Editing Quantum entanglement (section)

== Mathematical details ==
The following subsections use the formalism and theoretical framework developed in the articles [[bra–ket notation]] and [[mathematical formulation of quantum mechanics]].

=== Pure states ===
Consider two arbitrary quantum systems {{mvar|A}} and {{mvar|B}}, with respective [[Hilbert space]]s {{mvar|H<sub>A</sub>}} and {{mvar|H<sub>B</sub>}}. The Hilbert space of the composite system is the [[tensor product]]
: <math> H_A \otimes H_B.</math>

If the first system is in state <math>| \psi \rangle_A</math> and the second in state <math>| \phi \rangle_B</math>, the state of the composite system is
: <math>|\psi\rangle_A \otimes |\phi\rangle_B.</math>

States of the composite system that can be represented in this form are called separable states, or [[product state]]s. However, not all states of the composite system are separable. Fix a [[basis (linear algebra)|basis]] <math> \{|i \rangle_A\}</math> for {{mvar|H<sub>A</sub>}} and a basis <math> \{|j \rangle_B\}</math> for {{mvar|H<sub>B</sub>}}. The most general state in {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}} is of the form
: <math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.

This state is separable if there exist vectors <math> [c^A_i], [c^B_j]</math> so that <math> c_{ij}= c^A_i c^B_j,</math> yielding <math display="inline"> |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math display="inline"> |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math> [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math> c^A_i,c^B_j</math> we have <math> c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.<ref name="Rieffel2011"/>{{rp|218}}<ref name="Zwiebach2022"/>{{rp|§1.5}}

For example, given two basis vectors <math> \{|0\rangle_A, |1\rangle_A\}</math> of {{mvar|H<sub>A</sub>}} and two basis vectors <math> \{|0\rangle_B, |1\rangle_B\}</math> of {{mvar|H<sub>B</sub>}}, the following is an entangled state:
: <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right ).</math>

If the composite system is in this state, it is impossible to attribute to either system {{mvar|A}} or system {{mvar|B}} a definite [[pure state]]. Another way to say this is that while the [[von Neumann entropy]] of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". The above example is one of four [[Bell states]], which are (maximally) entangled pure states (pure states of the {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}} space, but which cannot be separated into pure states of each {{mvar|H<sub>A</sub>}} and {{mvar|H<sub>B</sub>}}).<ref name="Zwiebach2022"/>{{rp|§18.6}}

Now suppose Alice is an observer for system {{mvar|A}}, and Bob is an observer for system {{mvar|B}}. If in the entangled state given above Alice makes a measurement in the <math> \{|0\rangle, |1\rangle\}</math> eigenbasis of {{mvar|A}}, there are two possible outcomes, occurring with equal probability: Alice can obtain the outcome 0, or she can obtain the outcome 1. If she obtains the outcome 0, then she can predict with certainty that Bob's result will be 1. Likewise, if she obtains the outcome 1, then she can predict with certainty that Bob's result will be 0. In other words, the results of measurements on the two qubits will be perfectly anti-correlated. This remains true even if the systems {{mvar|A}} and {{mvar|B}} are spatially separated. This is the foundation of the EPR paradox.<ref name="Nielsen-2010" />{{Rp|pages=113–114}}

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see [[no-communication theorem]].

=== Ensembles ===
As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a [[density matrix]], which is a [[positive-semidefinite matrix]], or a [[trace class]] when the state space is infinite-dimensional, and which has trace 1. By the [[spectral theorem]], such a matrix takes the general form:
: <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>
where the ''w''<sub>''i''</sub> are positive-valued probabilities (they sum up to 1), the vectors {{math|''α''<sub>''i''</sub>}} are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret {{mvar|ρ}} as representing an ensemble where <math> w_i </math> is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.<ref name="Peres1993"/>{{rp|73–74}}<ref name="Holevo2001"/>{{rp|13–15}}<ref name="Zwiebach2022"/>{{rp|§22.2}}

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with spins aligned in the positive {{math|'''z'''}} direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative {{math|'''y'''}} direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Following the definition above, for a bipartite composite system, mixed states are just density matrices on {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}}. That is, it has the general form
: <math>\rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right]
</math>
where the ''w''<sub>''i''</sub> are positively valued probabilities, <math display="inline">\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<ref name=Laloe>{{cite journal|last=Laloe|first=Franck|year=2001|title=Do We Really Understand Quantum Mechanics|journal=American Journal of Physics |volume=69 |issue=6|pages=655–701 |arxiv=quant-ph/0209123 |bibcode=2001AmJPh..69..655L |doi=10.1119/1.1356698|s2cid=123349369 }}</ref>{{rp|131–132}}
: <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, </math>
where the {{math|''w''<sub>''i''</sub>}} are positively valued probabilities and the <math>\rho_i^A</math>s and <math>\rho_i^B</math>s are themselves mixed states (density operators) on the subsystems {{mvar|A}} and {{mvar|B}} respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{cite book |last=Gurvits |first=L. |title=Proceedings of the thirty-fifth annual ACM symposium on Theory of computing |year=2003 |isbn=978-1-58113-674-6 |page=10 |language=en |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |doi=10.1145/780542.780545 |arxiv=quant-ph/0303055 |s2cid=5745067}}</ref> For the {{math|2 × 2}} and {{math|2 × 3}} cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |vauthors=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |issue=1 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 |citeseerx=10.1.1.252.496 |s2cid=10580997 }}</ref>

