Negativity and Logarithmic Negativity

The Peres–Horodecki test asks a yes/no question: transpose one half of a bipartite density matrix and check whether the result is still a valid state. If a negative eigenvalue appears, the original was entangled. The negativity is what you get when you refuse to throw away the size of that violation — instead of reporting merely that $\rho^{T_B}$ has slipped below positivity, you add up how far it slipped, promoting a qualitative criterion into a quantitative entanglement measure. Two things make this the workhorse of the field. The first is computability. Almost every other entanglement measure — entanglement of formation, distillable entanglement, relative entropy of entanglement — hides an intractable optimization and is NP-hard in general. The negativity asks only that you diagonalize one Hermitian matrix of size $d_A d_B$ . That is why it dominates the places where entanglement is hardest to pin down: mixed many-body states, conformal field theories, and continuous-variable optics, where reduced-state entropies no longer measure entanglement at all. The second is honesty about its own blind spot. Because the negativity sees exactly what the partial transpose sees, it vanishes on the bound entangled states — entangled yet with a positive partial transpose. So $N=0$ certifies separability only in the small worlds ( $2\times2$ and $2\times3$ ) where the PPT criterion is airtight. Everywhere else, negativity is a faithful lower witness — never the whole story, but very often the only piece you can actually calculate.

Key takeaways

The negativity $N(\rho)=\tfrac12(\lVert\rho^{T_B}\rVert_1-1)=\sum_{\mu_i<0}\lvert\mu_i\rvert$ sums the magnitudes of the negative eigenvalues of the partial transpose — the amount by which $\rho^{T_B}$ fails to be a state.
Equivalently it is a trace-norm distance: $\lVert\rho^{T_B}\rVert_1=1+2N$ , so a valid state sits at trace norm one and every unit of negativity pushes it above.
The logarithmic negativity $E_N(\rho)=\log_2\lVert\rho^{T_B}\rVert_1=\log_2(2N+1)$ is additive and upper-bounds the distillable entanglement — the operational cousin of the negativity.
Both are entanglement monotones — non-increasing under LOCC — and both are efficiently computable, but both are blind to PPT (bound) entangled states, so $N=0$ implies separability only in $2\times2$ and $2\times3$ .
For a pure state $E_N$ is the Rényi- $\tfrac12$ entropy of the Schmidt spectrum, $E_N=2\log_2\!\big(\sum_i\sqrt{\lambda_i}\big)\ge S(\rho_A)$ , meeting the entropy only for a flat spectrum.
Key idea: take the partial transpose, measure how far it falls below positivity in trace norm — that distance is the entanglement the PPT criterion can see, packaged as a computable monotone.

The core insight

Take the partial transpose, measure how far it falls below positivity in trace norm — that distance is the entanglement the PPT criterion can see, packaged as a computable, LOCC-monotone number.

Where this lives today

The logarithmic negativity is now the default entanglement diagnostic wherever the reduced state is mixed — nearly everywhere interesting. In many-body physics and quantum field theory it is the tool of choice for the entanglement between two disjoint regions, or between a subregion and a thermal bath, where the entanglement entropy overcounts by mixing quantum correlation with classical thermal noise. Calabrese, Cardy and Tonni computed its universal scaling in $1{+}1$ -dimensional conformal field theories, and it is now a standard probe of quantum critical points, finite-temperature topological order, and thermalization after a quench, precisely because it stays meaningful where entropy-based measures fail. In continuous-variable quantum optics it has a closed form in the symplectic eigenvalues of the partially transposed covariance matrix, and it is the number reported to certify entanglement in two-mode squeezed light and optomechanical systems. Its additivity and computability make it a practical certificate in quantum-repeater engineering, where evaluating $E_N$ on a noisy shared state instantly bounds the distillable rate. And holographic constructions have imported the negativity into the study of black holes and the emergence of spacetime — a positivity test from 1996 now serving at the edge of quantum gravity.

