Entanglement distillation

The limits for entanglement dilution and distillation are due to C. H. Bennett, H. Bernstein, S. Popescu, and B. Schumacher,[2] who presented the first distillation protocols for pure states in 1996; entanglement distillation protocols for mixed states were introduced by Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters[3] the same year. 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.

A two qubit system can be written as a superposition of possible computational basis qubit states: ${\displaystyle |00\rangle ,|01\rangle ,|10\rangle ,|11\rangle }$, each with an associated complex coefficient ${\displaystyle \alpha \,\!}$:

$${\displaystyle |\psi \rangle =\alpha _{00}|00\rangle +\alpha _{01}|01\rangle +\alpha _{10}|10\rangle +\alpha _{11}|11\rangle }$$

As in the case of a single qubit, the probability of measuring a particular computational basis state ${\displaystyle |x\rangle }$ is the square of the modulus of its amplitude, or associated coefficient, ${\displaystyle |\alpha _{x}|^{2}\,\!}$, subject to the normalization condition ${\textstyle \sum _{x\in {0,1}}|\alpha _{x}|^{2}=1}$. 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: ${\textstyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )}$

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 ${\displaystyle |\psi \rangle }$ can then be defined as the ratio of ${\displaystyle n/m}$, called the distillable entanglement of a particular state ${\displaystyle |\phi \rangle }$, 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 ${\displaystyle S(p)}$ of the state, which is an extension of the concept of classical entropy for quantum systems. Mathematically, for a given density matrix ${\displaystyle p}$, the von Neumann entropy ${\displaystyle S(p)}$ is ${\displaystyle S(p)=-\mathrm {Tr} (p\ln p)}$. Entanglement can then be quantified as the entropy of entanglement, which is the von Neumann entropy of either ${\displaystyle p_{A}}$ or ${\displaystyle p_{B}}$ as:

$${\displaystyle E=-\mathrm {Tr} (p_{A}\ln p_{A})=-\mathrm {Tr} (p_{B}\ln p_{B}),}$$

Which ranges from 0 for a product state to ${\displaystyle \ln 2}$ for a maximally entangled state (if the ${\displaystyle \ln }$ is replaced by ${\displaystyle \log _{2}}$ then maximally entangled has a value of 1).

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 ${\displaystyle p_{1}\otimes p_{2}\otimes \cdots }$ where each ${\displaystyle p}$ is diagonal and can only be used as a one time input for a particular channel ${\displaystyle \epsilon }$.

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 ${\displaystyle \psi }$ is sent into a quantum channel emerges as the state represented by density matrix ${\displaystyle p}$, the fidelity of transmission is defined as ${\displaystyle F=\langle \psi |p|\psi \rangle }$.

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 ${\displaystyle \rho }$ into approximately ${\displaystyle S(\rho )N}$ 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.

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,[2] and requires Alice and Bob to know the bias (${\displaystyle \theta }$) 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:

$${\displaystyle \cos \theta \left|\uparrow _{A}\right\rangle \otimes \left|\downarrow _{B}\right\rangle -\sin \theta \left|\downarrow _{A}\right\rangle \otimes \left|\uparrow _{B}\right\rangle }$$

Assuming a collection of particles for which ${\displaystyle \theta }$ is known as being either less than or greater than ${\displaystyle \pi /4}$ 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 ${\displaystyle \tan ^{2}\theta }$ of the more likely outcome, are not absorbed or deflected. Therefore, if Alice possesses particles for which ${\displaystyle \theta \neq \pi /4}$, 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.

The purpose of an ${\displaystyle \left[n,k\right]}$ entanglement distillation protocol is to distill ${\displaystyle k}$ pure ebits from ${\displaystyle n}$ noisy ebits where ${\displaystyle 0\leq k\leq n}$. The yield of such a protocol is ${\displaystyle k/n}$. Two parties can then use the noiseless ebits for quantum communication protocols.

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

A one-way stabilizer entanglement distillation protocol uses a stabilizer code for the distillation procedure. Suppose the stabilizer ${\displaystyle {\mathcal {S}}}$ for an ${\displaystyle \left[n,k\right]}$ quantum error-correcting code has generators ${\displaystyle g_{1},\ldots ,g_{n-k}}$. The distillation procedure begins with Alice measuring the ${\displaystyle n-k}$ generators in ${\displaystyle {\mathcal {S}}}$. Let ${\displaystyle \left\{\mathbf {P} _{i}\right\}}$ be the set of the ${\displaystyle 2^{n-k}}$ projectors that project onto the ${\displaystyle 2^{n-k}}$ orthogonal subspaces corresponding to the generators in ${\displaystyle {\mathcal {S}}}$. The measurement projects ${\displaystyle \left\vert \Phi _{n}^{+}\right\rangle }$ randomly onto one of the ${\displaystyle i}$ subspaces. Each ${\displaystyle \mathbf {P} _{i}}$ commutes with the noisy operator ${\displaystyle \mathbf {A} }$ on Bob's side so that

