View source for Entanglement distillation

{{Use American English|date=January 2019}}
{{Short description|Process of "purifying" entangled quantum states}}

'''Entanglement distillation''' (also called '''entanglement purification''') is the transformation of ''N'' copies of an arbitrary [[Quantum entanglement|entangled state]] <math>\rho</math> into some number of approximately pure [[Bell pair]]s, using only [[LOCC|local operations and classical communication]]. Entanglement distillation can overcome the degenerative influence of noisy [[quantum channel]]s<ref name="BDSW96"/> by transforming previously shared, less-entangled pairs into a smaller number of [[Maximally entangled state|maximally-entangled]] pairs.

==History==
The limits for entanglement dilution and distillation are due to [[Charles H. Bennett (computer scientist)|C. H. Bennett]], H. Bernstein, [[Sandu Popescu|S. Popescu]], and [[Benjamin Schumacher|B. Schumacher]],<ref name="BBSP">{{cite journal | last1 = Bennett | first1 = Charles H. | last2 = Bernstein | first2 = Herbert J. | last3 = Popescu | first3 = Sandu | last4 = Schumacher | first4 = Benjamin | year = 1996 | title = Concentrating Partial Entanglement by Local Operations | journal = Phys. Rev. A | volume = 53 | issue = 4| pages = 2046–2052 | doi=10.1103/physreva.53.2046| pmid = 9913106 | arxiv = quant-ph/9511030 | bibcode = 1996PhRvA..53.2046B | s2cid = 8032709 }}</ref> who presented the first distillation protocols for [[pure state]]s in 1996;{{cn|reason=claims of "first" are original research unless sourced|date=December 2024}} entanglement distillation protocols for [[Mixed state (physics)|mixed state]]s were introduced by Bennett, [[Gilles Brassard|Brassard]], Popescu, Schumacher, [[John A. Smolin|Smolin]] and [[William Wootters|Wootters]]<ref name = "BBPSSW">{{cite journal | last1 = Bennett | first1 = Charles H. | last2 = Brassard | first2 = Gilles | last3 = Popescu | first3 = Sandu | last4 = Schumacher | first4 = Benjamin | last5 = Smolin | first5 = John A. | last6 = Wooters | first6 = William K. | year = 1996 | title = Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels | journal = Phys. Rev. Lett. | volume = 76 | issue = 5| pages = 722–725 | doi=10.1103/physrevlett.76.722 | bibcode=1996PhRvL..76..722B | pmid=10061534| arxiv = quant-ph/9511027 | s2cid = 8236531 }}</ref> the same year. Bennett, [[David P. DiVincenzo|DiVincenzo]], Smolin and Wootters<ref name="BDSW96">{{cite journal | last1 = Bennett | first1 = Charles H. | last2 = DiVincenzo | first2 = David P. | last3 = Smolin | first3 = John A. | last4 = Wooters | first4 = William K. | year = 1996 | title = Mixed State Entanglement and Quantum Error Correction | journal = Phys. Rev. A | volume = 54 | issue = 5| pages = 3824–3851 | doi=10.1103/physreva.54.3824 | pmid=9913930| arxiv = quant-ph/9604024 | bibcode = 1996PhRvA..54.3824B | s2cid = 3059636 }}</ref> established the connection to quantum error-correction in a ground-breaking paper published in August 1996, also in the journal of Physical Review, which has stimulated a lot of subsequent research.{{cn|reason=claims need secondary references|date=December 2024}}

==Motivation==

Suppose that two parties, [[Alice and Bob]], would like to communicate classical information over a noisy quantum channel. Either classical or quantum information can be transmitted over a quantum channel by encoding the information in a quantum state. With this knowledge, Alice encodes the classical information that she intends to send to Bob in a (quantum) product state, as a [[tensor product]] of reduced [[density matrices]] <math>p_{1} \otimes p_{2} \otimes \cdots</math> where each <math>p</math> is diagonal and can only be used as a one time input for a particular channel <math>\epsilon</math>.

The fidelity of the noisy quantum channel is a measure of how closely the output of a quantum channel resembles the input, and is therefore a measure of how well a quantum channel preserves information. If a pure state <math>\psi</math> is sent into a quantum channel emerges as the state represented by density matrix <math>p</math>, the fidelity of transmission is defined as <math>F = \langle\psi|p|\psi\rangle</math>.

The problem that Alice and Bob now face is that quantum communication over large distances depends upon successful distribution of highly entangled [[quantum states]], and due to unavoidable noise in quantum communication channels, the quality of entangled states generally decreases exponentially with channel length as a function of the fidelity of the channel. Entanglement distillation addresses this problem of maintaining a high degree of entanglement between distributed quantum states by transforming N copies of an arbitrary entangled state <math>\rho</math> into approximately <math>S(\rho)N</math> Bell pairs, using only local operations and classical communication. The objective is to share strongly correlated qubits between distant parties (Alice and Bob) in order to allow reliable [[quantum teleportation]] or [[quantum cryptography]].

== Entanglement entropy ==
Entanglement entropy quantifies entanglement. Several different definitions have been proposed.