=== Reduced density matrices ===
The idea of a reduced density matrix was introduced by [[Paul Dirac]] in 1930.<ref name="Dirac1930">
{{cite journal
 | last = Dirac | first = Paul Adrien Maurice | author-link = Paul Dirac
 | title = Note on exchange phenomena in the Thomas atom
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | volume = 26
 | number = 3
 | pages = 376–385
 | doi = 10.1017/S0305004100016108
 | bibcode=1930PCPS...26..376D
 | year = 1930
 | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6C5FF7297CD96F49A8B8E9E3EA50E412/S0305004100016108a.pdf/note-on-exchange-phenomena-in-the-thomas-atom.pdf
 | doi-access=free
}}</ref> Consider as above systems {{mvar|A}} and {{mvar|B}} each with a Hilbert space {{mvar|H<sub>A</sub>, H<sub>B</sub>}}. Let the state of the composite system be
: <math> |\Psi \rangle \in H_A \otimes H_B. </math>

As indicated above, in general there is no way to associate a pure state to the component system {{mvar|A}}. However, it still is possible to associate a density matrix. Let
: <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.
which is the [[projection operator]] onto this state. The state of {{mvar|A}} is the [[partial trace]] of {{mvar|ρ<sub>T</sub>}} over the basis of system {{mvar|B}}:
: <math>\rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j^{N_B} \left( I_A \otimes \langle j|_B \right) \left( |\Psi\rangle \langle\Psi| \right)\left( I_A \otimes |j\rangle_B \right) = \hbox{Tr}_B \; \rho_T.</math>

The sum occurs over <math>N_B := \dim(H_B)</math> and <math>I_A</math> the identity operator in <math>H_A</math>. {{mvar|ρ<sub>A</sub>}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}. Colloquially, we "trace out" or "trace over" system {{mvar|B}} to obtain the reduced density matrix on {{mvar|A}}.<ref name="Rieffel2011"/>{{rp|207–212}}<ref name="Rau2021"/>{{rp|133}}<ref name="Zwiebach2022"/>{{rp|§22.4}}

For example, the reduced density matrix of {{mvar|A}} for the entangled state
: <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),</math>
discussed above is<ref name="Zwiebach2022"/>{{rp|§22.4}}
: <math>\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right ).</math>

This demonstrates that the reduced density matrix for an entangled pure ensemble is a mixed ensemble. In contrast, the density matrix of {{mvar|A}} for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is<ref name="Nielsen-2010"/>{{rp|106}}
: <math>\rho_A = |\psi\rangle_A \langle\psi|_A,</math>
the projection operator onto <math>|\psi\rangle_A</math>.

In general, a bipartite pure state ''ρ'' is entangled if and only if its reduced states are mixed rather than pure.<ref name="Rau2021"/>{{rp|131}}

=== Entanglement as a resource ===
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.<ref name="Chitambar2019">
{{cite journal
 | last1 = Chitambar | first1 = Eric
 | last2 = Gour | first2 = Gilad
 | title = Quantum resource theories
 | journal = Reviews of Modern Physics
 | volume = 91
 | number = 2
 | pages = 025001
 | doi = 10.1103/RevModPhys.91.025001
 | arxiv = 1806.06107
 | year = 2019
 | bibcode = 2019RvMP...91b5001C
 | s2cid = 119194947
}}</ref><ref name="GG-2022">
{{cite journal
 | last1 = Georgiev
 | first1 = Danko D.
 | last2 = Gudder
 | first2 = Stanley P.
 | title = Sensitivity of entanglement measures in bipartite pure quantum states
 | journal = Modern Physics Letters B
 | volume = 36
 | number = 22
 | pages = 2250101–2250255
 | doi = 10.1142/S0217984922501019
 | arxiv = 2206.13180
 | year = 2022
 | bibcode = 2022MPLB...3650101G
 | s2cid = 250072286
}}</ref> The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary [[quantum operation]]s can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called [[LOCC]] (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. 