$${\displaystyle \left(\mathbf {P} _{i}\otimes \mathbf {I} \right)\left(\mathbf {I} \otimes \mathbf {A} \right)\left\vert \Phi _{n}^{+}\right\rangle =\left(\mathbf {I} \otimes \mathbf {A} \right)\left(\mathbf {P} _{i}\otimes \mathbf {I} \right)\left\vert \Phi _{n}^{+}\right\rangle .}$$

The following important Bell-state matrix identity holds for an arbitrary matrix ${\displaystyle \mathbf {M} }$:

$${\displaystyle \left(\mathbf {M} \otimes \mathbf {I} \right)\left\vert \Phi _{n}^{+}\right\rangle =\left(\mathbf {I} \otimes \mathbf {M} ^{T}\right)\left\vert \Phi _{n}^{+}\right\rangle .}$$

Then the above expression is equal to the following:

$${\displaystyle \left(\mathbf {I} \otimes \mathbf {A} \right)\left(\mathbf {P} _{i}\otimes \mathbf {I} \right)\left\vert \Phi _{n}^{+}\right\rangle =\left(\mathbf {I} \otimes \mathbf {A} \right)\left(\mathbf {P} _{i}^{2}\otimes \mathbf {I} \right)\left\vert \Phi _{n}^{+}\right\rangle =\left(\mathbf {I} \otimes \mathbf {A} \right)\left(\mathbf {P} _{i}\otimes \mathbf {P} _{i}^{T}\right)\left\vert \Phi _{n}^{+}\right\rangle .}$$

Therefore, each of Alice's projectors ${\displaystyle \mathbf {P} _{i}}$ projects Bob's qubits onto a subspace ${\displaystyle \mathbf {P} _{i}^{T}}$ corresponding to Alice's projected subspace ${\displaystyle \mathbf {P} _{i}}$. Alice restores her qubits to the simultaneous +1-eigenspace of the generators in ${\displaystyle {\mathcal {S}}}$. She sends her measurement results to Bob. Bob measures the generators in ${\displaystyle {\mathcal {S}}}$. Bob combines his measurements with Alice's to determine a syndrome for the error. He performs a recovery operation on his qubits to reverse the error. He restores his qubits ${\displaystyle {\mathcal {S}}}$. Alice and Bob both perform the decoding unitary corresponding to stabilizer ${\displaystyle {\mathcal {S}}}$ to convert their ${\displaystyle k}$ logical ebits to ${\displaystyle k}$ physical ebits.

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 ebits. The crucial assumption for an entanglement-assisted entanglement distillation protocol is that Alice and Bob possess ${\displaystyle c}$ noiseless ebits in addition to their ${\displaystyle n}$ noisy ebits. The total state of the noisy and noiseless ebits is

$${\displaystyle \left(\mathbf {I} ^{A}\otimes \left(\mathbf {A\otimes I} \right)^{B}\right)\left\vert \Phi _{n+c}^{+}\right\rangle }$$

where ${\displaystyle \mathbf {I} ^{A}}$ is the ${\displaystyle 2^{n+c}\times 2^{n+c}}$ identity matrix acting on Alice's qubits and the noisy Pauli operator ${\displaystyle \left(\mathbf {A\otimes I} \right)^{B}}$ affects Bob's first ${\displaystyle n}$ qubits only. Thus the last ${\displaystyle c}$ ebits are noiseless, and Alice and Bob have to correct for errors on the first ${\displaystyle n}$ ebits 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 ${\displaystyle n+c}$ qubits where the last ${\displaystyle c}$ qubits are noiseless.

We comment on the yield of this entanglement-assisted entanglement distillation protocol. An entanglement-assisted code has ${\displaystyle n-k}$ generators that each have ${\displaystyle n+c}$ Pauli entries. These parameters imply that the entanglement distillation protocol produces ${\displaystyle k+c}$ ebits. But the protocol consumes ${\displaystyle c}$ initial noiseless ebits as a catalyst for distillation. Therefore, the yield of this protocol is ${\displaystyle k/n}$.

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.

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 channel
• Quantum cryptography
• Quantum entanglement
• Quantum state
• Quantum teleportation
• LOCC
• Purification theorem