=== Von Neumann  Entropy ===

Von Neumann entropy is a measure of the "quantum uncertainty" or "quantum randomness" associated with a quantum state, analogous to the concept of [[Shannon entropy]] in classical information theory.<ref>{{Cite journal |last=Horodecki |first=Ryszard |date=2009 |title=Quantum entanglement |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.865 |journal=Reviews of Modern Physics |volume=81 |issue=2 |pages=865–942 |doi=10.1103/RevModPhys.81.865|arxiv=quant-ph/0702225 }}</ref>{{rp|880}} Von Neumann entropy measures how "mixed" or "pure" a quantum state is. Pure states (e.g., states that are entirely definite like 
<math>|\psi\rangle \langle \psi| </math>) have a von Neumann entropy of '''0'''. In pure states, there is no uncertainty about the system’s state. Mixed states  (e.g., probabilistic mixtures of pure states) have a positive entropy value, reflecting an inherent uncertainty in the system's state.

For a given quantum system, the von Neumann entropy <math>S</math> is defined as:<ref>{{Cite journal |last=Jozsa |first=Richard |date=2000 |title=Distinguishability of states and von Neumann entropy |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.62.012301 |journal=Physical Review A |volume=62 |issue=1 |doi=10.1103/PhysRevA.62.012301|arxiv=quant-ph/9911009 }}</ref>

<math>S(\rho)=-\textrm{Tr}~(\rho~log~\rho),</math>

where <math>\rho</math> is the density matrix representing the state of the quantum system and \textrm{Tr} denotes the trace operation, summing over the diagonal elements of a matrix. 

For a maximally mixed state (where all states are equally probable), von Neumann entropy is maximal. Von Neumann entropy is invariant under unitary transformations, meaning that if <math>\rho</math>  is transformed by a unitary matrix <math>U</math>, <math>S(U \rho~U^{\dagger})=S(\rho)</math>. 
It is widely used in quantum information theory to study entanglement, quantum thermodynamics, and the coherence of quantum systems.<ref>{{Cite journal |last=He |first=Huan |date=2018 |title=Entanglement entropy from tensor network states for stabilizer codes |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.125102 |journal=Physical Review B |volume=97 |issue=12 |doi=10.1103/PhysRevB.97.125102|arxiv=1710.04220 }}</ref>

=== Rényi entanglement entropy ===

[[Rényi entropy]] is a generalization of the various concepts of entropy, depending on a parameter <math>\alpha</math>, which adjusts the sensitivity of the entropy measure to different probabilities.<ref>{{Cite journal |last=Rakovszky |first=Tibor |date=2019 |title=Sub-ballistic Growth of Rényi Entropies due to Diffusion |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.250602 |journal=Physical Review Letters |volume=122 |issue=25 |doi=10.1103/PhysRevLett.122.250602|arxiv=1901.10502 }}</ref>

For a quantum state represented by a density matrix <math>\rho</math>, the Rényi entropy of order <math>\alpha</math> is defined as:<ref>{{Cite journal |last=Misra |first=Avijit |date=2015 |title=Quantum Rényi relative entropies affirm universality of thermodynamics |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.042161 |journal=Physical Review E |volume=92 |issue=4 |doi=10.1103/PhysRevE.92.042161|arxiv=1505.06980 }}</ref>

<math>S_{\alpha}(\rho)=\frac{1}{1-\alpha}log~Tr 
 ~(\rho^{\alpha})</math>

where <math>Tr~(\rho^{\alpha})</math> is the trace of <math>\rho</math> raised to the power <math>\alpha</math>.

Rényi entropy <math>H_{\alpha}(P)</math> is a non-increasing function of <math>\alpha</math>, meaning that higher values of 
<math>\alpha</math> emphasize the more probable outcomes more heavily, leading to a lower entropy value. Different values of <math>\alpha</math> allow Rényi entropy to highlight different aspects of the probability distribution (or quantum state), with higher <math>\alpha</math> emphasizing high-probability events. Rényi entropy is often used in contexts such as fractal dimensions, signal processing, and statistical mechanics, where a flexible measure of uncertainty or diversity is useful.<ref>{{Cite journal |last=Gour |first=Gilad |last2=Wilde |first2=Mark M. |date=2021-05-05 |title=Entropy of a quantum channel |url=https://link.aps.org/doi/10.1103/PhysRevResearch.3.023096 |journal=Physical Review Research |language=en |volume=3 |issue=2 |doi=10.1103/PhysRevResearch.3.023096 |issn=2643-1564|arxiv=1808.06980 }}</ref>

As an example of Renyi entropy, a two [[qubit]] system can be written as a superposition of possible computational basis qubit states: <math>|00\rangle, |01\rangle, |10\rangle, |11\rangle</math>, each with an associated complex coefficient <math>\alpha\,\!</math>:
<math display="block">|\psi\rangle =  \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle</math>

As in the case of a single qubit, the probability of measuring a particular computational basis state <math>|x\rangle</math> is the square of the modulus of  its amplitude, or associated coefficient, <math>|\alpha_{x}|^{2}\,\!</math>, subject to the normalization condition <math display="inline">\sum_{x \in {0,1}} |\alpha_{x}|^{2} = 1</math>.  The normalization condition guarantees that the sum of the probabilities add up to 1, meaning that upon measurement, one of the states will be observed. 

The Bell state is a particularly important example of a two qubit state: <math display="inline">\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)</math>

Bell states possess the property that measurement outcomes on the two qubits are correlated. As can be seen from the expression above, the two possible measurement outcomes are zero and one, both with probability of 50%. As a result, a measurement of the second qubit always gives the same result as the measurement of the first qubit.

