|E_N(\rho) - \mathbb{E}[E_N]| > \epsilon) \leq 2\exp(-c d \epsilon^2)$$ where $d$ is the Hilbert space dimension and $c$ is a constant.

Applications

• Quantum resource theory: Quantifying entanglement as a resource
• Quantum communication: Bounding channel capacities
• Many-body physics: Characterizing entanglement in quantum phases
• Quantum information protocols: Verifying entanglement resources for teleportation, superdense coding

Activation Keywords

• logarithmic negativity
• exact entanglement cost
• entanglement quantification
• concentration of measure
• quantum resource theory
• partial transpose norm
• 对数负性, 纠缠代价

Related Skills

• quantum-entanglement-detection — entanglement detection methods
• multipartite-entanglement-measures — multipartite entanglement quantification
• quantum-information-theory — broader quantum information framework