If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state <math>|\Psi\rangle</math> she has in her lab. Alice performs a joint measurement on <math>|\Psi\rangle</math> together with her half of the entangled state  and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to <math>|\Psi\rangle</math>. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state <math>|\Psi\rangle</math> is not copied, but transferred: it is said to be "[[quantum teleportation|teleported]]" to Bob's laboratory through this protocol.<ref name="Nielsen-2010">{{cite book |last1=Nielsen |first1=Michael A. |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10th anniversary|location=Cambridge}}</ref>{{rp|27}}<ref name="horodecki2007"/>{{rp|875}}<ref>{{cite journal|arxiv=1505.07831 |title=Advances in Quantum Teleportation |first1=S. |last1=Pirandola |first2=J. |last2=Eisert |first3=C. |last3=Weedbrook |first4=A. |last4=Furusawa |first5=S. L. |last5=Braunstein |journal=Nature Photonics |volume=9 |pages=641–652 |year=2015 |issue=10 |doi=10.1038/nphoton.2015.154|bibcode=2015NaPho...9..641P }}</ref>

[[File:Entanglement swapping.svg|thumb|Entanglement of states from independent sources can be swapped through Bell state measurement.<ref name="GuoReview2023">{{cite journal |last1=Hu |first1=Xiao-Min |last2=Guo |first2=Yu |last3=Liu |first3=Bi-Heng |last4=Li |first4=Chuan-Feng |last5=Guo |first5=Guang-Can |date=June 2023 |title=Progress in quantum teleportation |url=https://www.nature.com/articles/s42254-023-00588-x |journal=Nature Reviews Physics |language=en |volume=5 |issue=6 |pages=339–353 |doi=10.1038/s42254-023-00588-x |bibcode=2023NatRP...5..339H |issn=2522-5820}}</ref>{{rp|341}}]]
[[Entanglement swapping]] is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, A and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other.<ref name="horodecki2007"/><ref name="Pan1998">{{Cite journal |last1=Pan |first1=J.-W. |last2=Bouwmeester |first2=D. |last3=Weinfurter |first3=H. |last4=Zeilinger |first4=A. |author-link4=Anton Zeilinger |year=1998 |title=Experimental entanglement swapping: Entangling photons that never interacted |journal=[[Physical Review Letters]] |volume=80 |number=18 |pages=3891–3894 |doi=10.1103/PhysRevLett.80.3891 |bibcode=1998PhRvL..80.3891P }}</ref>

An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<ref name="horodecki2007" />

===Multipartite entanglement===
{{main|Multipartite entanglement}}
Quantum states describing systems made of more than two pieces can also be entangled. An example for a three-qubit system is the [[Greenberger–Horne–Zeilinger state|Greenberger–Horne–Zeilinger (GHZ) state]],
<math display="block">|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math>
Another three-qubit example is the [[W state]]:
<math display="block">|\mathrm{W}\rangle = \frac{|001\rangle + |010\rangle + |100\rangle}{\sqrt{3}}.</math>
Tracing out any one of the three qubits turns the GHZ state into a separable state, whereas the result of tracing over any of the three qubits in the W state is still entangled. This illustrates how multipartite entanglement is a more complicated topic than bipartite entanglement: systems composed of three or more parts can exhibit multiple qualitatively different types of entanglement.<ref name="Bengtsson2017"/>{{rp|493–497}} A single particle cannot be maximally entangled with more than a particle at a time, a property called [[Monogamy of entanglement|monogamy]].<ref>{{Cite book |last1=Bertlmann |first1=Reinhold |url=https://books.google.com/books?id=uzHaEAAAQBAJ&dq=monogamy+of+entanglement&pg=PA511 |title=Modern Quantum Theory: From Quantum Mechanics to Entanglement and Quantum Information |last2=Friis |first2=Nicolai |date=2023-10-05 |publisher=Oxford University Press |isbn=978-0-19-150634-5 |language=en |page=511}}</ref>

=== 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>

=== Entropy ===
In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

==== Definition ====
In classical [[information theory]] {{mvar|H}}, the [[Shannon entropy]], is associated to a probability distribution, <math>p_1, \cdots, p_n</math>, in the following way:<ref name="SE">{{cite journal |url=http://authors.library.caltech.edu/5516/1/CERpra97b.pdf#page=10 |title=Information-theoretic interpretation of quantum error-correcting codes |journal=Physical Review A |date=September 1997 |volume=56 |number=3 |pages=1721–1732 |arxiv=quant-ph/9702031 |doi=10.1103/PhysRevA.56.1721 |first1=Nicolas J. |last1=Cerf |first2=Richard |last2=Cleve |bibcode=1997PhRvA..56.1721C }}</ref>
: <math>H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i.</math>

Since a mixed state {{mvar|ρ}} is a probability distribution over an ensemble, this leads naturally to the definition of the [[von Neumann entropy]]:<ref name="Peres1993"/>{{rp|264}}
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right),</math>
which can be expressed in terms of the [[eigenvalue]]s of {{mvar|ρ}}: 
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.