Bell states can be used to quantify entanglement. Let ''m'' be the number of high-fidelity copies of a Bell state that can be produced using  local operations and classical communication ([[LOCC]]). Given a large number of Bell states the amount of entanglement present in a pure state <math>|\psi\rangle</math> can then be defined as the ratio of <math>n/m</math>, where <math>n</math> is the number of states transform to Bell state, called the distillable entanglement of a particular state <math>|\phi\rangle</math>, which gives a quantified measure of the amount of entanglement present in a given system. The process of entanglement distillation aims to saturate this limiting ratio. The number of copies of a pure state that may be converted into a maximally entangled state is equal to the von Neumann entropy <math>S(p)</math> of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix <math>p</math>, the von Neumann entropy <math>S(p)</math> is <math>S(p) = -\mathrm{Tr}(p \ln p)</math>. Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either <math>p_{A}</math> or <math>p_{B}</math> as:
<math display="block">E = -\mathrm{Tr}(p_{A} \ln p_{A}) = -\mathrm{Tr}(p_{B} \ln p_{B}),</math>

Which ranges from 0 for a product state to <math> \ln 2 </math>  for a maximally entangled state  (if the <math> \ln </math> is replaced by <math> \log_2 </math> then maximally entangled has a value of 1).

==Entanglement concentration==

===Pure states===
[[Image:Fidelitypure.png|thumb|The new fidelity after one iteration of the distillation protocol for pure states.]]

Given n particles in the [[singlet state]] shared between Alice and Bob, local actions and classical communication will suffice to prepare m arbitrarily good copies of <math>\phi</math> with a yield
{{block indent | em = 1.5 | text = <math>\frac{m}{n} \to \frac{1}{E(\phi)}</math> as <math>n \to \infty</math>.}}

Let an entangled state <math>|\psi\rangle</math> have a [[Schmidt decomposition]]:
<math display="block">|\psi\rangle = \sum_{x}\sqrt{p(x)}|x_{A}\rangle|x_{B}\rangle</math>
where coefficients p(x) form a [[probability distribution]], and thus are positive valued and sum to [[Unity (mathematics)|unity]]. The tensor product of this state is then,
<math display="block">|\psi\rangle^{\otimes m} = \sum_{x_{1},x_{2},\dots,x_{m}}\sqrt{p(x_{1}) p(x_{2}) \dots p(x_{m})}|x_{1A} x_{2A} \dots x_{mA}\rangle | x_{1B} x_{2B} \dots x_{mB}\rangle</math>

Now, omitting all terms <math>x_{1}, \dots, x_{m}</math> which are not part of any sequence which is likely to occur with high probability, known as the [[typical set]]: <math> A_{\epsilon}^{(n)} </math> the new state is
<math display="block">|\phi_{m}\rangle = \sum_{x \epsilon A_{\epsilon}^{(n)}} \sqrt{p(x_{1}) p(x_{2}) \dots p(x_{m})} |x_{1A} x_{2A} \dots x_{mA} \rangle | x_{1B} x_{2B} \dots x_{mB} \rangle</math>

And renormalizing,
<math display="block">|\phi_{m}'\rangle = \frac{|\phi_{m}\rangle}{\sqrt{\langle\phi_{m}|\phi_{m}\rangle}}</math>

Then the [[Fidelity of quantum states|fidelity]]
{{block indent | em = 1.5 | text = <math>F(|\psi\rangle ^{\otimes m}, |\phi_{m}^{'}\rangle) \to 1</math> as <math>m \to \infty</math>.}}

Suppose that Alice and Bob are in possession of m copies of <math>|\psi\rangle</math>. Alice can perform a measurement onto the typical set <math>A_{\epsilon}^{(n)}</math> subset of <math>p_{\psi}\,\!</math>, converting the state <math>|\psi\rangle^{\otimes m} \rightarrow |\phi_{m}\rangle</math> with high fidelity. The theorem of typical sequences then shows us that <math>1 - \delta</math> is the probability that the given sequence is part of the typical set, and may be made arbitrarily close to 1 for sufficiently large m, and therefore the Schmidt coefficients of the renormalized Bell state <math>|\phi_{m}'\rangle</math> will be at most a factor <math display="inline">{1}/{\sqrt{1-\delta}}</math> larger. Alice and Bob can now obtain a smaller set of n Bell states by performing LOCC on the state <math>|\phi_{m}'\rangle</math> with which they can overcome the noise of a quantum channel to communicate successfully.

===Mixed states===
[[Image:Fidelitymixed.png|thumb|The new fidelity after one iteration of the distillation protocol presented here for mixed states]]

Many techniques have been developed for doing entanglement distillation for mixed states, giving a lower bounds on the value of the distillable entanglement <math>D(p)</math> for specific classes of states <math>p</math>.

One common method involves Alice not using the noisy channel to transmit source states directly but instead  preparing a large number of Bell states, sending half of each Bell pair to Bob. The result from transmission through the noisy channel is to create the mixed entangled state <math>p</math>, so that Alice and Bob end up sharing <math>m</math> copies of <math>p</math>. Alice and Bob then perform entanglement distillation, producing <math>m \cdot D(p)</math> almost perfectly entangled states from the mixed entangled states <math>p</math> by performing local unitary operations and measurements on the shared entangled pairs, coordinating their actions through classical messages, and sacrificing some of the entangled pairs to increase the purity of the remaining ones. Alice can now prepare an <math>m \cdot D(p)</math> qubit state and teleport it to Bob using the <math>m \cdot D(p)</math> Bell pairs which they share with high fidelity. What Alice and Bob have then effectively accomplished is having simulated a noiseless quantum channel using a noisy one, with the aid of local actions and classical communication.

