Editing Quantum entanglement (section)

=== Classification of entanglement ===
Not all quantum states are equally valuable as a resource. One method to quantify this value is to use an [[#Entanglement measures|entanglement measure]] that assigns a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are:
* If two states can be transformed into each other by a local unitary operation, they are said to be in the same ''LU class''. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).<ref name="GRB1998">{{cite journal |author1=Grassl, M. |author2=Rötteler, M. |author3=Beth, T. |title=Computing local invariants of quantum-bit systems |journal=Phys. Rev. A |volume=58 |issue=3 |pages=1833–1839 |year=1998 |doi=10.1103/PhysRevA.58.1833 |arxiv=quant-ph/9712040|bibcode=1998PhRvA..58.1833G |s2cid=15892529 }}</ref><ref name="Kraus2010">{{cite journal |author=Kraus |first=Barbara |author-link=Barbara Kraus |year=2010 |title=Local unitary equivalence of multipartite pure states |journal=Physical Review Letters |volume=104 |issue=2 |page=020504 |arxiv=0909.5152 |bibcode=2010PhRvL.104b0504K |doi=10.1103/PhysRevLett.104.020504 |pmid=20366579 |s2cid=29984499}}</ref>
* If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states <math>\rho_1</math> and <math>\rho_2</math> in the same SLOCC class are equally powerful, since one can transform each into the other, but since the transformations <math>\rho_1\to\rho_2</math> and <math>\rho_2\to\rho_1</math> may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like <math>|00\rangle+0.01|11\rangle</math>) and the separable ones (i.e., product states like <math>|00\rangle</math>).<ref>{{cite journal |author=Nielsen |first=M. A. |year=1999 |title=Conditions for a Class of Entanglement Transformations |journal=Physical Review Letters |volume=83 |issue=2 |page=436 |arxiv=quant-ph/9811053 |bibcode=1999PhRvL..83..436N |doi=10.1103/PhysRevLett.83.436 |s2cid=17928003}}</ref><ref name="GoWa2010">{{cite journal |author1=Gour, G. |author2=Wallach, N. R. |title=Classification of Multipartite Entanglement of All Finite Dimensionality |journal=Phys. Rev. Lett. |volume=111 |issue=6 |page=060502 |year=2013 |doi=10.1103/PhysRevLett.111.060502 |pmid=23971544 |arxiv=1304.7259 |bibcode=2013PhRvL.111f0502G |s2cid=1570745}}</ref>
* Instead of considering transformations of single copies of a state (like <math>\rho_1\to\rho_2</math>) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when <math>\rho_1\to\rho_2</math> is impossible by LOCC, but <math>\rho_1\otimes\rho_1\to\rho_2</math> is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state <math>\rho</math> into at least one pure entangled state. States that have this property are called [[Entanglement distillation|distillable]]. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable; those that are not are called '[[Bound entanglement|bound entangled]]'.<ref name="HHH97">{{cite journal |author1=Horodecki, M. |author2=Horodecki, P. |author3=Horodecki, R. |title=Mixed-state entanglement and distillation: Is there a ''bound'' entanglement in nature? |journal=Phys. Rev. Lett. |volume=80 |issue=1998 |pages=5239–5242 |year=1998 |arxiv=quant-ph/9801069|doi=10.1103/PhysRevLett.80.5239 |bibcode=1998PhRvL..80.5239H |s2cid=111379972 }}</ref><ref name="horodecki2007" />

A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the ''[[Quantum nonlocality|non-local]] states'', which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the ''[[Quantum steering|steerable]] states'' that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.<ref name="WJD2007">{{cite journal |last1=Wiseman |first1=H. M. |last2=Jones |first2=S. J. |last3=Doherty |first3=A. C. |year=2007 |title=Steering, Entanglement, Nonlocality, and the Einstein–Podolsky–Rosen Paradox |journal=Physical Review Letters |volume=98 |issue=14 |page=140402 |arxiv=quant-ph/0612147 |bibcode=2007PhRvL..98n0402W |doi=10.1103/PhysRevLett.98.140402 |pmid=17501251 |s2cid=30078867}}</ref>