Entanglement distillation (also called ''entanglement purification'') is the transformation of ''N'' copies of an arbitrary
entangled state
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
into some number of approximately pure
Bell pairs, using only
local operations and classical communication.
Quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
distillation can in this way overcome the degenerative influence of noisy
quantum channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information i ...
s
by transforming previously shared less entangled pairs into a smaller number of
maximally entangled pairs.
History
The limits for entanglement dilution and distillation are due to
C. H. Bennett, H. Bernstein,
S. Popescu, and
B. Schumacher,
who presented the first distillation protocols for
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s in 1996; entanglement distillation protocols for
mixed states were introduced by Bennett,
Brassard
A brassard or armlet is an armband or piece of cloth or other material worn around the upper arm; the term typically refers to an item of uniform worn as part of military uniform or by police or other uniformed persons. Unit, role, rank b ...
, Popescu, Schumacher,
Smolin and
Wootters the same year. Bennett,
DiVincenzo, Smolin and Wootters
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.
Quantifying entanglement
A two
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
system can be written as a superposition of possible computational basis qubit states:
, each with an associated complex coefficient
:
As in the case of a single qubit, the probability of measuring a particular computational basis state
is the square of the modulus of its amplitude, or associated coefficient,
, subject to the normalization condition
. 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:
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 LOCC. Given a large number of Bell states the amount of entanglement present in a pure state
can then be defined as the ratio of
, called the distillable entanglement of a particular state
, 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
of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix
, the von Neumann entropy
is
. Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either
or
as:
Which ranges from 0 for a product state to
for a maximally entangled state (if the
is replaced by
then maximally entangled has a value of 1).
Motivation
Suppose that two parties,
Alice and Bob
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptographic systems and protocols, and in other science and engineering literature where there are several participants in a thought experiment. The A ...
, 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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of reduced
density matrices where each
is diagonal and can only be used as a one time input for a particular channel
.
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
is sent into a quantum channel emerges as the state represented by density matrix
, the fidelity of transmission is defined as
.
The problem that Alice and Bob now face is that quantum communication over large distances depends upon successful distribution of highly entangled
quantum states
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
, 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
into approximately
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
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
.
Entanglement concentration
Pure states
Given n particles in the
singlet state
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s=0. A ...
shared between Alice and Bob, local actions and classical communication will suffice to prepare m arbitrarily good copies of
with a yield
Let an entangled state
have a
Schmidt decomposition
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information the ...
:
where coefficients p(x) form a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, and thus are positive valued and sum to
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a h ...
. The tensor product of this state is then,
Now, omitting all terms
which are not part of any sequence which is likely to occur with high probability, known as the
typical set In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asympt ...
:
the new state is
And renormalizing,
Then the
fidelity
Fidelity is the quality of faithfulness or loyalty. Its original meaning regarded duty in a broader sense than the related concept of ''fealty''. Both derive from the Latin word ''fidēlis'', meaning "faithful or loyal". In the City of London f ...
Suppose that Alice and Bob are in possession of m copies of
. Alice can perform a measurement onto the typical set
subset of
, converting the state
with high fidelity. The theorem of typical sequences then shows us that
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
will be at most a factor
larger. Alice and Bob can now obtain a smaller set of n Bell states by performing LOCC on the state
with which they can overcome the noise of a quantum channel to communicate successfully.
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
for specific classes of states
.
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
, so that Alice and Bob end up sharing
copies of
. Alice and Bob then perform entanglement distillation, producing
almost perfectly entangled states from the mixed entangled states
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
qubit state and teleport it to Bob using the
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
be a general mixed state of two
spin-1/2
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
particles which could have resulted from the transmission of an initially pure singlet state
through a noisy channel between Alice and Bob, which will be used to distill some pure entanglement. The fidelity of
is a convenient expression of its purity relative to a perfect singlet. Suppose that M is already a pure state of two particles
for some
. The entanglement for
, as already established, is the von Neumann entropy
where
and likewise for
, represent the reduced density matrices for either particle. The following protocol is then used:
#Performing a random
bilateral rotation on each shared pair, choosing a random
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
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
and the three triplet states
and
:
The
Werner state A Werner state is a -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U \otimes U. That is, it is a bipartite quantum state \rho_ that satisfies
:\rho_ = (U \otimes U) \rho_ (U^\dagger \ot ...
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
, which has the effect of converting them from mainly
Werner states to mainly
states with a large component
of
while the components of the other three Bell states are equal.
#The two impure
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
states; and it is discarded otherwise.
#If the source pair has not been discarded it is converted back to a predominantly
state by a unilateral
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
from a collection ''M'' of input mixed states of purity
but with a yield tending to zero in the limit
. By performing another bilateral XOR operation, this time on a variable number
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
. This method can then be combined with others to obtain an even higher yield.
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,
and requires Alice and Bob to know the bias (
) of the n pairs in advance. The method derives its name from
Procrustes
In Greek mythology, Procrustes (; Greek: Προκρούστης ''Prokroustes'', "the stretcher ho hammers out the metal), also known as Prokoptas, Damastes (Δαμαστής, "subduer") or Polypemon, was a rogue smith and bandit from Attica ...
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:
Assuming a collection of particles for which
is known as being either less than or greater than
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
of the more likely outcome, are not absorbed or deflected. Therefore, if Alice possesses particles for which
, 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
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and ...
(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