Entanglement Diagnostic Table

This page is a lookup table for choosing an entanglement diagnostic. It does not replace the canonical pages on Schmidt rank, entropy, concurrence, PPT, witnesses, Gaussian states, many-body entanglement, or identical-particle cautions. Its job is narrower: identify which test is appropriate for a given kind of state and state clearly what the test proves.

The first step is always to specify the subsystem structure. Entanglement is not a property of a vector or density operator alone; it is a property relative to a chosen split such as $A\vert B$, a collection of modes, spatial regions, orbitals, or commuting observable algebras.

Read This First

Several diagnostic failures come from using a correct test in the wrong setting:

• For a bipartite pure state, Schmidt rank greater than one, nonzero entanglement entropy, and mixed one-party reduced states are equivalent ways to detect entanglement.
• For a mixed state, mixed reduced states do not prove entanglement. Classical correlations can make local states mixed.
• For two qubits, concurrence gives a necessary and sufficient entanglement test. It is not a universal mixed-state formula in larger dimensions.
• For $2\times2$ and $2\times3$ systems, the PPT criterion is necessary and sufficient for separability. In higher dimensions, PPT is necessary for separability but not sufficient.
• A negative entanglement-witness expectation proves entanglement, but a nonnegative value for one witness is usually inconclusive.
• Entropy scaling, mutual information, covariance-matrix criteria, and mode entanglement are meaningful only after the state class and subsystem split are fixed.

In the table below, “necessary and sufficient” means the diagnostic exactly characterizes entanglement for the stated setting. “Sufficient” means a positive detection proves entanglement, but failure to detect does not prove separability.

Diagnostic Table

Setting	State Type	Diagnostic	What It Proves	Caveats	Canonical Page
Finite bipartite system $A\vert B$	Pure vector $\lvert\psi\rangle$	Schmidt rank $R>1$	Necessary and sufficient for pure-state entanglement	Requires a chosen bipartition and finite Schmidt decomposition	Schmidt Rank
Finite bipartite system $A\vert B$	Pure vector $\lvert\psi\rangle$	Entanglement entropy $S(\rho_A)>0$	Necessary and sufficient for pure-state entanglement	Only an entanglement measure when the global state is pure	Entanglement Entropy
Finite bipartite system $A\vert B$	Pure vector $\lvert\psi\rangle$	Reduced state $\rho_A$ is mixed	Necessary and sufficient for pure-state entanglement	Mixed marginals alone do not certify entanglement when $\rho_{AB}$ is mixed	Subsystem Entropy
Two qubits	Mixed density operator $\rho_{AB}$	Concurrence $C(\rho)>0$	Necessary and sufficient for two-qubit entanglement	The closed formula is special to two qubits	Concurrence for Two Qubits
$2\times2$ or $2\times3$ bipartite system	Mixed density operator $\rho_{AB}$	PPT test: $\rho^{T_B}\ge0$	PPT is necessary and sufficient for separability	The implication changes in higher dimensions	Negativity and PPT Criterion
Higher-dimensional bipartite system	Mixed density operator $\rho_{AB}$	Negativity $\mathcal N(\rho)>0$ or negative partial transpose	Sufficient for entanglement	PPT entangled states can evade negativity	Negativity and PPT Criterion
Experimental or partially reconstructed state	Density operator or measured ensemble	Witness expectation $\operatorname{Tr}(W\rho)<0$	Sufficient for entanglement under the witness assumptions	Nonnegative value for one witness is inconclusive; calibration and statistical errors matter	Entanglement Witnesses
Many-body lattice or field-inspired model	Pure state with spatial or site cut	Subsystem entropy $S_A$ and scaling with subsystem size	Diagnoses correlations, area laws, criticality, and bipartite entanglement across a cut	Not a complete phase classifier; mixed states need other measures	Entanglement in Many-Body Physics
Any bipartite system	Pure or mixed state	Mutual information $I(A:B)$	Measures total correlation, including classical correlation	Nonzero mutual information is not by itself an entanglement test	Mutual Information
Continuous-variable Gaussian system	Gaussian density operator	Covariance-matrix PPT, Simon, or Duan criteria	Detects Gaussian entanglement in common two-mode settings	Depends on quadrature conventions, Gaussian assumptions, and the chosen mode split	Gaussian States Preview
Identical bosons or fermions	Symmetric, antisymmetric, or Fock-space state	Mode, orbital, region, or algebra-specific reduced states	Can diagnose entanglement after a physical subsystem choice	Exchange symmetry by itself is not the same as usable entanglement	Identical-Particle Entanglement Cautions
Multipartite systems	Pure or mixed states on three or more subsystems	Reduced-state pattern, separability class, witnesses, stabilizer structure	Distinguishes some inequivalent entanglement structures	No single scalar classifies multipartite entanglement	Multipartite Systems

