Entanglement distillation - AbsoluteAstronomy.com

Entanglement distillation is the transformation of N copies of an arbitrary entangled state

into some number of approximately pure Bell pairs, using only local operations and classical communication (LOCC

LOCC

LOCC, or Local Operations and Classical Communication, is a method in quantum information theory where a local operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed...

). Entanglement

Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

distillation can in this way overcome the degenerative influence of noisy quantum channel

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...

s by transforming previously shared less entangled pairs into a smaller number of maximally entangled pairs (Bell state

Bell state

The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. An EPR pair is a pair of qubits which jointly are in a Bell state, that is, entangled...

s).

History

The limits for entanglement dilution and distillation are due to Bennett, Bernstein, Popescu and Schumacher. Entanglement distillation protocols for pure states were originally presented in a paper by C. H. Bennett

Charles H. Bennett (computer scientist)

Charles H. Bennett is an IBM Fellow at IBM Research. Bennett's recent work at IBM has concentrated on a re-examination of the physical basis of information, applying quantum physics to the problems surrounding information exchange...

, H. Bernstein, S. Popescu, and B. Schumacher while Entanglement distillation protocols for mixed states were introduced by Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters in a paper later that year in the same journal. 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

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

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 amplitude of its associated coefficient

, subject to the normalization condition

.

The Bell state is a particularly important example of a two qubit state:

Bell states possess the amazing property that measurement outcomes of a Bell state 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 S(p) of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix p, the von Neumann entropy S(p) is

. Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either

as:

Which ranges from 0 for a product state to 1 for a maximally entangled state.

Motivation

Suppose that two parties, Alice and Bob

Alice and Bob

The names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party...

, 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

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

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, 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

Quantum teleportation

Quantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...

or quantum cryptography

Quantum cryptography

Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...

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

with a yield

approaching

.

Let an entangled state

have a Schmidt decomposition

Schmidt decomposition

In linear algebra, the Schmidt decomposition refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has applications in quantum information theory and plasticity....

where coefficients p(x) form a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

, and thus are positive valued and sum to unity. 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

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 asymptotic equipartition property which is a kind of law...

the new state is

And renormalizing,

Then the fidelity

Fidelity of quantum states

In quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.- Motivation :...

.

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 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 M

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) rotation independently for each pair and applying it locally to both members of the pair transforms the initial general two-spin matrix state M into a rotationally symmetric mixture of the singlet state and the three triplet states and :

The Werner state
Werner state
A Werner state is a -dimensional bipartite quantum state that is invariant under all unitary operators of the form U \otimes U. That is, it is a quantum state ρ that satisfies\rho = \rho...

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 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

Procrustes

In Greek mythology Procrustes or "the stretcher [who hammers out the metal]", also known as Prokoptas or Damastes "subduer", was a rogue smith and bandit from Attica who physically attacked people by stretching them or cutting off their legs, so as to force them to fit the size of an iron bed...

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

POVM

In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...

(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.

Entanglement Distillation with a Stabilizer Code

The purpose of an

entanglement distillation protocol is to distill

pure ebit

Bell state

s from

noisy ebit

Bell state

s where

.
The yield of such a protocol is

. Two parties can then use the noiseless
ebit

Bell state

s for quantum communication protocols.

The two parties establish a set of shared noisy ebit

Bell state

s in the following way.
The sender Alice first prepares

Bell state

locally. She sends the second qubit

Qubit

of each
pair over a noisy quantum channel

Quantum channel

to a receiver Bob. Let

be the state

rearranged so that all of Alice's qubit

Qubit

s are on the left and all
of Bob's qubit

Qubit

s are on the right. The noisy quantum channel

Quantum channel

applies a Pauli error in
the error set

to the set of

qubit

Qubit

s sent over
the channel. The sender and receiver then share a set of

noisy ebit

Bell state

s of
the form

where the identity

acts on Alice's qubit

Qubit

s and

is some Pauli operator in

acting on Bob's qubit

Qubit

s.

A one-way stabilizer entanglement distillation protocol uses a stabilizer code

Stabilizer code

The theory of quantum error correction plays a prominent role in the practical realization and engineering ofquantum computing and quantum communication devices. The first quantumerror-correcting codes are strikingly similar to classical block codes in their...

for the distillation procedure. Suppose the stabilizer

for an

quantum error-correcting code has generators

. The
distillation procedure begins with Alice measuring the

generators in

. Let

be the set of the

projector

Projector

Projector may refer to:*Image projector, a device that projects an image on a surface** Video projector, a device that projects a video signal from computer, home theater system etc.** Movie projector, a device that projects moving pictures from a filmstrip...

s that project onto the

orthogonal subspaces
corresponding to the generators in

. The measurement projects

randomly onto one of the

subspaces. Each

commute

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

s with the noisy operator

on Bob's side so that

The following important Bell-state matrix identity holds for an arbitrary matrix

Then the above expression is equal to the following:

Therefore each of Alice's projectors

projects Bob's qubit

Qubit

s
onto a subspace

Subspace

-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...

corresponding to Alice's projected
subspace

. Alice restores her qubit

Qubit

s to the simultaneous
+1-eigenspace of the generators in

. She sends her measurement
results to Bob. Bob measures the generators in

. Bob combines his
measurements with Alice's to determine a syndrome

Syndrome

In medicine and psychology, a syndrome is the association of several clinically recognizable features, signs , symptoms , phenomena or characteristics that often occur together, so that the presence of one or more features alerts the physician to the possible presence of the others...

for the error. He performs a
recovery operation on his qubit

Qubit

s to reverse the error. He restores his qubit

Qubit

. Alice
and Bob both perform the decoding unitary

Unitary

Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...

corresponding to stabilizer

Stabilizer code

to convert their

logical ebit

Bell state

s to

physical ebit

Bell state

Entanglement Distillation with an 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 ebit

Bell state

s. The crucial assumption for
an entanglement-assisted entanglement distillation protocol is that Alice and
Bob possess

noiseless ebit

Bell state

s in addition to their

noisy ebit

Bell state

s. The
total state of the noisy and noiseless ebit

Bell state

s is

where

is the

identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

acting on
Alice's qubit

Qubit

s and the noisy Pauli operator

affects Bob's first

qubit

Qubit

s only. Thus the last

ebit

Bell state

s
are noiseless, and Alice and Bob have to correct for errors on the first

ebit

Bell state

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

qubit

Qubit

s
where the last

qubit

Qubit

s are noiseless.

We comment on the yield of this entanglement-assisted entanglement
distillation protocol. An entanglement-assisted code has

generators that
each have

Pauli entries. These parameters imply that the entanglement
distillation protocol produces

ebits. But the protocol consumes

initial noiseless ebit

Bell state

s as a catalyst for distillation. Therefore the yield of
this protocol is

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

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

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

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

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.

History

Quantifying Entanglement

Motivation

Pure States

Mixed States

Procrustean Method

Entanglement Distillation with a Stabilizer Code

Entanglement Distillation with an Entanglement-Assisted Stabilizer Code

Entanglement Dilution

Applications

Quantum Error Correction

Quantum Cryptography

Quantum Teleportation

See also