Entanglement frontier map · Deep map
Classifying bipartite entanglement
Every idea used to sort two-party states: what counts as a party, what operations are free, what tests can see, what a state can do, and how much of it there is.
114 ideas and questions · 7 on the frontier · checked 2026-10-30
Every idea, in outline
What counts as a party Topic
Before any state can be classified, the system has to be split in two. That split is usually taken for granted, but it is a choice, and it is not always a tensor product of two Hilbert spaces.
A tensor product structure is induced by accessible observables Established
Which subsystems exist depends on which observables can be controlled and measured, so entanglement is relative to that choice rather than a property of a state vector alone.
Slater rank replaces Schmidt rank for two fermions Established
Two-fermion pure states have a Slater decomposition and a Slater rank, analogous to the Schmidt decomposition and rank. Mixed states are then sorted by their Slater number, the smallest Slater rank needed to build them.
Mode entanglement and particle entanglement differ Established
With indistinguishable particles, entanglement can be assigned to modes or restricted to what parties can actually extract. The operational quantity is bounded by the mode-occupation value, and a single particle shared across two modes has none of it.
Infinite-dimensional systems need more careful definitions Established
In infinite dimensions, states with infinite entropy of entanglement are trace-norm dense, so every product state has arbitrarily entangled neighbours. A bound on mean energy restores sensible behaviour.
In quantum field theory, regions do not factorise Established
Entanglement here is a property of the algebra of observables, not only of states. Regions and their complements do not give a tensor product of two Hilbert spaces, so entropy needs more careful treatment.
Tensor-product and commuting-operator correlations differ Established
Separation can be modelled by a tensor product or by commuting operators on one space. MIP* = RE implies the two give different correlation sets, a negative answer to Tsirelson's problem.
Pure states are classified by their Schmidt coefficients Topic
A bipartite pure state is summarised by its Schmidt coefficients. Each notion of equivalence reads a different amount from that list: all of it, its length, its majorization order, or only its entropy.
R. Horodecki 2007 · M. A. Nielsen 1998
Local unitaries preserve exactly the Schmidt coefficients Established
Two bipartite pure states are related by local unitaries exactly when they have the same Schmidt coefficients. This is the finest classification on the page.
Stochastic LOCC classes are labelled by Schmidt rank Established
If conversion only has to succeed with non-zero probability, a pure state can reach another exactly when its Schmidt rank is at least as large. Mutual convertibility therefore means equal rank.
Deterministic LOCC gives a partial order Topic
With LOCC and certainty required, Schmidt coefficients are compared by majorization. The result is an order, not a list of classes.
Majorization decides deterministic pure state conversion Established
A pure state converts to another by LOCC exactly when its Schmidt coefficients are majorized by those of the target.
Some pure states cannot be converted in either direction Established
Majorization is only a partial order, so there are pairs of states neither of which can deterministically reach the other. No single number can capture single-copy convertibility.
A finite set of monotones gives the optimal conversion probability Established
When conversion cannot be certain, its best probability is set by a finite family of monotones built from the Schmidt coefficients. These quantify the entanglement of a single copy.
Catalysts coarsen the order Topic
Allowing an entangled state that is returned at the end enlarges what is reachable. The order relaxes in stages, ending at a single number.
D. Jonathan 1999 · T. V. Kondra 2021
Catalysts enable conversions majorization forbids Established
Some transformations ruled out by Nielsen's theorem become possible with a catalyst that is recovered intact. The resulting relation between Schmidt vectors is called trumping.
Explicit inequalities decide the trumping relation Established
Turgut and Klimesh independently gave complete families of inequalities that decide whether one Schmidt vector can trump another.
Asymptotically, entropy is the only invariant Established
With many copies, pure states convert reversibly at a rate set by the entropy of the reduced state. All the structure of the Schmidt list collapses to one number.
The geometry of mixed states Topic
Mixed states form a convex body with the separable states inside it. Classifying a state can mean locating it in that body: by the sets it belongs to, its orbit under local operations, or its distance from the boundary.