The physical picture

A density matrix's eigenvalues are the weights of the states in some ensemble, so they must all be non-negative and sum to one. The partial transpose $\rho^{T_B}$ — transpose the indices of subsystem $B$ only — is a perfectly legal shuffle of matrix entries, but it is positive, not completely positive. Apply it to a separable state and every eigenvalue survives as a non-negative weight, because a separable state is a mixture of things the local transpose handles one term at a time. Apply it to an entangled state and the shuffle can manufacture a negative eigenvalue — a "probability" below zero, the unmistakable signature that no classical mixture could have produced these correlations. The negativity is the total weight of that impossible, negative probability. The first figure makes the accounting concrete. Transpose preserves the trace, so the eigenvalues of $\rho^{T_B}$ still sum to one; if one dips negative, the positive ones must overshoot to compensate. That is why the trace norm — the sum of absolute values — climbs above one by exactly twice the buried negativity: $\lVert\rho^{T_B}\rVert_1=1+2N$ .

Figure

Partial transpose of the two-qubit Werner state at mixing p = 0.6. Its four eigenvalues are 0.4, 0.4, 0.4 and −0.2: three stay positive but one is driven below zero. The lone negative eigenvalue gives negativity N = 0.2, and the trace norm sum of magnitudes is 3 times 0.4 plus 0.2 = 1.4 = 1 + 2N — the amount by which the state fails positivity

The negativity responds smoothly to how entangled the state is. As the Werner state is diluted with white noise its one dangerous eigenvalue climbs back toward zero, and at a sharp threshold the state becomes PPT and the negativity switches off. The second figure traces that switch-on: below $p=1/3$ , $N=0$ ; above it the negativity rises linearly to its maximum $N=\tfrac12$ at the pure singlet, where $E_N$ reads exactly one ebit — entanglement as a tunable, computable function of a physical parameter.

Figure

Negativity of the two-qubit Werner state as the singlet fraction p is tuned. Below the threshold p = 1/3 the partial transpose stays positive and N = 0; above it the negativity rises linearly as N = (3p − 1)/4, reaching N = 1/2 at the pure singlet p = 1, where the logarithmic negativity equals one ebit. The dot at p = 0.6 marks N = 0.2, the state dissected in the first figure

The core equations

The definition is the whole idea in one line: collect the negative eigenvalues of the partial transpose. The two forms below are identical — one spectral, one a trace-norm excess — and the derivation that follows proves they agree from trace preservation alone.

N(\rho)=\frac{\lVert\rho^{T_B}\rVert_1-1}{2}=\sum_{\mu_i<0}\lvert\mu_i\rvert \qquad (1)

where: $\rho$ — bipartite density operator on $\mathcal H_A\otimes\mathcal H_B$ ; $\rho^{T_B}$ — partial transpose over $B$ ; $\lVert X\rVert_1=\mathrm{Tr}\sqrt{X^\dagger X}=\sum_i\lvert\mu_i\rvert$ — the trace norm; $\mu_i$ — eigenvalues of $\rho^{T_B}$ , the sum over negative ones. Read it as: how far below the positive cone $\rho^{T_B}$ has fallen, in trace-norm units. Taking the logarithm converts the multiplicative trace norm into an additive quantity — losing convexity, gaining additivity and an operational meaning. This is the logarithmic negativity, in ebits.

E_N(\rho)=\log_2\lVert\rho^{T_B}\rVert_1=\log_2\!\big(2N(\rho)+1\big) \qquad (2)

where: $E_N$ — logarithmic negativity in ebits; $N$ — the negativity, tied to it through $\lVert\rho^{T_B}\rVert_1=2N+1$ . Because $\log_2$ is monotone, $E_N$ and $N$ vanish together and order states identically. Everything rests on what the partial transpose does to matrix elements: swap the two indices of $B$ , leave $A$ untouched. In components, in any fixed product basis,

(\rho^{T_B})_{ia,\,jb}=\rho_{ib,\,ja} \qquad (3)

where: $i,j$ — basis labels of subsystem $A$ ; $a,b$ — basis labels of subsystem $B$ , exchanged by the transpose. The map is basis-dependent, but its spectrum — and hence $N$ — is not, since a local change of basis on $B$ merely conjugates $\rho^{T_B}$ by a transposed unitary. For a pure state the whole spectrum of $\rho^{T_B}$ is built from products of Schmidt coefficients, and the trace norm collapses to a perfect square — so $E_N$ becomes a Rényi entropy of the Schmidt spectrum, the cleanest closed form in the subject.

