Logarithmic NegativityEdit
Logarithmic Negativity is a widely used quantitative measure of quantum entanglement, especially in the field of quantum information. Grounded in the mathematics of density matrices and partial transposition, it provides a computable diagnostic for whether a given bipartite state is entangled and, in many common cases, how much entanglement is present. Because real systems are never perfectly pure, researchers rely on measures that work for mixed states; logarithmic negativity is prized for its relative simplicity and its solid theoretical footing as an entanglement monotone. In practical terms, it serves as a reliable benchmark for experiments in photonic qubits, superconducting devices, and even complex many-body systems in condensed matter physics. See Density matrix and Entanglement for foundational context, and consider how Partial transpose and Trace norm underpin its definition.
Definition
Let a quantum state ρ describe a bipartite system composed of subsystems A and B. The partial transpose with respect to subsystem B, denoted ρ^{T_B}, is formed by transposing only the indices associated with B. The trace norm, written as ||X||_1, is the sum of the singular values of a matrix X (equivalently, the sum of absolute values of its eigenvalues when X is Hermitian).
Logarithmic Negativity, often written as Log2(||ρ^{T_B}||_1), is defined as
Logarithmic Negativity = log2(||ρ^{T_B}||_1).
A related quantity is the (non-logarithmic) negativity, E_N(ρ) = (||ρ^{T_B}||_1 − 1)/2, which serves as an entanglement measure in its own right. The use of the partial transpose and the trace norm makes the logarithmic negativity a practical tool, since ρ and ρ^{T_B} can be computed from state tomography or inferred from experimental data. See Partial transpose and Trace norm for the mathematical underpinnings, and Density matrix for the language of quantum states.
In two-qubit and qubit–qutrit systems (the 2×2 and 2×3 cases), a convenient heuristic holds: the state is entangled if and only if ρ^{T_B} has a negative eigenvalue, which yields a positive logarithmic negativity. In higher dimensions, there exist entangled states that remain PPT (positive under partial transpose), so Logarithmic Negativity can be zero even when entanglement is present in more subtle forms; this highlights the difference between a practical diagnostic and a fully operational, universal measure of distillable entanglement. See PPT criterion for the broader criteria and Distillable entanglement for the notion of extractable entanglement.
Properties
Entanglement monotone under LOCC: Logarithmic Negativity does not increase on average under local operations and classical communication, aligning with the general idea that entanglement is a resource that cannot be generated by local actions alone. See LOCC and Entanglement.
Additivity: Logarithmic Negativity is additive under tensor products, i.e., Log2(||(ρ⊗σ)^{T_B}||_1) = Log2(||ρ^{T_B}||_1) + Log2(||σ^{T_B}||_1). This makes it convenient for analyzing composite systems and scaling behavior in many-body contexts. Compare with other measures such as Entanglement of formation or Distillable entanglement to see different scaling properties.
Clear operational connection in simple cases: For many commonly studied states, especially in low dimensions, a positive logarithmic negativity corresponds to entanglement, and its magnitude gives a straightforward sense of how much entanglement is present. See examples in Bell state and Werner state for concrete illustrations.
Limitations in higher dimensions: While PPT states are separable in 2×2 and 2×3 systems, in larger systems there exist entangled states with PPT that yield LN = 0 despite nonclassical correlations; therefore LN is a sufficient but not always necessary detector of entanglement in the general case. See PPT criterion and Distillable entanglement for broader context.
Calculation and examples
General recipe: Given a density matrix ρ on AB, compute ρ^{T_B}, obtain its eigenvalues, take the sum of absolute eigenvalues to get the trace norm, and apply the logarithm base 2. This yields Logarithmic Negativity. The procedure is straightforward to implement once a state tomography or a good model of ρ is available. See Density matrix and Partial transpose.
Bell state: For the maximally entangled Bell state |Φ+⟩ = (|00⟩ + |11⟩)/√2, the logarithmic negativity equals 1, reflecting a full unit of entanglement in base-2 units. This simple benchmark is often used when testing measurement and state-preparation procedures. See Bell state.
Werner state in 2×2 systems: The Werner state ρ_W(p) = p|Φ+⟩⟨Φ+| + (1−p)I/4 is entangled for p > 1/3, and logarithmic negativity tracks this transition, becoming positive as soon as the state becomes entangled. This illustrates how LN behaves for a family of mixed states that interpolate between a maximally mixed state and a maximally entangled one. See Werner state.
Higher-dimensional caveat: In a 3×3 or larger system, there are states that are entangled yet have LN = 0 due to PPT entanglement, underscoring that LN detects certain kinds of entanglement more readily than others. See PPT criterion and Distillable entanglement for related concepts.
Applications
Diagnostic in quantum experiments: Because Logarithmic Negativity is computable from a density matrix, it serves as a practical benchmark for experiments in quantum optics, superconducting circuits, and other platforms where entanglement needs to be quantified under realistic noise. See Quantum information and Quantum communication for the broader context of entanglement as a resource.
Tool in condensed matter and many-body physics: LN is used to explore entanglement structure in ground states and thermal states, providing insight into quantum phase transitions and the robustness of entanglement under thermal fluctuations. See Quantum information and Many-body physics for related topics.
Resource-theoretic perspective: As an entanglement monotone, logarithmic negativity informs how much entanglement is present as a resource that could be converted, under LOCC, into other tasks like teleportation or superdense coding. See Distillable entanglement and Teleportation for linked ideas.
Controversies and debates
Interpretational scope: Proponents value LN for its mathematical transparency and computational practicality, especially in mixed-state settings. Critics point out that LN is not a perfect proxy for operational capability in all tasks, since it provides an upper bound on distillable entanglement and can miss PPT-entangled states in higher dimensions. This tension mirrors a broader discussion in which researchers weigh ease of calculation against the sharpness of operational meaning. See Distillable entanglement for the operational angle and Entanglement measure for the landscape of alternatives.
Choice of metric in policy or engineering contexts: In settings where engineers and policymakers seek clear performance metrics for quantum technologies, LN’s straightforward computation is appealing. Some scholars argue that multiple measures should be reported to capture different aspects of entanglement (e.g., LN for computability, entanglement of formation for irreversibility, distillable entanglement for practical task performance). The balance between a single, easy-to-interpret number and a fuller characterization of quantum correlations is a live topic in the field. See Quantum information for the broader framework.
Cultural critiques and methodological debates: As with many scientific metrics, some critics external to the physics community attempt to frame measures in political terms. Those critiques tend to miss the core point that Logarithmic Negativity is a mathematical construct with clear definitions and limitations. When taken seriously, the discussion focuses on the metric’s usefulness for real-world tasks and its relation to other entanglement concepts, rather than on speculative political narratives.