Pure Bipartite Shortcut

For a normalized pure state on $\mathcal H_A\otimes\mathcal H_B$, compute the Schmidt decomposition:

$$\lvert\psi\rangle = \sum_{r=1}^{R} s_r \lvert r_A\rangle \lvert r_B\rangle, \qquad s_r>0, \qquad \sum_{r=1}^{R}s_r^2=1.$$

The following statements are equivalent:

• $\lvert\psi\rangle$ is a product state across $A\vert B$.
• The Schmidt rank is $R=1$.
• The reduced state $\rho_A$ is pure.
• The reduced state $\rho_B$ is pure.
• The entanglement entropy is zero:

$$S(\rho_A) = - \operatorname{Tr}(\rho_A\log\rho_A) = 0.$$

Equivalently, the following statements are equivalent:

• $\lvert\psi\rangle$ is entangled across $A\vert B$.
• The Schmidt rank is $R>1$.
• The reduced state $\rho_A$ is mixed.
• The entanglement entropy is positive:

$$S(\rho_A) = - \sum_{r=1}^{R} s_r^2\log s_r^2 > 0.$$

This is the cleanest entanglement-detection setting in ordinary finite-dimensional quantum mechanics. It is also the setting where it is easiest to overgeneralize. Once the global state is mixed, reduced-state mixedness and entropy no longer distinguish quantum entanglement from classical correlation.

Mixed-State Diagnostics

For mixed states, the definition of separability is a convex-decomposition condition:

$$\rho_{AB} = \sum_k p_k\, \rho_A^{(k)} \otimes \rho_B^{(k)}, \qquad p_k\ge0, \qquad \sum_k p_k=1.$$

The state is entangled if no such decomposition exists. This makes mixed-state entanglement a global property of the density operator, not a property of either marginal alone.

The common diagnostics have different logical status:

• Concurrence: for two qubits, $C(\rho)>0$ if and only if $\rho$ is entangled.
• PPT in low dimensions: for $2\times2$ and $2\times3$, $\rho^{T_B}\ge0$ if and only if $\rho$ is separable.
• Negativity: $\mathcal N(\rho)>0$ proves entanglement, but $\mathcal N(\rho)=0$ does not prove separability in dimensions where PPT entangled states exist.
• Witnesses: $\operatorname{Tr}(W\rho)<0$ proves entanglement for a valid witness $W$, but $\operatorname{Tr}(W\rho)\ge0$ for one witness usually proves only that this witness did not detect the state.

These tests are often combined. For example, in a two-qubit experiment, one might reconstruct an approximate density matrix, test positivity and trace normalization, compute concurrence, check PPT, and report a witness value with error bars. The diagnostic conclusion should match the assumptions: “entangled if the reconstructed two-qubit state and error model are valid” is stronger and more honest than a context-free claim that an apparatus “measured entanglement.”