E_N(|\psi\rangle)=\log_2\!\Big(\sum_i\sqrt{\lambda_i}\Big)^{\!2}=2\log_2\!\Big(\sum_i\sqrt{\lambda_i}\Big),\qquad |\psi\rangle=\sum_i\sqrt{\lambda_i}\,|i\rangle_A|i\rangle_B \qquad (4)

where: $\lambda_i$ — Schmidt eigenvalues (squared coefficients), $\sum_i\lambda_i=1$ ; $\sqrt{\lambda_i}$ — the Schmidt coefficients. The right-hand side is exactly the Rényi- $\tfrac12$ entropy $S_{1/2}(\rho_A)$ , which sits above the von Neumann entanglement entropy. The reason $N$ deserves to be called an entanglement measure is that it cannot be forged by local operations and classical communication — it is an LOCC monotone: no local manipulation can raise it on average.

N(\rho)\ \ge\ N\big(\Lambda_{\mathrm{LOCC}}(\rho)\big),\qquad N\Big(\sum_k p_k\,\rho_k\Big)\le\sum_k p_k\,N(\rho_k) \qquad (5)

where: $\Lambda_{\mathrm{LOCC}}$ — any local-operations-and-classical-communication protocol; the second inequality is convexity. Together they certify that $N$ measures a resource free operations cannot create — the defining property of an entanglement monotone. The logarithmic negativity adds two properties the negativity lacks, exactly the ones a resource theory wants: it is additive across independent copies, and it bounds the rate at which pure entanglement can be distilled from the state.

E_N(\rho_1\otimes\rho_2)=E_N(\rho_1)+E_N(\rho_2),\qquad E_N(\rho)\ \ge\ E_D(\rho) \qquad (6)

where: $\rho_1,\rho_2$ — independent bipartite systems; $E_D$ — distillable entanglement, the maximal asymptotic ebit rate extractable by LOCC. No protocol can distill faster than $E_N$ permits. Finally, the family that makes this concrete: the two-qubit Werner state, a singlet mixed with white noise, dissected in both figures above. Its negativity has a closed form with a sharp entanglement threshold.

N\big(\rho_W(p)\big)=\max\!\Big(0,\ \frac{3p-1}{4}\Big),\qquad \rho_W(p)=p\,|\Psi^-\rangle\langle\Psi^-|+\frac{1-p}{4}\,\mathbb 1 \qquad (7)

where: $p$ — singlet fraction, $0\le p\le1$ ; $|\Psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt2$ — the singlet; $\mathbb 1$ — the $4\times4$ identity. The state is entangled ( $N>0$ ) precisely for $p>1/3$ , the PPT threshold for this family.

Derivation AWhy the trace norm counts twice the negativity

The goal is to prove that the two expressions in equation (1) are the same number — that the spectral sum of negative eigenvalues equals the trace-norm excess over one — using only two facts: the partial transpose preserves the trace, and the trace norm of a Hermitian matrix is the sum of absolute eigenvalues. This is the identity that lets a computation report either the eigenvalue sum or the trace norm and mean the same thing, and it is why $\lVert\rho^{T_B}\rVert_1=1+2N$ is the natural bookkeeping for negativity.

Step 1. $\rho^{T_B}$ is Hermitian, so it has real eigenvalues $\mu_i$ , and transposition preserves the trace: $\sum_i\mu_i=\mathrm{Tr}\,\rho^{T_B}=\mathrm{Tr}\,\rho=1$ . Why: the total weight is conserved even as eigenvalues are rearranged into possibly-negative values — the negativity is paid out of the positive part.

Step 2. Split the spectrum and write the trace norm as $\lVert\rho^{T_B}\rVert_1=\sum_i\lvert\mu_i\rvert=\sum_{\mu_i\ge0}\mu_i-\sum_{\mu_i<0}\mu_i$ . Why: the absolute value flips only the negative eigenvalues, so the two contributions add rather than cancel.