Let <math>M</math> be a general mixed state of two [[spin-1/2]] particles which could have resulted from the transmission of an initially pure singlet state
<math display="block">\psi^{-} = (\uparrow\downarrow-\downarrow\uparrow)/\sqrt{2}</math>
through a noisy channel between Alice and Bob, which will be used to distill some pure entanglement. The fidelity of {{math|''M''}}
<math display="block">F = \langle\psi^{-}|M|\psi^{-}\rangle</math>
is a convenient expression of its purity relative to a perfect singlet. Suppose that M is already a pure state of two particles <math>M = |\phi\rangle\langle\phi|</math> for some <math>\phi</math>. The entanglement for <math>\phi</math>, as already established, is the von Neumann entropy <math>E(\phi) = S(p_{A}) = S(p_{B})</math> where
<math display="block">p_{A} = \operatorname{tr}^{}_{B}(|\phi\rangle\langle\phi|),</math>
and likewise for <math>p_{B}</math>, represent the reduced density matrices for either particle. The following protocol is then used:<ref name="BBPSSW" />

#Performing a random [[bilateral rotation]] on each shared pair, choosing a random [[SU(2)]] rotation independently for each pair and applying it locally to both members of the pair transforms the initial general two-spin mixed state M into a rotationally symmetric mixture of the singlet state <math>\psi^{-}</math> and the three triplet states <math>\psi^{+}</math> and <math>\phi^{\pm}</math>: <math display="block">W_{F} = F \cdot |\psi^{-}\rangle\langle\psi^{-}| + \frac{1-F}{3}|\phi^{+}\rangle\langle\phi^{+}| + \frac{1-F}{3}|\psi^{+}\rangle\langle\psi^{+}| + \frac{1-F}{3}|\phi^{-}\rangle\langle\phi^{-}|</math> The [[Werner state]] <math>W_{F}</math> has the same purity F as the initial mixed state M from which it was derived due to the singlet's invariance under bilateral rotations.
#Each of the two pairs is then acted on by a unilateral rotation, which we can call <math>\sigma_{y}</math>, which has the effect of converting them from mainly <math>\psi^{-}</math> Werner states to mainly <math>\phi^{+}</math> states with a large component <math>F > \frac{1}{2}</math> of <math>\phi^{+}</math> while the components of the other three Bell states are equal.
#The two impure <math>\phi^{+}</math> states are then acted on by a bilateral [[XOR]], and afterwards the target pair is locally measured along the z axis. The unmeasured source pair is kept if the target pair's spins come out parallel as in the case of both inputs being true <math>\phi^{+}</math> states; and it is discarded otherwise.
#If the source pair has not been discarded it is converted back to a predominantly <math>\psi^{-}</math> state by a unilateral <math>\sigma_{y}</math> rotation, and made rotationally symmetric by a random bilateral rotation.

Repeating the outlined protocol above will distill Werner states whose purity may be chosen to be arbitrarily high <math>F_\text{out} < 1</math> from a collection ''M'' of input mixed states of purity <math display="inline">F_\text{in} > \frac{1}{2}</math> but with a yield tending to zero in the limit <math>F_\text{out} \to 1</math>. By performing another bilateral XOR operation, this time on a variable number <math display="inline">k(F) \approx \frac{1}{\sqrt{1-F}}</math> of source pairs, as opposed to 1, into each target pair prior to measuring it, the yield can be made to approach a positive limit as <math>F_\text{out} \to 1</math>. This method can then be combined with others to obtain an even higher yield.

==Distillation Protocols==

===BBPSSW Protocol===

The BBPSSW protocol <ref>
{{cite journal
 | last1 = Bennett
 | first1 = C. H.
 | last2 = Brassard
 | first2 = G.
 | last3 = Popescu
 | first3 = S.
 | last4 = Schumacher
 | first4 = B.
 | last5 = Smolin
 | first5 = J. A.
 | last6 = Wootters
 | first6 = W. K.
 | title = Purification of noisy entanglement and faithful teleportation via noisy channels
 | journal = Physical Review Letters
 | volume = 76
 | issue = 5
 | pages = 722–725
 | year = 1996
 | doi = 10.1103/PhysRevLett.76.722
 | pmid = 10062366
| arxiv = quant-ph/9511027
 }}
</ref> is one of the simplest protocols that uses CNOT (Controlled-NOT) gates and measurements to probabilistically increase the entanglement of Bell states (standard maximally entangled two-qubit states). Here’s a step-by-step example:

'''Setup:'''<br>
#Suppose Alice and Bob share many copies of a noisy Bell state, represented by the density matrix: <math>\rho_{AB} = p \, |\Phi^{+}\rangle \langle \Phi^{+}| + \frac{1 - p}{3} \sum_{i \neq 0} |\Phi_{i} \rangle \langle \Phi_{i}|</math> where <math>|\Phi^{+}\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)</math>, and <math>|\Phi_{i}\rangle</math> are the other Bell states: <math>|\Phi^{-}\rangle = \frac{1}{\sqrt{2}} (|00\rangle - |11\rangle)</math>, <math>|\Psi^{+}\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle)</math>, <math>|\Psi^{-}\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle)</math>.
#The parameter <math>p</math> represents the fidelity of <math>\rho_{AB}</math> with respect to <math>|\Phi^{+}\rangle</math>, and the goal is to increase <math>p</math> closer to 1 through distillation.