Many-Body and Field-Theoretic Uses

In many-body physics, entanglement diagnostics are often used less as yes-or-no separability tests and more as structural probes. Given a pure state $\lvert\Psi\rangle$ and a spatial region or site subset $A$, the subsystem entropy

$$S_A = - \operatorname{Tr}(\rho_A\log\rho_A), \qquad \rho_A = \operatorname{Tr}_{\bar A} \lvert\Psi\rangle\langle\Psi\rvert$$

measures bipartite entanglement across the cut $A\vert\bar A$. Its scaling can reveal short-range entanglement, critical behavior, matrix-product-state efficiency, or topological contributions. The diagnostic is powerful precisely because it uses the subsystem cut as part of the question.

For mixed many-body states, thermal states, open systems, and continuum fields, entropy of a region is not automatically an entanglement measure. Mutual information, negativity, relative entropy, algebraic methods, and model-specific witnesses may be more appropriate. The Entanglement in QFT Preview discusses the extra complications from local algebras, UV behavior, and gauge constraints.

Gaussian and Continuous-Variable States

Continuous-variable systems add an additional layer: the Hilbert spaces are infinite-dimensional, and common laboratory states are often described by quadratures and covariance matrices rather than finite coefficient arrays.

For Gaussian states, first moments and the covariance matrix

$$V_{ij} = \frac12 \langle \Delta R_i\Delta R_j + \Delta R_j\Delta R_i \rangle$$

determine the state. Partial transposition corresponds to a sign flip of one momentum quadrature in a common convention. For two-mode Gaussian states, covariance-matrix PPT criteria give a practical entanglement test. Criteria such as the Simon and Duan tests are exact under their stated Gaussian and mode assumptions, but the units and normalization of the quadratures matter.

This is why the table says “Gaussian state” rather than “continuous-variable state” without qualification. Non-Gaussian states can have entanglement that is invisible to purely covariance-based diagnostics.

Identical Particles

For identical particles, the most common false diagnostic is:

$$\text{antisymmetric wavefunction} \quad\Longrightarrow\quad \text{entanglement}.$$

That implication is not a safe physical statement. Antisymmetry is a kinematic constraint on fermionic states. Entanglement becomes a physical question only after one has specified accessible subsystems or algebras: modes, spin-orbitals, spatial regions, detector channels, occupation-number partitions, or other operationally meaningful degrees of freedom.

For example, a single Slater determinant is antisymmetric, but it is often treated as unentangled relative to the fermionic mode structure used in mean-field theory. A superposition of determinants can carry mode or orbital entanglement. The diagnostic therefore belongs to the chosen mode, region, or algebra, not to the particle labels that were introduced only to write antisymmetric wavefunctions.

What the Table Cannot Do

The table is a triage tool, not a universal algorithm. It cannot:

• find the best separable decomposition of an arbitrary high-dimensional mixed state;
• classify all multipartite entanglement types;
• decide whether a noisy experimental data set supports entanglement without an error model;
• turn total correlation into quantum entanglement;
• remove the need to specify a subsystem decomposition.

When the state class is unclear, use the most conservative route: identify the Hilbert space, the subsystem split, whether the state is pure or mixed, the finite or infinite-dimensional setting, and the operational observables. Then choose the diagnostic whose assumptions match that setting.

Common Mistakes

• Using mixed marginals as a mixed-state entanglement test. A separable state can have mixed local reductions.
• Saying PPT means separable in all dimensions. PPT is complete only in $2\times2$ and $2\times3$ finite-dimensional bipartite systems.
• Treating mutual information as an entanglement measure. Mutual information includes both classical and quantum correlations.
• Ignoring the subsystem split. A state can be product in one tensor-product structure and entangled in another.
• Equating exchange symmetry with entanglement. Symmetry or antisymmetry is not enough; the physically accessible split matters.
• Reporting a witness as if it were complete. A witness detects states it was designed to detect; one failed witness does not establish separability.