Step 3. Eliminate the positive sum with Step 1, $\sum_{\mu_i\ge0}\mu_i=1-\sum_{\mu_i<0}\mu_i$ , giving $\lVert\rho^{T_B}\rVert_1=1-2\sum_{\mu_i<0}\mu_i=1+2\sum_{\mu_i<0}\lvert\mu_i\rvert$ . Why: the trace constraint fixes the entire trace norm from the negative eigenvalues alone.

Step 4. Rearrange: $N\equiv\sum_{\mu_i<0}\lvert\mu_i\rvert=\tfrac12(\lVert\rho^{T_B}\rVert_1-1)$ , recovering both forms of equation (1); a valid state has $\rho^{T_B}\ge0$ , $\lVert\rho^{T_B}\rVert_1=1$ , $N=0$ . Why: the negativity is literally a distance — how far $\rho^{T_B}$ has been pushed outside the set of density matrices, in trace norm.

Go deeper · Derivation B · 1 minNegativity of the two-qubit Werner state

Derivation BNegativity of the two-qubit Werner state

The goal is to compute equation (7) from scratch by diagonalizing the partial transpose of the Werner state, and to locate the entanglement threshold $p=1/3$ as the exact point where a single eigenvalue crosses zero. This is the canonical worked case of the subject — a one-parameter family from maximally mixed noise to a pure singlet — on which every claim about negativity can be checked by hand.

Step 1. In the basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ the singlet contributes $\tfrac12$ on the $01,10$ diagonal and $-\tfrac12$ on their cross terms; adding $\tfrac{1-p}{4}\mathbb 1$ gives diagonal $\tfrac{1-p}{4},\ \tfrac{1+p}{4},\ \tfrac{1+p}{4},\ \tfrac{1-p}{4}$ with off-diagonal $-\tfrac{p}{2}$ linking $|01\rangle$ and $|10\rangle$ . Why: the singlet lives entirely in the $\{01,10\}$ sector, so only that block carries coherence.

Step 2. Partial-transpose on $B$ : diagonal entries are untouched, but the coherence $-\tfrac{p}{2}\,|01\rangle\langle10|$ has its $B$ index swapped, becoming $-\tfrac{p}{2}\,|00\rangle\langle11|$ (and its conjugate). Why: the transpose moves the off-diagonal weight into the $\{00,11\}$ sector — the single act that can create a negative eigenvalue.

Step 3. The result is block-diagonal: the $\{01,10\}$ block has both entries $\tfrac{1+p}{4}$ ; the $\{00,11\}$ block $\begin{pmatrix}\frac{1-p}{4}&-\frac{p}{2}\\-\frac{p}{2}&\frac{1-p}{4}\end{pmatrix}$ has eigenvalues $\tfrac{1-p}{4}\pm\tfrac{p}{2}=\tfrac{1+p}{4}$ and $\tfrac{1-3p}{4}$ . Why: a symmetric off-diagonal block splits into sum and difference eigenvalues; the difference branch is the one that can go negative.

Step 4. The spectrum is $\tfrac{1+p}{4}$ (thrice) and $\tfrac{1-3p}{4}$ (once). The lone eigenvalue is negative exactly when $p>\tfrac13$ , so $N=\tfrac{3p-1}{4}$ there and $N=0$ otherwise. Why: only one eigenvalue ever dips below zero, and its sign change is the sharp PPT threshold of the second figure.

Step 5. Endpoints: at $p=1$ (pure singlet) $N=\tfrac12$ , $E_N=\log_2 2=1$ ebit; at $p=\tfrac13$ both vanish continuously. Why: recovering one ebit for the Bell state and a clean switch-off confirms the calculation against known limits.

Go deeper · Derivation C · 1 minPure-state log-negativity is a Rényi one-half entropy

Derivation CPure-state log-negativity is a Rényi one-half entropy

The goal is to derive equation (4) — that a pure state's logarithmic negativity equals twice the log of the sum of Schmidt coefficients — and to show it is the Rényi- $\tfrac12$ entropy of the reduced spectrum, which always sits at or above the von Neumann entanglement entropy. This is why, for pure states, negativity and entropy agree only at maximal entanglement. Start from the Schmidt form and build $\rho^{T_B}$ block by block.