R. F. Werner 1989 · K. Zyczkowski 1998
The separable states are a convex set Established
Mixtures of product states are closed under further mixing, so any entangled state can be separated from the set by a hyperplane. That geometric fact is what makes entanglement witnesses exist.
How large is the separable set? Topic
The separable set has non-zero volume, but how much of state space it fills depends strongly on dimension and on how states are drawn.
The separable states have positive volume Established
Separable states are not a negligible set: they occupy a positive fraction of the state space in any finite dimension.
A ball of separable states surrounds the maximally mixed state Established
All states close enough to the maximally mixed state are separable. Gurvits and Barnum bound the largest such ball.
Random induced states change from entangled to separable sharply Established
For a random state on two d-dimensional systems, obtained by tracing out an environment of dimension s, there is a sharp threshold of order roughly d³. Below it the state is very probably entangled, above it very probably separable.
What can the eigenvalues alone decide? Topic
Some states are separable however they are rotated globally. For those, the eigenvalues are enough, and no entangling unitary can help.
M. Kus 2000 · S. Arunachalam 2014
Absolutely separable states stay separable under every global unitary Established
Kuś and Życzkowski defined this class, in which every state with the same spectrum is separable.
Absolutely PPT states have a spectral characterisation Established
Hildebrand showed that whether every rotation of a state remains PPT can be decided from its eigenvalues, using a finite set of linear matrix inequalities.
For qubit-qudit systems, absolutely separable equals absolutely PPT Established
In 2×n systems, the two spectral classes coincide, so the question is answered by a simple condition on the eigenvalues.
Does absolutely PPT imply absolutely separable in every dimension? Open
The two sets coincide in the qubit-qudit case, and absolutely separable states are always absolutely PPT. Whether the converse holds generally is the question.
Still unknown: The general equivalence has not been proved or refuted. Arunachalam, Johnston and Russo show that a wide range of witnesses and positive maps cannot separate the two sets.
Sorting mixed states by local orbits and normal forms Topic
Two mixed states count as the same if local operations relate them. Invariants and normal forms label these orbits, in place of the Schmidt list that pure states enjoy.
M. Kus 2000 · F. Verstraete 2001
Local orbits can differ in dimension Established
Explicit methods compute the dimension of the local orbit of any mixed state, which characterises the set of effectively different states.
Polynomial invariants label local unitary classes Established
Local unitary classes of mixed states can be separated by polynomial invariants of the density matrix. Makhlin gave a full set for mixed states of two qubits.
Local filtering brings states to a normal form Established
Invertible local filters generalise the singular value decomposition to mixed states. For two qubits this is a Lorentz singular value decomposition, which gives a concrete analysis of separability.
F. Verstraete 2001 · F. Verstraete 2001 · J. M. Leinaas 2006
A complete picture of two-qubit states with maximally mixed subsystems Established
Written in the Hilbert-Schmidt basis, two-qubit states with maximally disordered subsystems admit a complete characterisation of which are inseparable.
Every state splits into a separable part and an entangled remainder Established
A state can be decomposed with the largest possible separable weight, leaving an entangled remainder. This gives a way to compare states by how much separable part they contain.
The Schmidt number sorts mixed states into nested sets Topic
Schmidt rank extends to mixed states as a ladder of nested convex sets, in which separable states are only the first rung.
The Schmidt number extends rank to mixed states Established
A mixed state's Schmidt number is the smallest k such that it can be built from pure states of Schmidt rank at most k, minimised over all decompositions. Separable states have Schmidt number 1.
k-positive maps detect Schmidt number Established
A state has Schmidt number at most k exactly when k-positive maps send it to a positive operator. This extends the positive-map test for separability up the ladder.
Witnesses can certify a lower bound on Schmidt number Established
Sanpera, Bruß and Lewenstein give a canonical form for Schmidt-number witnesses, along with methods to optimise them.
Two-qutrit PPT states have Schmidt number at most 2 Established
Sanpera, Bruß and Lewenstein gave strong evidence for this. Yang, Leung and Tang, and independently Choi, proved that every 2-positive map from M3 to M3 is decomposable, which is the dual statement.