'''Protocol Steps:'''

#'''CNOT Operation:''' Alice and Bob each take two qubits, say <math>\rho_{AB}^{(1)}</math> and <math>\rho_{AB}^{(2)}</math>, and apply a CNOT gate between the pairs, with one qubit of each state as the control and the other as the target:<math>U_{\text{CNOT}} |x\rangle |y\rangle = |x\rangle |x \oplus y\rangle</math>, where <math>\oplus</math> is addition modulo 2. This step correlates the two copies.
#'''Measurement and Postselection:''' Alice and Bob each measure the target qubits in the <math>Z</math>-basis (measuring 0 or 1). If both measure same output (i.e., 0 or 1), they keep the control qubits and discard the targets; otherwise, they discard both pairs. This postselection step has a probability of success but increases the fidelity of the remaining entangled pair.

'''Example Calculation:'''

* After one round, if the initial fidelity <math>p</math> of <math>|\Phi^{+}\rangle</math> was 0.6, the protocol can increase it to around 0.8 with some probability. 
* If multiple rounds are performed on the surviving pairs, <math>p</math> can approach 1, producing a near-perfect <math>|\Phi^{+}\rangle</math> state.

===DEJMPS Protocol===

The DEJMPS protocol <ref>
{{cite journal
 | last1 = Deutsch
 | first1 = D.
 | last2 = Ekert
 | first2 = A.
 | last3 = Jozsa
 | first3 = R.
 | last4 = Macchiavello
 | first4 = C.
 | last5 = Popescu
 | first5 = S.
 | last6 = Sanpera
 | first6 = A.
 | title = Quantum privacy amplification and the security of quantum cryptography over noisy channels
 | journal = Physical Review Letters
 | volume = 77
 | issue = 13
 | pages = 2818–2821
 | year = 1996
 | doi = 10.1103/PhysRevLett.77.2818
 | pmid = 10062370
| arxiv = quant-ph/9604039
 }}
</ref> is an optimized version of BBPSSW and works especially well for Bell diagonal states.

'''Setup:''' <br>
Assume the initial state is in the form: <math>\rho_{AB} = p_1 |\Phi^{+}\rangle \langle \Phi^{+}| + p_2 |\Psi^{+}\rangle \langle \Psi^{+}| + p_3 |\Phi^{-}\rangle \langle \Phi^{-}| + (1 - p_1 - p_2 - p_3) |\Psi^{-}\rangle \langle \Psi^{-}|</math> where <math>p_1 + p_2 + p_3 + p_4 = 1</math>, and we assume <math>p_1</math> is the largest coefficient.

'''Protocol Steps:'''

#'''Apply Local Unitaries:''' Alice and Bob apply unitary operations on their qubits to transform the state into a form where <math>p_1</math> can be maximized. This involves bit and phase flips to swap the Bell states without affecting the target state, <math>|\Phi^{+}\rangle</math>.
#'''CNOT Operations:''' Similar to the BBPSSW protocol, Alice and Bob each apply a CNOT operation between their pairs.
#'''Basis Measurement:''' After the CNOT, Alice and Bob measure the target qubits in the Bell basis, postselecting on successful outcomes.

'''Example Calculation:'''<br>
*If the initial fidelity of <math>|\Phi^{+}\rangle</math> is 0.6, a single round of DEJMPS can increase it more effectively than BBPSSW, pushing fidelity closer to 0.9, depending on the values of <math>p_2</math>, <math>p_3</math>, and <math>p_4</math>.

===Filtering protocol===

Filtering protocols <ref>
{{cite journal
 | last1 = Rozpędek
 | first1 = Filip
 | last2 = Schiet
 | first2 = Thomas
 | last3 = Thinh
 | first3 = Le Phuc
 | last4 = Elkouss
 | first4 = David
 | last5 = Doherty
 | first5 = Andrew C.
 | last6 = Wehner
 | first6 = Stephanie
 | title = Optimizing practical entanglement distillation
 | journal = Physical Review A
 | volume = 97
 | issue = 6
 | pages = 062333
 | year = 2018
 | doi = 10.1103/PhysRevA.97.062333
| arxiv = 1803.10111
 }}
</ref> apply local filtering operations to probabilistically enhance entanglement without requiring multiple pairs. This approach is useful when operations are limited, such as in photon-based quantum communication.

'''Protocol Steps:'''

#Consider a noisy entangled state: <math>\rho_{AB} = p |\Phi^{+}\rangle \langle \Phi^{+}| + (1 - p) |00\rangle \langle 00|</math> where <math>p < 1</math>.
#'''Local Filtering Operators:''' Alice and Bob apply filtering operators <math>F_A</math> and <math>F_B</math>: <math>F_A = \begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}, \quad F_B = \begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}</math>.
#'''Normalization and Success Probability:''' After applying the filters, the resulting state is re-normalized: <math>\rho_{AB'} = \frac{(F_A \otimes F_B) \rho_{AB} (F_A \otimes F_B)^{\dagger}}{\text{Tr}((F_A \otimes F_B) \rho_{AB} (F_A \otimes F_B)^{\dagger})}</math>. <br>The probability of successful filtering (success probability) is: <math>P_{\text{success}} = \text{Tr}((F_A \otimes F_B) \rho_{AB} (F_A \otimes F_B)^{\dagger})</math>.
#'''Resulting Fidelity:''' If <math>p = 0.6</math> initially, filtering can increase the fidelity to 0.8 or higher, but it reduces the probability of obtaining this result due to the probabilistic nature of the filter.