Step 1. In the Schmidt basis $\rho=\sum_{i,j}\sqrt{\lambda_i\lambda_j}\,|ii\rangle\langle jj|$ . Transposing the $B$ index sends $|ii\rangle\langle jj|\mapsto|ij\rangle\langle ji|$ , so $\rho^{T_B}=\sum_{i,j}\sqrt{\lambda_i\lambda_j}\,|ij\rangle\langle ji|$ . Why: the transpose converts diagonal Schmidt correlations into off-diagonal swaps between distinct index pairs.

Step 2. Read off the blocks: diagonal terms $i=j$ give eigenvalues $\lambda_i$ ; each pair $i\ne j$ forms a $2\times2$ block with eigenvalues $\pm\sqrt{\lambda_i\lambda_j}$ . Why: an off-diagonal symmetric block has eigenvalues plus and minus its entry, so every distinct Schmidt pair contributes one negative eigenvalue $\sqrt{\lambda_i\lambda_j}$ — the origin of pure-state negativity.

Step 3. Sum the magnitudes: $\lVert\rho^{T_B}\rVert_1=\sum_i\lambda_i+2\sum_{i<j}\sqrt{\lambda_i\lambda_j}=\big(\sum_i\sqrt{\lambda_i}\big)^2$ , completing the square via $\sum_i\lambda_i=1$ ; then $E_N=2\log_2\big(\sum_i\sqrt{\lambda_i}\big)$ . Why: the diagonal weights and doubled off-diagonal magnitudes are exactly a squared sum — the closed form of equation (4).

Step 4. Identify the entropy: $S_{1/2}(\rho_A)=\tfrac{1}{1-1/2}\log_2\big(\sum_i\lambda_i^{1/2}\big)=2\log_2\big(\sum_i\sqrt{\lambda_i}\big)=E_N$ , and since Rényi entropies decrease in order, $S_{1/2}\ge S_1=S(\rho_A)$ , equal only for a flat spectrum $\lambda_i\in\{0,1/r\}$ . Why: pure-state log-negativity is a Rényi entropy of the Schmidt weights, so it upper-bounds the entanglement entropy and meets it only when all nonzero coefficients are equal.

Limiting cases

The negativity specializes cleanly across the regimes that matter, from its blind spot to its saturation:

Case	Result	Meaning
Separable or PPT state	$\rho^{T_B}\ge0\ \Rightarrow\ N=0,\ E_N=0$	Vanishes for every separable state — but also for bound (PPT) entangled states, the blind spot inherited from the PPT criterion.
Maximally entangled qudit	$E_N(\|\Phi_d\rangle)=\log_2 d,\quad N=\tfrac{d-1}{2}$	Saturates the scale; a Bell pair ( $d=2$ ) gives $E_N=1$ ebit, $N=\tfrac12$ .
Pure state, Schmidt spectrum $\{\lambda_i\}$	$E_N=2\log_2\!\big(\sum_i\sqrt{\lambda_i}\big)=S_{1/2}(\rho_A)\ge S(\rho_A)$	A Rényi- $\tfrac12$ entropy of the Schmidt spectrum, generally exceeding the von Neumann entanglement entropy.
Two-qubit Werner state $\rho_W(p)$	$N=\max\!\big(0,\tfrac{3p-1}{4}\big)$	Detects entanglement exactly for $p>\tfrac13$ , matching the PPT threshold for this family.
Gaussian state (continuous variable)	$E_N=\sum_k\max\!\big(0,-\log_2\tilde\nu_k\big)$	Closed form from the symplectic eigenvalues $\tilde\nu_k$ of the partially transposed covariance matrix — the CV workhorse.

Common trapReading N = 0 as a certificate of separability

The single most common misuse of negativity is to conclude " $N(\rho)=0$ , therefore the state is separable." That inference is valid only in $2\times2$ and $2\times3$ systems, where the Horodecki theorem makes PPT necessary and sufficient. In every larger system there exist bound entangled states — genuinely entangled, yet with a positive partial transpose — for which $N=0$ despite non-separability. Negativity cannot see them, because it sees exactly and only what the PPT criterion sees. So a nonzero negativity is always a rock-solid certificate of entanglement, but a vanishing negativity certifies separability only in those two smallest cases. Everywhere else, $N=0$ means "PPT," which is strictly weaker than "separable" — and treating the negativity as faithful in higher dimensions silently discards the entire zoo of bound entanglement.