What separability tests can see Topic
Every test divides states into those it flags and those it misses, so the tests themselves classify states. Deciding separability exactly is NP-hard, so the useful question is which test sees which states.
Positive maps classify entanglement by what they can detect Topic
A positive but not completely positive map applied to one party can turn an entangled state into something with a negative eigenvalue. The structure of these maps mirrors the structure of entanglement.
Positive maps collectively decide separability Established
A state is separable exactly when it stays positive under every positive map applied to one party. The test is complete, but no general description of all positive maps exists.
The transpose gives the partial transpose test Established
The transpose is the most familiar positive map. It splits states into PPT and NPT, and PPT states cannot be distilled. Whether every NPT state can be is still open.
The reduction criterion ties detection to distillation Established
The reduction map gives a separability test, and a state that violates it can be distilled. Cerf, Adami and Gingrich reached the same criterion through a conditional operator.
Decomposable maps cannot detect PPT entanglement Established
A decomposable map adds nothing beyond the partial transpose. Detecting PPT entangled states requires an indecomposable map, the first example of which came from Choi. In 2×2 and 2×3, every positive map is decomposable.
Families of indecomposable maps reveal bound entanglement Established
Explicit families of non-decomposable positive maps detect PPT entangled states in even dimensions and in 2^N×2^N systems. Each family reveals a different slice of them.
Witnesses can be optimised against the separable set Established
Lewenstein and coauthors give necessary and sufficient conditions for a witness to be optimal. They construct non-decomposable witnesses from edge states, which ties the structure of witnesses to that of PPT entanglement.
What is the general structure of positive maps? Open
A full description of positive maps would solve the separability problem through the Horodecki criterion, so the two questions are closely tied.
Still unknown: No general characterisation of positive maps is known in dimensions where indecomposable maps exist. Since deciding separability is NP-hard, a simple one is not expected.
Tests from rearranged matrices and correlations Topic
A second family of tests needs no positive map. It bounds a norm or a covariance that separable states cannot exceed, and often catches PPT entangled states.
The computable cross norm and realignment criterion Established
Realigning the density matrix and summing its singular values gives a quantity at most 1 on separable states.
Covariance matrices unify several criteria Established
Conditions on covariance matrices of local observables give strong criteria that detect many bound entangled states, are necessary and sufficient for two qubits, and clarify how earlier criteria are related.
Violating local uncertainty relations signals entanglement Established
Separable states inherit uncertainty bounds from their local parts. Hofmann and Takeuchi show that violating such a sum bound is a signature of entanglement.
The correlation tensor gives a separability condition Established
Writing a state in terms of local generators gives a correlation tensor. Bounding its norm yields separability conditions and lower bounds on concurrence.
Tests comparing global and local disorder Topic
Separable states cannot be more ordered globally than locally. Entropy and majorization turn that intuition into tests.
Negative conditional entropy implies entanglement Established
Inseparability is linked to conditional α-entropy inequalities: a state more disordered locally than globally cannot be separable.
Separable states are more disordered globally than locally Established
For a separable state, the global spectrum is majorized by each local spectrum. This is stronger than the entropy inequality alone.
Tests based on range, rank and product vectors Topic
A separable state's range is spanned by product vectors whose local conjugates span the range of its partial transpose. Rank and range thus classify PPT states that spectral tests cannot reach.
The range criterion detects PPT entangled states Established
The range condition gave the first explicit PPT entangled states, in 3×3 and 2×4.
Unextendible product bases build bound entangled states Established
A set of orthogonal product vectors with no further product vector orthogonal to it leaves a complement containing no product states. The normalised projector onto that complement is bound entangled.
Low-rank PPT states are separable Established
For an M×N state with M ≤ N and rank at most N, PPT is necessary and sufficient for separability, so there are no rank-3 PPT entangled states.
Edge states sit at the boundary of PPT entanglement Established
An edge state is a PPT entangled state from which no product vector can be subtracted while staying PPT. Lewenstein and coauthors use them to build optimal witnesses.