Cross-Links

• Formula Sheet
• Common Composite States
• Computational Notebooks
• Entangled States
• Separable Mixed States
• Entanglement Depends on a Decomposition
• Schmidt Decomposition
• Schmidt Rank
• Entanglement Measures
• Entanglement Entropy
• Concurrence for Two Qubits
• Negativity and PPT Criterion
• Entanglement Witnesses
• Mutual Information
• Subsystem Entropy
• Gaussian States Preview
• Identical-Particle Entanglement Cautions

References

• A. Peres, “Separability Criterion for Density Matrices”, Physical Review Letters 77, 1413-1415, 1996, doi:10.1103/PhysRevLett.77.1413.
• M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions”, Physics Letters A 223, 1-8, 1996, doi:10.1016/S0375-9601(96)00706-2.
• W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits”, Physical Review Letters 80, 2245-2248, 1998, doi:10.1103/PhysRevLett.80.2245.
• G. Vidal and R. F. Werner, “Computable measure of entanglement”, Physical Review A 65, 032314, 2002, doi:10.1103/PhysRevA.65.032314.
• O. Guhne and G. Toth, “Entanglement detection”, Physics Reports 474, 1-75, 2009, doi:10.1016/j.physrep.2009.02.004.
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement”, Reviews of Modern Physics 81, 865-942, 2009, doi:10.1103/RevModPhys.81.865.
• R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems”, Physical Review Letters 84, 2726-2729, 2000, doi:10.1103/PhysRevLett.84.2726.
• L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems”, Physical Review Letters 84, 2722-2725, 2000, doi:10.1103/PhysRevLett.84.2722.
• L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems”, Reviews of Modern Physics 80, 517-576, 2008, doi:10.1103/RevModPhys.80.517.

Exercises

1. Pure-state shortcut. For

$$\lvert\Phi^+\rangle = \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2},$$

find the Schmidt rank and the one-qubit entanglement entropy.

Solution

The state is already in Schmidt form with coefficients

$$s_1=s_2=\frac{1}{\sqrt2}.$$

The Schmidt rank is $R=2$, so the state is entangled across the two-qubit split. The reduced density operator is

$$\rho_A = \frac12 \lvert0\rangle\langle0\rvert + \frac12 \lvert1\rangle\langle1\rvert.$$

Therefore

$$S(\rho_A) = - 2 \left( \frac12\log\frac12 \right) = \log2.$$

2. Mixed marginals are not enough. Consider

$$\rho_{\mathrm{cc}} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

The one-qubit reductions are maximally mixed. Does that prove entanglement?

Solution

No. The state is explicitly a convex mixture of product states:

$$\rho_{\mathrm{cc}} = \frac12 \bigl( \lvert0\rangle\langle0\rvert \otimes \lvert0\rangle\langle0\rvert \bigr) + \frac12 \bigl( \lvert1\rangle\langle1\rvert \otimes \lvert1\rangle\langle1\rvert \bigr).$$

It is separable. Its mixed marginals show that each subsystem alone is uncertain, and its joint density operator shows classical correlation, but neither fact proves entanglement for a mixed global state.

3. PPT logic. In a two-qubit problem, what does a negative eigenvalue of $\rho^{T_B}$ prove? What does $\rho^{T_B}\ge0$ prove?

Solution

A negative eigenvalue of $\rho^{T_B}$ proves that $\rho$ is entangled. For two qubits, the converse is also complete: $\rho^{T_B}\ge0$ proves that $\rho$ is separable. This second statement relies on the $2\times2$ dimension. It is not true in all larger bipartite dimensions.

4. Witness logic. A valid entanglement witness $W$ gives $\operatorname{Tr}(W\rho)=-0.03$ for a state $\rho$. What can be concluded? What if the same witness gives $+0.02$?

Solution

With the sign convention on this page, $\operatorname{Tr}(W\rho)<0$ proves that $\rho$ is entangled, assuming $W$ is a valid witness and the state estimate is reliable. If the same witness gives $+0.02$, this witness has not detected entanglement. That result alone does not prove that $\rho$ is separable.