Worked examples

1 · Negativity of a noisy Werner state, end to end (fully worked)

A noisy channel leaves the Werner state $\rho_W(p)$ with singlet fraction $p=0.6$ . Compute the partial-transpose spectrum, negativity, logarithmic negativity, and cross-check against the two-qubit concurrence.

Step 1. The partial-transpose eigenvalues are $\tfrac{1+p}{4}$ (thrice) and $\tfrac{1-3p}{4}$ (once). At $p=0.6$ : $\mathbf{0.4}$ (three of them) and $\mathbf{-0.2}$ , summing to $1.0$ , the conserved trace.

Step 2. Only one eigenvalue is negative, so $N=\lvert-0.2\rvert=\mathbf{0.2}$ . Equivalently $\lVert\rho^{T_B}\rVert_1=3(0.4)+0.2=1.4$ and $N=\tfrac12(1.4-1)=0.2$ — the two routes agree.

Step 3. Logarithmic negativity: $E_N=\log_2 1.4=\mathbf{0.485\ \text{ebit}}$ , well below the one ebit of a pure singlet — the noise has eaten most, though not all, of the entanglement.

Step 4. Cross-check with concurrence. For the Werner family $C(\rho_W)=\max\!\big(0,\tfrac{3p-1}{2}\big)=\mathbf{0.4}$ , so $N=C/2$ exactly — a relation holding for every two-qubit pure state and for the Werner family. The state is entangled with $N=0.2$ , $E_N=0.485$ ebit, $C=0.4$ ; it becomes PPT-separable only at $p=1/3$ .

2 · A partially entangled pure state: negativity versus entropy

Take $|\psi\rangle=\sqrt{0.8}\,|00\rangle+\sqrt{0.2}\,|11\rangle$ , already in Schmidt form. Find its negativity, logarithmic negativity, and entanglement entropy, and confirm $E_N$ exceeds the entropy.

Step 1. The Schmidt eigenvalues are $\lambda_1=0.8,\ \lambda_2=0.2$ with Schmidt coefficients $\sqrt{0.8}=0.8944$ and $\sqrt{0.2}=0.4472$ . Their sum is $\sum_i\sqrt{\lambda_i}=1.3416$ .

Step 2. Trace norm of the partial transpose: $\lVert\rho^{T_B}\rVert_1=\big(\sum_i\sqrt{\lambda_i}\big)^2=(1.3416)^2=\mathbf{1.8}$ . Hence the negativity $N=\tfrac12(1.8-1)=\mathbf{0.4}$ and $E_N=\log_2 1.8=\mathbf{0.848\ \text{ebit}}$ .

Step 3. Von Neumann entanglement entropy: $S=-0.8\log_2 0.8-0.2\log_2 0.2=\mathbf{0.722\ \text{ebit}}$ . As predicted, $E_N=0.848>0.722=S$ — the log-negativity is the larger Rényi- $\tfrac12$ entropy, and the two coincide only for a flat ( $0.5,0.5$ ) spectrum.

Step 4. Concurrence check: $C=2\sqrt{\lambda_1\lambda_2}=2\sqrt{0.16}=0.8$ , and indeed $N=\sqrt{\lambda_1\lambda_2}=0.4=C/2$ , as in Example 1. A non-maximal pure state: $N=0.4$ , $E_N=0.848$ ebit, $S=0.722$ ebit, $C=0.8$ — negativity and entropy measure the same entanglement on different scales.

3 · The maximally entangled qudit benchmark

Evaluate $N$ and $E_N$ for $|\Phi_d\rangle=\tfrac{1}{\sqrt d}\sum_{i=1}^{d}|ii\rangle$ , then specialize to a qutrit.

Step 1. All Schmidt coefficients equal $1/\sqrt d$ , so $\sum_i\sqrt{\lambda_i}=\sqrt d$ and $\lVert\rho^{T_B}\rVert_1=d$ ; hence $E_N=\log_2 d$ and $N=\tfrac12(d-1)$ . For the qutrit $d=3$ this is $E_N=\log_2 3=\mathbf{1.585\ \text{ebit}}$ and $N=\mathbf{1.0}$ . The maximally entangled qudit sets the ceiling $E_N=\log_2 d$ , with the Bell pair the $d=2$ corner ( $E_N=1$ ebit, $N=1/2$ ).