Two-qutrit rank-four PPT entangled states all come from UPBs Established
Every two-qutrit PPT entangled state of rank four can be built from an unextendible product basis. Chen and Đoković also determine the closure of this set.
Extendibility, and the limits of efficient testing Topic
Entanglement cannot be freely shared. Asking how widely a state can be extended yields a complete test, along with a measure of how hard the problem is.
A hierarchy of symmetric extensions detects all entanglement Established
Separable states have symmetric extensions at every level. An entangled state fails at some finite level, and each level is a semidefinite program.
Monogamy limits how far correlations can be shared Established
Koashi and Winter give an identity trading entanglement with one party against classical correlation with another, from which rigorous monogamy relations follow.
No efficient complete test is expected Established
Deciding separability is NP-hard, even within an inverse polynomial distance of the separable set. Any complete classification by easy tests is therefore not expected unless P = NP.
Moments of the partial transpose can be measured directly Established
Randomised local measurements estimate moments of the partial transpose without full tomography. Yu, Imai and Gühne find the optimal entanglement certification these moments allow.
Free and bound entanglement Topic
Can a state be turned back into pure entangled pairs? This operational question divides entangled states into free and bound, and leaves one major boundary unsettled.
M. Horodecki 1998 · B. C. Hiesmayr 2024
Distillable states can be purified into singlets Established
Given many sufficiently pure noisy pairs, LOCC can extract fewer pairs of arbitrarily high fidelity. This defines the free entangled class.
Every entangled two-qubit state is distillable Established
In the smallest case, the classes coincide: entangled and free mean the same thing.
PPT entangled states cannot be distilled Established
A distillable state must violate the partial transpose test. PPT entangled states therefore form a class of bound entanglement.
Is every NPT state distillable? Topic
Bound entanglement is known on the PPT side. Whether it also exists on the NPT side is the central open question of this classification.
D. P. DiVincenzo 1999 · W. Dür 1999
The question reduces to Werner states Established
Local filtering and twirling take any NPT state to an NPT Werner state. If all NPT Werner states are distillable, then so are all NPT states.
Do bound entangled states with a negative partial transpose exist? Open
Every distillable state is NPT. Evidence going back to 1999 suggests the converse may fail in dimensions above 2×2, with Werner states as the key test family.
Still unknown: The question remains listed among the five open problems of Horodecki, Rudnicki and Życzkowski, and was still called long-standing in 2024. A 2006 preprint claimed existence, but later work continues to treat the question as open.
D. P. DiVincenzo 1999 · W. Dür 1999 · P. Horodecki 2020 · S.-Y. Qi 2024 · R. Simon 2006
NPT Werner states are two-copy undistillable in every dimension Claimed
A July 2026 preprint proves a partial-trace matrix inequality and uses it to claim two-copy undistillability of NPT Werner states in arbitrary local dimension.
Still unknown: No independent confirmation had been found by the check date. Even if correct, it covers two copies, while NPT bound entanglement requires every number of copies.
PPT-preserving channels distil every NPT state Established
All NPT states are distillable under channels that preserve the partial transpose's positivity. The open question therefore turns on the gap between separable and PPT-preserving operations.
Sorting states by the correlations they support Topic
Entanglement is one rung in a ladder of correlations, with discord below it and steering and Bell nonlocality above. The picture changes once local filtering, many copies or side resources are allowed.
H. M. Wiseman 2006 · H. Ollivier 2001
Quantum discord separates classical from quantum correlation Established
Discord compares two forms of mutual information that agree classically. Its vanishing is a stricter condition than separability, so some separable states still carry it.
Steering lies strictly between entanglement and Bell nonlocality Established
Wiseman, Jones and Doherty define steering operationally. Bell nonlocal states are steerable and steerable states are entangled, and neither inclusion can be reversed.
Can entangled states have a local model? Topic
Entangled states need not violate a Bell inequality directly. What looks local can however become nonlocal once more resources are allowed.
R. F. Werner 1989 · J. Barrett 2001
Some entangled states admit a local hidden-variable model Established
Werner built a model for projective measurements on some entangled states. Barrett extended the conclusion to general POVM measurements.
