Entanglement distillation
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. Entanglement distillation can overcome the degenerative influence of noisy quantum channels 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 states in 1996; entanglement distillation protocols for mixed states were introduced by Bennett, Gilles Brassard, Popescu, Schumacher, John A. Smolin and William Wootters the same year. Bennett, David 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.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 product state, as a tensor product 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 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
The 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. Von Neumann entropy measures how "mixed" or "pure" a quantum state is. Pure states have a von Neumann entropy of 0. In pure states, there is no uncertainty about the system's state. Mixed states have a positive entropy value, reflecting an inherent uncertainty in the system's state.For a given quantum system, the von Neumann entropy is defined as:
where is the density matrix representing the state of the quantum system and \textrm denotes the trace operation, summing over the diagonal elements of a matrix.
For a maximally mixed state, von Neumann entropy is maximal. Von Neumann entropy is invariant under unitary transformations, meaning that if is transformed by a unitary matrix,.
It is widely used in quantum information theory to study entanglement, quantum thermodynamics, and the coherence of quantum systems.
Rényi entanglement entropy
is a generalization of the various concepts of entropy, depending on a parameter, which adjusts the sensitivity of the entropy measure to different probabilities.For a quantum state represented by a density matrix, the Rényi entropy of order is defined as:
where is the trace of raised to the power.
Rényi entropy is a non-increasing function of, meaning that higher values of
emphasize the more probable outcomes more heavily, leading to a lower entropy value. Different values of allow Rényi entropy to highlight different aspects of the probability distribution, with higher 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.
As an example of Renyi entropy, a two qubit 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 local operations and classical communication. Given a large number of Bell states the amount of entanglement present in a pure state can then be defined as the ratio of, where is the number of states transform to Bell state, 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 .
Entanglement concentration
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 yieldLet an entangled state have a Schmidt decomposition:
where coefficients p form a probability distribution, 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: the new state is
And renormalizing,
Then the fidelity
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
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 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 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.