Physics of the portrait

The negativity has no single inventor — it is a relay across four groups over a decade, from a positivity test, to a volume measure, to a proven monotone, to an additive operational bound. The concept card below tracks that relay.

Origins · 1 minnegativity and how this was discovered

negativity

the computable entanglement monotone N = (‖ρᵀᴮ‖₁ − 1)/2 · entanglement measures from the partial transpose

The negativity was assembled, not discovered in one stroke. Its foundation is Asher Peres's 1996 observation that a separable state must stay positive under partial transposition — a checkable necessary condition that the Horodecki family (Michał, Paweł and Ryszard) proved the same year to be also sufficient in the smallest cases, $2\times2$ and $2\times3$ . That made "does a negative eigenvalue appear?" a decision procedure, but it was still a yes/no test. Two years later Karol Życzkowski, Paweł Horodecki, Anna Sanpera and Maciej Lewenstein, studying the geometry of the separable set, defined the negativity as the amount of that violation. The measure earned its credentials in 2002, when Guifré Vidal and Reinhard F. Werner proved that both the negativity and its logarithm are entanglement monotones — and stressed the property that made them famous: unlike almost every rival measure, they are computable, needing only one diagonalization. Finally, in 2005 Martin Plenio isolated the logarithmic negativity as a full monotone that is additive and upper-bounds the distillable entanglement, giving it a clean operational role even though it fails to be convex. What began as a positivity check ended as the most-used quantitative measure of mixed-state entanglement in physics.

History

1996

— Peres proposes the partial-transpose positivity test for separability

1996

— the Horodecki family proves PPT is necessary and sufficient in 2×2 and 2×3

1998

— Życzkowski, Horodecki, Sanpera and Lewenstein introduce the negativity as a measure

2002

— Vidal and Werner prove negativity and log-negativity are computable entanglement monotones

2005

— Plenio establishes the logarithmic negativity as additive and an upper bound on distillable entanglement

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §12.5 (mixed-state entanglement) · Vidal & Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002) · Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95, 090503 (2005) · Watrous, The Theory of Quantum Information, Ch. 6 · Wilde, Quantum Information Theory, Ch. 11 · Horodecki, Horodecki, Horodecki & Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).

Provenance. The definition and monotonicity follow Vidal & Werner (2002); the additivity and distillable-entanglement bound follow Plenio (2005); the negativity as a measure was introduced by Życzkowski, Horodecki, Sanpera & Lewenstein, Phys. Rev. A 58, 883 (1998), on the PPT foundation of Peres (1996) and Horodecki³ (1996). A verifier should recompute: (1) the identity $N=\tfrac12(\lVert\rho^{T_B}\rVert_1-1)$ from trace preservation; (2) the Werner spectrum $\{(1+p)/4\ (\times3),\,(1-3p)/4\}$ with threshold $p=1/3$ , and at $p=0.6$ the values $N=0.2$ , $E_N=\log_2 1.4=0.485$ , $C=0.4$ ; (3) the pure-state result $\lVert\rho^{T_B}\rVert_1=(\sum_i\sqrt{\lambda_i})^2$ and, for $\{0.8,0.2\}$ , $N=0.4$ , $E_N=\log_2 1.8=0.848>S=0.722$ ; (4) the benchmarks $E_N(|\Phi_d\rangle)=\log_2 d$ , $N=(d-1)/2$ , giving $E_N=\log_2 3=1.585$ at $d=3$ . All numbers are standard and textbook-verifiable.

Prerequisites. PPT Criterion and the Peres–Horodecki Test · Separable versus Entangled States · Schmidt Decomposition · Entanglement Entropy of Pure States. Leads to. Entanglement Witnesses · Concurrence and Wootters' Formula · Bell States and EPR Pairs.

Ask the tutor: Hand me any density matrix — a noisy Bell pair, a Werner state, a two-mode squeezed vacuum — and I will take its partial transpose, find the negative eigenvalues, and walk you through what its negativity and logarithmic negativity say about how entangled it is.