Copies of a local state can violate a Bell inequality Established
Palazuelos showed that tensoring a local state with itself can produce a Bell violation, so nonlocality is not fixed by the behaviour of a single copy.
All entangled states are nonlocal in semi-quantum games Established
Buscemi shows that a suitable class of nonlocal games, in which referees ask quantum questions, witnesses every entangled state. So what counts as nonlocal depends on the test.
Asymptotic CHSH violation is equivalent to distillability Established
If local processing over arbitrarily many copies is allowed, violating CHSH is the same as being distillable. Bound entangled states do not violate it in this asymptotic sense.
Do nonlocality and bound entanglement exclude each other? Topic
Peres conjectured that PPT states would admit local models. Both the original form and its steering-based strengthening were later disproved.
T. Vértesi 2014 · T. Moroder 2014
Bound entangled states never violate a Bell inequality Refuted
False. Vértesi and Brunner found a bound entangled state violating a Bell inequality, so nonlocality implies neither distillability nor a negative partial transpose.
Bound entangled states can never be used for steering Refuted
False. Moroder and coauthors gave bound entangled states that show steering, a counterexample to the stronger Peres conjecture.
Sorting states by what they can do Topic
A task splits states into useful and useless ones. Teleportation and key distillation draw lines that differ from distillability, and from each other.
P. Horodecki 1998 · K. Horodecki 2003
Teleportation fidelity is fixed by singlet fraction Established
Best teleportation fidelity is directly tied to the maximal singlet fraction attainable by trace-preserving LOCC, giving a task-based threshold on states.
States useful for teleportation are nonlocal resources Established
Cavalcanti, Acín, Brunner and Vértesi show every entangled state useful for teleportation yields a Bell violation in a suitable scheme, linking the two classes.
Every entangled state can enhance teleportation of some state Established
Masanes proved that any bipartite entangled state, even a bound one, can raise the teleportation power of some other state.
Bound entangled states can yield a secret key Established
Horodecki, Horodecki, Horodecki and Oppenheim characterise states containing a private key and show key can be distilled from bound entangled states, so privacy draws a different line from distillability.
Classification depends on the allowed operations Topic
Each class of free operations gives a different resource theory, with different equivalences and conversions. Enlarging the class often simplifies the mathematics, while raising the question of what physical meaning remains.
E. Chitambar 2018 · E. Chitambar 2012
Entanglement is a resource theory of free states and operations Established
Choosing the free states and free operations sets the rules, and comparing theories becomes a way to see which features depend on those rules.
LOCC is physically natural yet mathematically hard Established
LOCC captures what separated labs can do, but its structure is complex enough that even characterising the class is difficult, which motivates the larger classes below.
Separable operations strictly contain LOCC Established
Rains studied entanglement purification under separable superoperators. Nonlocality without entanglement shows orthogonal product states that LOCC cannot reliably distinguish, a task separable measurements manage.
PPT-preserving operations make many quantities computable Topic
Relaxing LOCC to operations that preserve the partial transpose's positivity turns many optimisations into semidefinite programs, and gives bounds for LOCC.
PPT distillation fidelity is a semidefinite program Established
Rains shows the maximum fidelity of a PPT distillation protocol solves a semidefinite program. This gives new upper bounds on LOCC-distillable entanglement.
PPT entanglement cost is tied to logarithmic negativity Established
Audenaert, Plenio and Eisert relate PPT cost directly to log-negativity, and show that cost and distillation coincide for the antisymmetric Werner state.
Entanglement is irreversible under PPT operations Established
Wang and Duan resolved this question with a computable additive bound on the regularised relative entropy of entanglement, showing a gap between cost and distillation for some states.
Under which operations is entanglement reversible? Topic
A second law of entanglement would mean one quantity decides all conversions. Whether this holds depends sharply on the free operations.
F. G. S. L. Brandão 2007 · L. Lami 2021
Entanglement manipulation under LOCC is irreversible Established
Creating a bound entangled state costs entanglement, yet none can be recovered, so LOCC has no single entanglement currency for mixed states.