Since an event of probability 0 should not contribute to the entropy, and given that
: <math> \lim_{p \to 0} p \log p = 0,</math>
the convention {{math|0 log(0) {{=}} 0}} is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle is {{math|log(2)}}, which can be shown to be the maximum entropy for {{math|2 × 2}} mixed states.<ref name="Holevo2001"/>{{rp|15}}

==== As a measure of entanglement ====
Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<ref name="Plenio">{{cite journal|last1=Plenio |first1=Martin B. |first2=Shashank |last2=Virmani|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quant. Inf. Comp. |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P}}</ref><ref name="Vedral2002">{{cite journal | last = Vedral | first = Vlatko |author-link = Vlatko Vedral | doi = 10.1103/RevModPhys.74.197 | arxiv = quant-ph/0102094 | bibcode=2002RvMP...74..197V | volume=74 | issue = 1 | title=The role of relative entropy in quantum information theory | year=2002 | journal=Reviews of Modern Physics | pages=197–234 | s2cid = 6370982 }}</ref> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.<ref>{{cite journal |last1=Hill |first1=S |last2=Wootters |first2=W. K. |title=Entanglement of a Pair of Quantum Bits |journal=Phys. Rev. Lett. |arxiv=quant-ph/9703041 |doi =10.1103/PhysRevLett.78.5022 |year=1997 |volume=78 |issue=26 |pages=5022–5025 |bibcode=1997PhRvL..78.5022H |s2cid=9173232 }}</ref>

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {{mset|1/''n'', ..., 1/''n''}}.<ref name="Nielsen-2010"/>{{rp|505}} Therefore, a bipartite pure state {{math|''ρ'' ∈ ''H''<sub>A</sub> ⊗ ''H''<sub>B</sub>}} is said to be a ''maximally entangled state'' if the reduced state of each subsystem of {{mvar|ρ}} is the diagonal matrix<ref>{{Cite journal |last1=Enríquez |first1=M. |last2=Wintrowicz |first2=I. |last3=Życzkowski |first3=K. |author-link3=Karol Życzkowski |date=March 2016 |title=Maximally Entangled Multipartite States: A Brief Survey |journal=Journal of Physics: Conference Series |volume=698 |issue=1 |pages=012003 |doi=10.1088/1742-6596/698/1/012003 |bibcode=2016JPhCS.698a2003E |issn=1742-6588|doi-access=free }}</ref>
: <math>\begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}.</math>

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.<ref name="Bengtsson2017"/>{{rp|471}}

[[Rényi entropy]] also can be used as a measure of entanglement.<ref name="Bengtsson2017"/>{{rp|447,480}}<ref>{{cite journal |last1=Wang |first1=Yu-Xin |last2=Mu |first2=Liang-Zhu |last3=Vedral |first3=Vlatko |last4=Fan |first4=Heng |date=17 February 2016 |title=Entanglement Rényi α entropy |url=https://link.aps.org/doi/10.1103/PhysRevA.93.022324 |journal=Physical Review A |language=en |volume=93 |issue=2 |page=022324 |arxiv=1504.03909 |doi=10.1103/PhysRevA.93.022324 |bibcode=2016PhRvA..93b2324W |issn=2469-9926}}</ref>

=== Entanglement measures ===
Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, [[entropy of entanglement|entanglement entropy]] is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature<ref name="Plenio" /> and no single one is standard.
* Entanglement cost
* [[entanglement distillation|Distillable entanglement]]
* [[Entanglement of formation]]
* [[Concurrence (quantum computing)|Concurrence]]
* [[quantum relative entropy|Relative entropy of entanglement]]
* [[Squashed entanglement]]
* [[Negativity (quantum mechanics)#Logarithmic negativity|Logarithmic negativity]]
Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult ([[NP-hard]]) to compute for mixed states as the dimension of the entangled system grows.<ref>{{cite journal|last1=Huang|first1=Yichen|title=Computing quantum discord is NP-complete|journal=New Journal of Physics|date=21 March 2014|volume=16|issue=3|pages=033027|doi=10.1088/1367-2630/16/3/033027|bibcode=2014NJPh...16c3027H|arxiv = 1305.5941 |s2cid=118556793}}</ref>

=== Quantum field theory ===
The [[Reeh–Schlieder theorem]] of [[quantum field theory]] is sometimes interpreted as saying that entanglement is omnipresent in the [[quantum vacuum]].<ref>{{cite book|first=Stephen J. |last=Summers |chapter=Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State |arxiv=0802.1854 |title=Deep Beauty: Understanding the Quantum World through Mathematical Innovation |pages=317–341 |editor-first=Hans |editor-last=Halvorson |publisher=Cambridge University Press |year=2011 |isbn=9781139499224}}</ref>