Entanglement distillation
into some number of approximately pure Bell pairs, using only local operations and classical communication.
Bennett, DiVincenzo, Smolin and Wootters[1] 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.
[citation needed] Suppose that two parties, Alice and Bob, would like to communicate classical information over a noisy quantum channel.
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.
[4]: 880 Von Neumann entropy measures how "mixed" or "pure" a quantum state is.
emphasize the more probable outcomes more heavily, leading to a lower entropy value.
allow Rényi entropy to highlight different aspects of the probability distribution (or quantum state), with higher
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.
As in the case of a single qubit, the probability of measuring a particular computational basis state
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.
Bell states possess the property that measurement outcomes on the two qubits are correlated.
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 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
where coefficients p(x) form a probability distribution, and thus are positive valued and sum to unity.
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
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
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.
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.
She sends the second qubit of each pair over a noisy quantum channel to a receiver Bob.
The crucial assumption for an entanglement-assisted entanglement distillation protocol is that Alice and Bob possess
identity matrix acting on Alice's qubits and the noisy Pauli operator
The only difference is that Alice and Bob measure the generators in an entanglement-assisted stabilizer code.
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.
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
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.
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.