Entanglement is reversible under all non-entangling operations Refuted
False. Lami and Regula showed that entanglement theory stays irreversible even under all non-entangling transformations.
Reversibility under asymptotically non-entangling operations Established
Once operations may generate vanishing entanglement, reversibility holds with the regularised relative entropy as the single quantity. The proof rests on the generalised quantum Stein's lemma, whose original proof had a gap repaired in 2024.
F. G. S. L. Brandão 2007 · M. Berta 2022 · M. Hayashi 2024 · L. Lami 2024
The Choi correspondence reads states as channels Topic
Bipartite states and channels are two views of the same object, so classifications of one carry over to the other.
M.-D. Choi 1975 · M. Horodecki 2003
Separable Choi states correspond to entanglement-breaking channels Established
A channel destroys all entanglement with a reference exactly when its Choi state is separable.
Is the composition of two PPT maps always entanglement breaking? Conjectured
Equivalently, can entanglement swapping across two PPT links ever produce entanglement? The conjecture says it cannot.
Still unknown: It has been proved in dimension 3 and for Gaussian channels. Higher finite dimensions remain open.
Ordering states by quantity Topic
Measures replace classes with numbers. For mixed states they split by construction, and they disagree about which state has more.
Measures are bracketed by distillation and formation Established
A quantity qualifies as a measure by not increasing under LOCC. Any measure suited to the regime of many pairs lies between distillable entanglement and entanglement of formation.
Different measures can disagree on which state is more entangled Established
Eisert and Plenio compared measures directly. Virmani and Plenio showed that any two good asymptotic measures are either identical or order some states differently.
Measures built from pure-state decompositions Topic
Extend a pure state measure to mixed states by taking the cheapest decomposition. These are physically direct but generally hard to compute.
Entanglement of formation has a two-qubit closed form Established
Wootters, building on Hill and Wootters, gave an explicit formula for any two-qubit state. Beyond this, computing it generally needs optimisation.
Concurrence extends to arbitrary dimensions Established
Introduced for two qubits by Hill and Wootters, concurrence was generalised to higher dimensions through universal state inversion by Rungta and coauthors.
Geometric measure: overlap with the nearest product state Established
On pure states this measure uses the largest overlap with a product state, and extends to mixed states by a convex roof. Barnum and Linden studied related monotones.
Entanglement of formation is additive Refuted
False. Shor tied this to other additivity questions, and Hastings's counterexample settled all of them, so one-copy formation does not give the cost.
Entanglement cost is regularised entanglement of formation Established
The asymptotic preparation cost equals the many-copy limit of formation, which is exact but hard to evaluate.
Measures of distance to the separable set Topic
Measure how far a state lies from separable states, or how much noise it takes to reach them.
V. Vedral 1997 · G. Vidal 1998
Relative entropy measures distinguishability from separable states Established
Vedral and coauthors proposed distance-based measures, of which relative entropy is central. Its regularisation is the quantity in the generalised Stein's lemma.
Relative entropy of entanglement is additive Refuted
False. Vollbrecht and Werner gave a counterexample, which is why asymptotic theorems use the regularised version.
Robustness measures the noise needed to remove entanglement Established
Vidal and Tarrach define it as the minimal mixing with locally prepared states that washes out all entanglement.
Smoothed distance measures give one-shot rates Established
Brandão and Datta derive one-shot conversion rates under non-entangling maps from a smoothed max-relative entropy, and Buscemi and Datta study entanglement cost in a similar setting.
Measures defined by tasks or by what can be calculated Topic
Some measures come from a task, while others are chosen because they can be evaluated or have useful properties. Rarely is a measure all three.
Negativity and log-negativity can be computed for any state Established
These come directly from the partial transpose. Log-negativity is a monotone despite not being convex. Being zero on all PPT states, neither sees bound entanglement.
Squashed entanglement is additive and faithful Established
Built from conditional mutual information minimised over extensions, it is additive and is strictly positive on all entangled states.
