In the theory of quantum communication, the entanglement-assisted stabilizer formalism is a method for protecting quantum information with the help of entanglement shared between a sender and receiver before they transmit quantum data over a quantum communication channel. It extends the standard stabilizer formalism by including shared entanglement (Brun et al. 2006). The advantage of entanglement-assisted stabilizer codes is that the sender can exploit the error-correcting properties of an arbitrary set of Pauli operators. The sender's Pauli operators do not necessarily have to form an Abelian subgroup of the Pauli group
Contents
Definition
We review the construction of an entanglement-assisted code (Brun et al. 2006). Suppose that there is a nonabelian subgroup
The decomposition of
We can partition the nonabelian group
The elements of the entanglement subgroup
Entanglement-assisted stabilizer code error correction conditions
The two subgroups
Operation
The operation of an entanglement-assisted code is as follows. The sender performs an encoding unitary on her unprotected qubits, ancilla qubits, and her half of the ebits. The unencoded state is a simultaneous +1-eigenstate of the following Pauli operators:
The Pauli operators to the right of the vertical bars indicate the receiver's half of the shared ebits. The encoding unitary transforms the unencoded Pauli operators to the following encoded Pauli operators:
The sender transmits all of her qubits over the noisy quantum channel. The receiver then possesses the transmitted qubits and his half of the ebits. He measures the above encoded operators to diagnose the error. The last step is to correct for the error.
Rate of an entanglement-assisted code
We can interpret the rate of an entanglement-assisted code in three different ways (Wilde and Brun 2007b). Suppose that an entanglement-assisted quantum code encodes
Which interpretation is most reasonable depends on the context in which we use the code. In any case, the parameters
Example of an entanglement-assisted code
We present an example of an entanglement-assisted code that corrects an arbitrary single-qubit error (Brun et al. 2006). Suppose the sender wants to use the quantum error-correcting properties of the following nonabelian subgroup of
The first two generators anticommute. We obtain a modified third generator by multiplying the third generator by the second. We then multiply the last generator by the first, second, and modified third generators. The error-correcting properties of the generators are invariant under these operations. The modified generators are as follows:
The above set of generators have the commutation relations given by the fundamental theorem of symplectic geometry:
The above set of generators is unitarily equivalent to the following canonical generators:
We can add one ebit to resolve the anticommutativity of the first two generators and obtain the canonical stabilizer:
The receiver Bob possesses the qubit on the left and the sender Alice possesses the four qubits on the right. The following state is an eigenstate of the above stabilizer
where
The receiver measures the above generators upon receipt of all qubits to detect and correct errors.
Encoding algorithm
We continue with the previous example. We detail an algorithm for determining an encoding circuit and the optimal number of ebits for the entanglement-assisted code---this algorithm first appeared in the appendix of (Wilde and Brun 2007a) and later in the appendix of (Shaw et al. 2008). The operators in the above example have the following representation as a binary matrix (See the stabilizer code article):
Call the matrix to the left of the vertical bar the "
The algorithm consists of row and column operations on the above matrix. Row operations do not affect the error-correcting properties of the code but are crucial for arriving at the optimal decomposition from the fundamental theorem of symplectic geometry. The operations available for manipulating columns of the above matrix are Clifford operations. Clifford operations preserve the Pauli group
The algorithm begins by computing the symplectic product between the first row and all other rows. We emphasize that the symplectic product here is the standard symplectic product. Leave the matrix as it is if the first row is not symplectically orthogonal to the second row or if the first row is symplectically orthogonal to all other rows. Otherwise, swap the second row with the first available row that is not symplectically orthogonal to the first row. In our example, the first row is not symplectically orthogonal to the second so we leave all rows as they are.
Arrange the first row so that the top left entry in the
Perform CNOTs to clear the entries in the
We perform the above operations for our example. Perform a Hadamard on qubits two and three. The matrix becomes
Perform a CNOT from qubit one to qubit two and from qubit one to qubit three. The matrix becomes
The first row is complete. We now proceed to clear the entries in the second row. Perform a Hadamard on qubits one and four. The matrix becomes
Perform a CNOT from qubit one to qubit two and from qubit one to qubit four. The matrix becomes
The first two rows are now complete. They need one ebit to compensate for their anticommutativity or their nonorthogonality with respect to the symplectic product.
Now we perform a "Gram-Schmidt orthogonalization" with respect to the symplectic product. Add row one to any other row that has one as the leftmost entry in its
The first two rows are now symplectically orthogonal to all other rows per the fundamental theorem of symplectic geometry. We proceed with the same algorithm on the next two rows. The next two rows are symplectically orthogonal to each other so we can deal with them individually. Perform a Hadamard on qubit two. The matrix becomes
Perform a CNOT from qubit two to qubit three and from qubit two to qubit four. The matrix becomes
Perform a phase gate on qubit two:
Perform a Hadamard on qubit three followed by a CNOT from qubit two to qubit three:
Add row three to row four and perform a Hadamard on qubit two:
Perform a Hadamard on qubit four followed by a CNOT from qubit three to qubit four. End by performing a Hadamard on qubit three:
The above matrix now corresponds to the canonical Pauli operators. Adding one half of an ebit to the receiver's side gives the canonical stabilizer whose simultaneous +1-eigenstate is the above state. The above operations in reverse order take the canonical stabilizer to the encoded stabilizer.