Is the negativity the same thing as the PPT criterion?

They are two readings of the same object. The PPT criterion asks a yes/no question — does $\rho^{T_B}$ have any negative eigenvalue? — and answers "entangled" if it does. The negativity answers the quantitative follow-up: how much negativity is there, summed over all negative eigenvalues? The criterion is the sign of a violation; the negativity is its size. Because they share the same underlying test, they share the same blind spot: a state with $\rho^{T_B}\ge0$ passes the criterion and has $N=0$ , whether it is truly separable or bound entangled. The negativity adds a magnitude to the PPT verdict without changing what that verdict can detect.

Why does the negativity miss bound entangled states?

Because it is built entirely on the partial transpose, and bound entangled states are defined by having a positive partial transpose despite being entangled. If $\rho^{T_B}\ge0$ , none of its eigenvalues are negative, so $N=0$ — the measure has nothing to add up. This is not a flaw; it is the exact scope of the PPT criterion showing through. In $2\times2$ and $2\times3$ systems no PPT entangled states exist, so the negativity is faithful there. In higher dimensions they do exist, and detecting them requires different tools — entanglement witnesses tailored to the state, or the range criterion — none as cheap to compute as the negativity.

The negativity is convex but the logarithmic negativity is not — does that matter?

It matters for what each is good at. The negativity $N$ inherits convexity from the triangle inequality of the trace norm, so it never increases when you mix states — a natural property for a measure. The logarithmic negativity $E_N=\log_2(2N+1)$ loses convexity, because the log of a convex function need not be convex. In exchange it gains additivity: $E_N$ of $n$ independent copies is exactly $n$ times $E_N$ of one, which convex measures generally fail. Additivity is what lets $E_N$ serve as a clean asymptotic bound on distillable entanglement and entanglement cost. So the two are a deliberate trade: convexity ( $N$ ) or additivity and operational meaning ( $E_N$ ), but not both in one quantity.

How does the pure-state negativity compare with the entanglement entropy?

For a pure state both are functions of the Schmidt spectrum $\{\lambda_i\}$ , but different functions. The entanglement entropy is $S=-\sum_i\lambda_i\log_2\lambda_i$ , while $E_N=2\log_2\big(\sum_i\sqrt{\lambda_i}\big)$ is exactly the Rényi- $\tfrac12$ entropy of the same spectrum. Because Rényi entropies decrease with their order, the order- $\tfrac12$ value sits at or above the order-1 value, so $E_N\ge S$ for every pure state. Equality holds only for a flat spectrum — all nonzero coefficients equal, the maximally entangled case within a rank. For any partially entangled pure state, such as the $\{0.8,0.2\}$ example, the log-negativity strictly overshoots the entanglement entropy.

Can I use the negativity for continuous-variable or Gaussian states?

Yes, and it is the standard measure there. A Gaussian state is captured entirely by its covariance matrix, and partial transposition acts as a simple sign flip on the momentum quadratures of one subsystem. You then compute the symplectic eigenvalues $\tilde\nu_k$ of the partially transposed covariance matrix, and $E_N=\sum_k\max\big(0,-\log_2\tilde\nu_k\big)$ — a closed form needing no diagonalization of an infinite-dimensional density matrix. A symplectic eigenvalue below one signals entanglement, exactly as a negative eigenvalue does in the discrete case. This is why the log-negativity, not the entropy, is the figure of merit quoted for two-mode squeezed vacua and optomechanical entanglement.

Why is negativity used in many-body physics instead of the entanglement entropy?

The entanglement entropy only measures entanglement for pure states. In a many-body system at finite temperature, or between two regions $A$ and $B$ that do not together make up the whole system, the reduced state $\rho_{AB}$ is mixed, and its von Neumann entropy counts classical and thermal correlations along with the quantum ones. The logarithmic negativity is a genuine entanglement monotone for mixed states, so it isolates the quantum part. That is decisive for the entanglement between two disjoint intervals in a critical chain, or its survival above a phase transition, where entropy-based diagnostics mislead. Its universal scaling in conformal field theory, computed via a replica trick on the partial transpose, has made it a primary probe of quantum criticality and thermalization.