==Procrustean method==

The Procrustean method of entanglement concentration can be used for as little as one partly entangled pair, being more efficient than the Schmidt projection method for entangling less than 5 pairs,<ref name="BBSP" /> and requires Alice and Bob to know the bias (<math>\theta</math>) of the n pairs in advance. The method derives its name from [[Procrustes]] because it produces a perfectly entangled state by chopping off the extra probability associated with the larger term in the partial entanglement of the pure states:
<math display="block">\cos\theta \left|\uparrow_{A}\right\rangle \otimes \left|\downarrow_{B}\right\rangle - \sin\theta \left|\downarrow_{A}\right\rangle \otimes \left|\uparrow_{B}\right\rangle</math>

Assuming a collection of particles for which <math>\theta</math> is known as being either less than or greater than <math>\pi / 4</math> the Procrustean method may be carried out by keeping all particles which, when passed through a polarization-dependent absorber, or a polarization-dependent-reflector, which absorb or reflect a fraction <math>\tan^{2}\theta</math> of the more likely outcome, are not absorbed or deflected. Therefore, if Alice possesses particles for which <math>\theta \neq \pi/4</math>, she can separate out particles which are more likely to be measured in the up/down basis, and left with particles in maximally mixed state of spin up and spin down. This treatment corresponds to a [[POVM]] (positive-operator-valued measurement). To obtain a perfectly entangled state of two particles, Alice informs Bob of the result of her generalized measurement while Bob doesn't measure his particle at all but instead discards his if Alice discards hers.

==Stabilizer protocol==

The purpose of an <math>\left[  n,k\right]</math> entanglement distillation protocol is to distill <math>k</math> pure [[Bell state|ebit]]s from <math>n</math> noisy [[Bell state|ebit]]s where <math>0\leq k\leq n</math>. The yield of such a protocol is <math>k/n</math>. Two parties can then use the noiseless [[Bell state|ebit]]s for [[quantum communication]] protocols.

The two parties establish a set of shared noisy [[Bell state|ebit]]s in the following way. The sender Alice first prepares <math>n</math> [[Bell state]]s <math>\left\vert \Phi^{+}\right\rangle ^{\otimes n}</math> locally. She sends the second [[qubit]] of each pair over a noisy [[quantum channel]] to a receiver Bob. Let <math>\left\vert \Phi_{n}^{+}\right\rangle</math> be the state <math>\left\vert \Phi^{+}\right\rangle^{\otimes n}</math> rearranged so that all of Alice's [[qubit]]s are on the left and all of Bob's [[qubit]]s are on the right. The noisy [[quantum channel]] applies a Pauli error in the error set <math>\mathcal{E}\subset\Pi^{n}</math> to the set of <math>n</math> [[qubit]]s sent over the channel. The sender and receiver then share a set of <math>n</math> noisy [[Bell state|ebit]]s of the form <math>\left(  \mathbf{I}\otimes\mathbf{A}\right)  \left\vert \Phi_{n}^{+}\right\rangle</math> where the identity <math>\mathbf{I}</math> acts on Alice's [[qubit]]s and <math>\mathbf{A}</math> is some [[Pauli operator]] in <math>\mathcal{E}</math> acting on Bob's [[qubit]]s.

A one-way stabilizer entanglement distillation protocol uses a [[stabilizer code]] for the distillation procedure. Suppose the stabilizer <math>\mathcal{S}</math> for an <math>\left[  n,k\right]</math> [[quantum error-correcting code]] has generators <math>g_{1},\ldots,g_{n-k}</math>. The distillation procedure begins with Alice [[quantum measurement|measuring]] the <math>n-k</math> generators in <math>\mathcal{S}</math>. Let <math>\left\{  \mathbf{P}_{i}\right\}</math> be the set of the <math>2^{n-k}</math> [[Projection (linear algebra)|projector]]s that project onto the <math>2^{n-k}</math> orthogonal subspaces corresponding to the generators in <math>\mathcal{S}</math>. The [[quantum measurement|measurement]] projects <math>\left\vert \Phi_{n}^{+}\right\rangle</math> randomly onto one of the <math>i</math> subspaces. Each <math>\mathbf{P}_{i}</math> [[Commutativity|commute]]s with the noisy operator <math>\mathbf{A}</math> on Bob's side so that
<math display="block">
\left(  \mathbf{P}_{i}\otimes\mathbf{I}\right)  \left(  \mathbf{I}
\otimes\mathbf{A}\right)  \left\vert \Phi_{n}^{+}\right\rangle =\left(
\mathbf{I}\otimes\mathbf{A}\right)  \left(  \mathbf{P}_{i}\otimes
\mathbf{I}\right)  \left\vert \Phi_{n}^{+}\right\rangle .
</math>