Can distillable entanglement be computed in general? Open
The one-way rate has a coding theorem, but it is a regularised optimisation over many copies.
Still unknown: Single-letter formulas are known only in special classes, with new non-degradable families added in 2026. No general tractable formula is known.
Where the classification can be completed Topic
Structure and symmetry make some families fully solvable. They are the test beds for every general conjecture on this page.
Two qubits are classified completely Established
PPT decides separability, every entangled state is distillable, and formation has a closed form. Here the main classifications agree.
Twirling reduces states to one-parameter symmetric families Established
Averaging over U⊗U gives Werner states and over U⊗U* gives isotropic states, each with one parameter. Their symmetry makes measures and distillation calculations far simpler.
R. F. Werner 1989 · K. G. H. Vollbrecht 2000 · M. Horodecki 1997
Bipartite stabilizer states reduce to EPR pairs Established
Local unitaries reduce any bipartite stabilizer state to EPR pairs and local states, so the pair count fully sets its entanglement.
Gaussian states are classified by covariance matrices Topic
For Gaussian states of continuous-variable systems, separability and distillability reduce to conditions on a finite covariance matrix.
R. Simon 1999 · R. F. Werner 2000
PPT decides separability for 1×N mode Gaussian states Established
Simon and Duan and coauthors gave two-mode criteria. Werner and Wolf extended the PPT result to one mode against any number of modes.
Bound entangled Gaussian states exist in 2×2 modes Established
With two modes on each side, Werner and Wolf give a family of explicit counterexamples: PPT no longer implies separable.
A computable criterion decides separability for all Gaussian states Established
Giedke and coauthors give a necessary and sufficient, operational test for any number of modes. It is independent of partial transposition and strictly stronger.
All inseparable two-mode Gaussian states are distillable Established
Giedke, Duan, Cirac and Zoller show that every entangled Gaussian state of two infinite-dimensional systems can be distilled. Their protocol includes a non-Gaussian step, a measurement of total particle number.
Gaussian operations alone cannot distil Gaussian states Established
Giedke and Cirac characterise Gaussian operations and show they cannot distil Gaussian states. Eisert, Scheel and Plenio rule out the corresponding two-copy iterative protocols, so distillation needs a non-Gaussian step.
Sorting many-body states by how entanglement grows Topic
For a large system, the class of a state is how entanglement across a cut grows with the region. This classifies states by physics and by how hard they are to simulate.
Gapped 1D ground states obey an area law Established
Hastings proved that ground states of gapped one-dimensional Hamiltonians have block entropy bounded by a constant, independent of block size.
Do gapped 2D ground states obey an area law? Partial
Anshu, Arad and Gosset prove an area law for locally gapped frustration-free 2D spin systems, with respect to a vertical bipartition of the lattice.
Still unknown: The result does not yet cover general gapped two-dimensional ground states or arbitrary regions.
Critical 1D systems have logarithmic corrections Established
Calabrese and Cardy show that entanglement entropy in 1+1-dimensional conformal field theories grows logarithmically with interval length, with a coefficient set by the central charge.
Random pure states are nearly maximally entangled Established
Page computed the average subsystem entropy of a random pure state, which is close to maximal. Hayden, Leung and Winter show this concentrates sharply, and that large subspaces contain only almost maximally entangled states.
A constant correction detects topological order Established
Kitaev and Preskill, and Levin and Wen, independently extract a universal constant from the entropy of a region, which detects topological order in a ground state.
The entanglement spectrum holds more than entropy Established
Li and Haldane propose the full spectrum of the reduced state as a generalisation of entropy, one that can identify topological order.
Limited entanglement allows efficient classical simulation Established
Vidal shows that low Schmidt rank across cuts allows efficient simulation. Schuch and coauthors show even a von Neumann area law is not enough for matrix product state approximation, so Rényi entropies matter.
Sources were checked against arXiv or Crossref records, and most statements against the source’s abstract. Standard results were written from the wider literature, so treat the cited paper as the place to confirm them. The map is a growing selection, so a missing idea means it has not been mapped yet.