Entanglement Measures Table

An entanglement measure must be matched to a subsystem split, state class, and allowed operations. This table fixes the bipartition $A\vert B$, finite-dimensional states, base-two logarithms, and asymptotically vanishing error whenever a conversion rate is discussed. Entanglement Measures explains how to choose a measure for a task; the linked Composite Systems pages develop the mathematical constructions.

Bipartite measures and their domains

Let $S(\rho)=-\operatorname{Tr}(\rho\log_2\rho)$, with $0\log_2 0=0$. A Bell pair has one ebit. Here LOCC means local operations and classical communication, with messages allowed in both directions unless stated otherwise. The trace norm is $\lVert M\rVert_1=\operatorname{Tr}\sqrt{M^\dagger M}$, and $T_B$ denotes partial transposition in a fixed local basis.

Quantity	Definition, meaning, and limitations
Entanglement entropy	$E(\psi)=S(\rho_A)=S(\rho_B)$ for a pure bipartite state Reversible asymptotic Bell-pair rate under LOCC; subsystem entropy alone fails as a mixed-state entanglement measure.
Negativity	$\mathcal N(\rho)=(\lVert\rho^{T_B}\rVert_1-1)/2$ Sum of absolute negative partial-transpose eigenvalues; zero for all PPT states, including PPT entangled states.
Logarithmic negativity	$E_N(\rho)=\log_2\lVert\rho^{T_B}\rVert_1$ Computable upper bound on distillable entanglement; generally not the achievable distillation rate.
Entanglement of formation	$E_F(\rho)=\inf_{\{p_i,\psi_i\}}\sum_i p_iS(\rho_A^{(i)})$ Infimum over pure-state decompositions of the given state; a convex-roof quantity, not generally the asymptotic cost without regularization.
Distillable entanglement	$E_D(\rho)$ is the optimal Bell-pair yield per input copy under LOCC Requires the communication class and approximation criterion; mixed entangled states can have zero yield.
Entanglement cost	$E_C(\rho)=\lim_{n\to\infty}E_F(\rho^{\otimes n})/n$ Bell-pair preparation cost under asymptotic LOCC; may exceed distillable entanglement.
Squashed entanglement	$E_{\mathrm{sq}}(\rho)=\frac12\inf_{\rho_{ABE}}I(A:B\mid E)$ Optimization over all extensions of the given state; useful structural bound, difficult to compute in general.

The convex-roof decomposition satisfies $\rho=\sum_i p_i\lvert\psi_i\rangle\langle\psi_i\rvert$. For squashed entanglement, the extension satisfies $\operatorname{Tr}_E\rho_{ABE}=\rho_{AB}$ and

$$I(A:B\mid E)={}S(AE)+S(BE) -S(E)-S(ABE).$$

See Entanglement Entropy for pure states and Negativity and PPT for the partial-transpose test. Vidal and Werner establish the computable negativity framework; Christandl and Winter introduce squashed entanglement.

Two-qubit concurrence

For a two-qubit state define its spin flip in the computational basis,

$$\widetilde\rho=(Y\otimes Y)\rho^*(Y\otimes Y).$$

Let $\lambda_1\geq\lambda_2\geq\lambda_3\geq\lambda_4$ be the nonnegative square roots of the eigenvalues of $\rho\widetilde\rho$. Wootters gives

$$\begin{aligned} C(\rho)&=\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4),\\ E_F(\rho)&=h_2\!\left(\frac{1+\sqrt{1-C(\rho)^2}}2\right), \end{aligned}$$

where $h_2(x)=-x\log_2x-(1-x)\log_2(1-x)$. The four-eigenvalue formula is specific to two qubits; it cannot be reused by padding or truncating a larger state. See Concurrence for Two Qubits.

Normalization checks

For $0\leq p\leq1$, let $\lvert\psi_p\rangle=\sqrt p\lvert00\rangle+\sqrt{1-p}\lvert11\rangle$. Its partial-transpose spectrum is $p$, $1-p$, and $\pm\sqrt{p(1-p)}$. Consequently,

$$\begin{aligned} E(\psi_p)&=h_2(p),\\ C(\psi_p)&=2\sqrt{p(1-p)},\\ \mathcal N(\psi_p)&=\sqrt{p(1-p)},\\ E_N(\psi_p)&=\log_2[1+2\sqrt{p(1-p)}]. \end{aligned}$$

At $p=1/2$, entropy, concurrence, and logarithmic negativity are each 1, but negativity is $1/2$. Away from maximal entanglement, even the quantities normalized to 1 generally disagree. At $p=0$ or 1 they all vanish.

As a mixed-state countercheck, $(\lvert00\rangle\langle00\rvert+\lvert11\rangle\langle11\rvert)/2$ is explicitly separable, although both subsystem entropies are one bit. The entropy measures local uncertainty here, not entanglement.

For multipartite states, first specify which bipartition is being tested. A collection of bipartite values does not automatically quantify genuine multipartite entanglement. Infinite-dimensional states additionally require energy and finiteness conditions; the finite-dimensional table alone does not supply them.

References

• Christandl, Matthias, and Andreas Winter, “Squashed entanglement: An additive entanglement measure,” Journal of Mathematical Physics 45, 829–840 (2004), arXiv:quant-ph/0308088 — conditional-information construction and bounds.
• Vidal, Guifré, and Reinhard F. Werner, “A computable measure of entanglement,” Physical Review A 65, 032314 (2002), arXiv:quant-ph/0102117 — negativity and distillation bounds.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, author manuscript and errata — bipartite entanglement definitions and asymptotic qualifications.
• Wootters, William K., “Entanglement of formation of an arbitrary state of two qubits,” Physical Review Letters 80, 2245–2248 (1998), doi:10.1103/PhysRevLett.80.2245 — exact two-qubit formula.