The following important ''Bell-state matrix identity'' holds for an arbitrary matrix <math>\mathbf{M}</math>:
<math display="block">
\left( \mathbf{M}\otimes\mathbf{I}\right)  \left\vert \Phi_{n}^{+}
\right\rangle =\left(  \mathbf{I}\otimes\mathbf{M}^{T}\right)  \left\vert
\Phi_{n}^{+}\right\rangle .
</math>

Then the above expression is equal to the following:
<math display="block">
\left(  \mathbf{I}\otimes\mathbf{A}\right)  \left(  \mathbf{P}_{i}
\otimes\mathbf{I}\right)  \left\vert \Phi_{n}^{+}\right\rangle    =\left(
\mathbf{I}\otimes\mathbf{A}\right)  \left(  \mathbf{P}_{i}^{2}\otimes
\mathbf{I}\right)  \left\vert \Phi_{n}^{+}\right\rangle
  =\left(  \mathbf{I}\otimes\mathbf{A}\right)  \left(  \mathbf{P}_{i}
\otimes\mathbf{P}_{i}^{T}\right)  \left\vert \Phi_{n}^{+}\right\rangle .
</math>
Therefore, each of Alice's projectors <math>\mathbf{P}_{i}</math> projects Bob's [[qubit]]s onto a [[linear subspace|subspace]] <math>\mathbf{P}_{i}^{T}</math> corresponding to Alice's projected subspace <math>\mathbf{P}_{i}</math>. Alice restores her [[qubit]]s to the simultaneous +1-[[eigenspace]] of the generators in <math>\mathcal{S}</math>. She sends her measurement results to Bob. Bob measures the generators in <math>\mathcal{S}</math>. Bob combines his measurements with Alice's to determine a [[syndrome]] for the error. He performs a recovery operation on his [[qubit]]s to reverse the error. He restores his [[qubit]]s <math>\mathcal{S}</math>. Alice and Bob both perform the decoding [[Unitary transformation|unitary]] corresponding to [[Stabilizer code|stabilizer]] <math>\mathcal{S}</math> to convert their <math>k</math> logical [[Bell state|ebit]]s to <math>k</math> physical [[Bell state|ebit]]s.

== Entanglement-assisted stabilizer code ==

Luo and Devetak provided a straightforward extension of the above protocol (Luo and Devetak 2007). Their method converts an [[entanglement-assisted stabilizer code]] into an entanglement-assisted entanglement distillation protocol.

Luo and Devetak form an entanglement distillation protocol that has entanglement assistance from a few noiseless [[Bell state|ebit]]s. The crucial assumption for an entanglement-assisted entanglement distillation protocol is that Alice and Bob possess <math>c</math> noiseless [[Bell state|ebit]]s in addition to their <math>n</math> noisy [[Bell state|ebit]]s. The total state of the noisy and noiseless [[Bell state|ebit]]s is
<math display="block">
\left(\mathbf{I}^{A}\otimes\left(  \mathbf{A\otimes I}\right)  ^{B}\right)\left\vert
\Phi_{n+c}^{+}\right\rangle
</math>
where <math>\mathbf{I}^{A}</math> is the <math>2^{n+c}\times2^{n+c}</math> [[identity matrix]] acting on Alice's [[qubit]]s and the noisy [[Pauli operator]] <math>\left(  \mathbf{A \otimes I}\right)  ^{B}</math> affects Bob's first <math>n</math> [[qubit]]s only. Thus the last <math>c</math> [[Bell state|ebit]]s are noiseless, and Alice and Bob have to correct for errors on the first <math>n</math> [[Bell state|ebit]]s only.

The protocol proceeds exactly as outlined in the previous section. The only difference is that Alice and Bob measure the generators in an [[entanglement-assisted stabilizer code]]. Each generator spans over <math>n+c</math> [[qubit]]s where the last <math>c</math> [[qubit]]s are noiseless.

We comment on the yield of this entanglement-assisted entanglement distillation protocol. An [[entanglement-assisted code]] has <math>n-k</math> generators that each have <math>n+c</math> Pauli entries. These parameters imply that the entanglement distillation protocol produces <math>k+c</math> ebits. But the protocol consumes <math>c</math> initial noiseless [[Bell state|ebit]]s as a catalyst for distillation. Therefore, the yield of this protocol is <math>k/n</math>.

==Entanglement dilution==

The reverse process of entanglement distillation is entanglement dilution, where large copies of the Bell state  are converted into less entangled states using LOCC with high fidelity. The aim of the entanglement dilution process, then, is to saturate the inverse ratio of n to m, defined as the distillable entanglement.

==Applications==

Besides its important application in quantum communication, entanglement purification also plays a crucial role in [[error correction]] for [[quantum computation]], because it can significantly increase the quality of logic operations between different qubits. The role of entanglement distillation is discussed briefly for the following applications.

===Quantum error correction===
{{main article|Quantum error correction}}

Entanglement distillation protocols for mixed states can be used as a type of error-correction for quantum communications channels between two parties Alice and Bob, enabling Alice to reliably send mD(p) qubits of information to Bob, where D(p) is the distillable entanglement of p, the state that results when one half of a Bell pair is sent through the noisy channel <math>\epsilon</math> connecting Alice and Bob.

In some cases, entanglement distillation may work when conventional quantum error-correction techniques fail. Entanglement distillation protocols are known which can produce a non-zero rate of transmission D(p) for channels which do not allow the transmission of quantum information due to the property that entanglement distillation protocols allow classical communication between parties as opposed to conventional  error-correction which prohibits it.

===Quantum cryptography===
{{main article|Quantum cryptography}}

The concept of correlated measurement outcomes and entanglement is central to quantum key exchange, and therefore the ability to successfully perform entanglement distillation to obtain maximally entangled states is essential for quantum cryptography.

If an entangled pair of particles is shared between two parties, anyone intercepting either particle will alter the overall system, allowing their presence (and the amount of information they have gained) to be determined so long as the particles are in a maximally entangled state. Also, in order to share a secret key string, Alice and Bob must perform the techniques of privacy amplification and information reconciliation to distill a shared secret key string. Information reconciliation is error-correction over a public channel which reconciles errors between the correlated random classical bit strings shared by Alice and Bob while limiting the knowledge that a possible eavesdropper Eve can have about the shared keys. After information reconciliation is used to reconcile possible errors between the shared keys that Alice and Bob possess and limit the possible information Eve could have gained, the technique of privacy amplification is used to distill a smaller subset of bits maximizing Eve's uncertainty about the key.

===Quantum teleportation===
{{main article|Quantum teleportation}}

In quantum teleportation, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Quantum teleportation is able to achieve faithful transmission of quantum information by substituting classical communication and prior entanglement for a direct quantum channel. Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally-entangled qubits shared between sender and receiver, and a 2-bit classical message from the sender to the receiver. Quantum teleportation requires a noiseless quantum channel for sharing perfectly entangled particles, and therefore entanglement distillation satisfies this requirement by providing the noiseless quantum channel and maximally entangled qubits.

==See also==
*[[Quantum channel]]
*[[Quantum cryptography]]
*[[Quantum entanglement]]
*[[Quantum state]]
*[[Quantum teleportation]]
*[[LOCC]]
*[[Purification theorem (physics)|Purification theorem]]


==Notes and references==
{{reflist}}
{{refbegin}}
* {{citation|author1-link=Paul Kwiat|first1=Paul G.|last1=Kwiat|first2=Salvador|last2=Barraza-Lopez|first3=André|last3=Stefanov| first4=Nicolas|last4=Gisin| title=Experimental entanglement distillation and 'hidden' non-locality|journal=Nature|volume=409| year=2001|issue=6823|pages=1014–1017|doi=10.1038/35059017|pmid=11234004|bibcode=2001Natur.409.1014K|s2cid=4430054}}
* {{citation|first1=Takashi|last1=Yamamoto|first2=Masato|last2=Koashi|first3=Şahin Kaya|last3=Özdemir| first4=Nobuyuki|last4=Imoto| title=Experimental extraction of an entangled photon pair from two identically decohered pairs| journal=Nature|volume=421|year=2003|issue=6921|pages=343–346|doi=10.1038/nature01358|pmid=12540894| bibcode=2003Natur.421..343Y| s2cid=20824150}}.
* {{citation|first1=Jian-Wei|last1=Pan|first2=Sara|last2=Gasparoni| first3=Rupert|last3=Ursin|first4=Gregor|last4=Weihs| first5=Anton|last5=Zeilinger| title=Experimental entanglement purification of arbitrary unknown states|journal=Nature| volume=423|year=2003|issue=6938| pages=417–422|doi=10.1038/nature01623 |bibcode = 2003Natur.423..417P|pmid=12761543| s2cid=4393391}}.
* {{citation|first1=Jian-Wei|last1=Pan|first2=Christoph|last2=Simon| first3=Časlav|last3=Brunker|first4=Anton|last4=Zeilinger| title=Entanglement purification for quantum communication|journal=Nature|volume=410|year=2001|pages=1067–1070| pmid=11323664|issue=6832|doi=10.1038/35074041|arxiv = quant-ph/0012026 |bibcode = 2001Natur.410.1067P |s2cid=4424450}}.
* {{citation|first1=M.A.|last1=Nielsen|author-link=Nielsen M.A.|first2=I.L.|last2=Chuang|author-link2=Chuang I.L.|title=Quantum Computation and Quantum Information|year=2000|publisher=Cambridge University Press|isbn=0521635039}}
* {{citation|first1=Dirk|last1=Bouwmeester|author-link=Dirk Bouwmeester|first2=Artur|last2=Ekert|author-link2=Artur Ekert| first3=Anton|last3=Zeilinger|author-link3=Anton Zeilinger|title=The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation|year=2000|publisher=Springer|isbn=3540667784}}
* {{citation|first=I.|last=Newton|title=Principia Mathematica|publisher=Cambridge University Press|volume=1|year=1687}}
* {{citation|doi=10.1103/PhysRevA.75.010303|first1=Zhicheng|last1=Luo|first2=Igor|last2=Devetak|title=Efficiently implementable codes for quantum key expansion|journal=Physical Review A|volume=75|issue=1|pages=010303|year=2007|arxiv = quant-ph/0608029 |bibcode = 2007PhRvA..75a0303L |s2cid=119491901}}
* Mark M. Wilde, [https://arxiv.org/abs/1106.1445 "From Classical to Quantum Shannon Theory", arXiv:1106.1445].

{{refend}}

{{Quantum computing}}

[[Category:Quantum information science]]
[[Category:Statistical mechanics]]

{{DEFAULTSORT:Entanglement Distillation}}