Entanglement-assisted stabilizer formalism
Encyclopedia
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
over 
qubit
s.
The sender can make clever use of her shared
ebit
s so that the global stabilizer is abelian
and thus forms a valid
quantum error-correcting code.
there is a nonabelian subgroup
of size 
.
Application of the fundamental theorem of symplectic geometry (Lemma 1 in the first external reference)
states that there exists a minimal set of independent generators
for
with the following commutation
relations:
The decomposition of
into the above minimal generating set
determines that the code requires
ancilla qubits and 
ebit
s. The code
requires an ebit
for every anticommuting pair in the minimal generating set.
The simple reason for this requirement is that an ebit
is a simultaneous
-eigenstate of the Pauli operators 
. The second qubit
in the ebit
transforms the anticommuting pair
into a
commuting
pair
. The above decomposition also
minimizes the number of ebit
s required for the code---it is an optimal decomposition.
We can partition the nonabelian group
into two subgroup
s: the
isotropic subgroup
and the entanglement subgroup
. The isotropic subgroup 
is a commuting
subgroup of
and thus corresponds to ancilla
qubits:
.
The elements of the entanglement subgroup
come in
anticommuting pairs and thus correspond to ebit
s:
.
and 
play a role in the
error-correcting conditions for the entanglement-assisted stabilizer
formalism. An entanglement-assisted code corrects errors in a set
if for all 
,
performs an encoding unitary on her unprotected qubits, ancilla qubits, and
her half of the ebit
s. 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 ebit
s. The encoding unitary transforms the unencoded Pauli operators
to the following encoded Pauli operators:
The sender transmits all of her qubit
s over the noisy quantum channel
. The
receiver then possesses the transmitted qubits and his half of the ebit
s. He
measures the above encoded operators to diagnose the error. The last step is
to correct for the error.
in three different ways (Wilde and Brun 2007b).
Suppose that an entanglement-assisted quantum code encodes
information
qubits into
physical qubits with the help of 
ebits.
Which interpretation is most reasonable depends on the context in which we use
the code. In any case, the parameters
, 
, and 
ultimately govern
performance, regardless of which definition of the rate we use to interpret
that performance.
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
is a qubit that the sender wants to
encode. The encoding unitary then rotates the canonical stabilizer to the following set of globally commuting
generators:
The receiver measures the above generators upon receipt of all qubits to
detect and correct errors.
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 "
matrix" and the matrix to the right of the vertical bar the
"
matrix."
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
under conjugation. The
CNOT gate, the Hadamard gate, and the Phase gate generate the Clifford group.
A CNOT gate from qubit
to qubit 
adds column 
to column 
in the
matrix and adds column 
to column 
in the 
matrix. A Hadamard
gate on qubit
swaps column 
in the 
matrix with column 
in the
matrix and vice versa. A phase gate on qubit 
adds column 
in the
matrix to column 
in the 
matrix. Three CNOT gates implement a
qubit swap operation. The effect of a swap on qubits
and 
is to swap columns 
and 
in both the 
and 
matrix.
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
matrix is one. A
CNOT, swap, Hadamard, or combinations of these operations can achieve this
result. We can have this result in our example by swapping qubits one and two.
The matrix becomes
Perform CNOTs to clear the entries in the
matrix in the top row to the
right of the leftmost entry. These entries are already zero in this example so
we need not do anything. Proceed to the clear the entries in the first row of
the
matrix. Perform a phase gate to clear the leftmost entry in the first
row of the
matrix if it is equal to one. It is equal to zero in this case
so we need not do anything. We then use Hadamards and CNOTs to clear the other
entries in the first row of the
matrix.
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
matrix. Add row two to any other row that has one as the leftmost entry in its
matrix. For our example, we add row one to row four and we add row two to
rows three and four. The matrix becomes
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.
by including shared 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...
(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
Abelian
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of the Pauli group
over qubitQubit
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....
s.
The sender can make clever use of her shared
ebit
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...
s so that the global stabilizer is abelian
Abelian
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...
and thus forms a valid
quantum error-correcting code.
Definition
We review the construction of an entanglement-assisted code (Brun et al. 2006). Suppose thatthere is a nonabelian subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of size .Application of the fundamental theorem of symplectic geometry (Lemma 1 in the first external reference)
states that there exists a minimal set of independent generators
for
with the following commutationCommutativity
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...
relations:
The decomposition of
into the above minimal generating setdetermines that the code requires
ancilla qubits and ebitBell 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. The code
requires an ebit
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...
for every anticommuting pair in the minimal generating set.
The simple reason for this requirement is that an ebit
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...
is a simultaneous
-eigenstate of the Pauli operators . The second qubitQubit
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....
in the ebit
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...
transforms the anticommuting pair
into acommuting
Commuting
Commuting is regular travel between one's place of residence and place of work or full time study. It sometimes refers to any regular or often repeated traveling between locations when not work related.- History :...
pair
. The above decomposition alsominimizes the number of ebit
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 required for the code---it is an optimal decomposition.
We can partition the nonabelian group
Nonabelian group
In mathematics, a non-abelian group, also sometimes called a non-commutative group, is a group in which there are at least two elements a and b of G such that a * b ≠ b * a...
into two subgroupSubgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s: the
isotropic subgroup
and the entanglement subgroup. The isotropic subgroup is a commutingsubgroup of
and thus corresponds to ancillaqubits:
.The elements of the entanglement subgroup
come inanticommuting pairs and thus correspond to ebit
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:
.Entanglement-assisted stabilizer code error correction conditions
The two subgroups and play a role in theerror-correcting conditions for the entanglement-assisted stabilizer
formalism. An entanglement-assisted code corrects errors in a set
if for all ,Operation
The operation of an entanglement-assisted code is as follows. The senderperforms an encoding unitary on her unprotected qubits, ancilla qubits, and
her half of the ebit
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. 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 ebit
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. The encoding unitary transforms the unencoded Pauli operators
to the following encoded Pauli operators:
The sender transmits all of her 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....
s over the 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...
. The
receiver then possesses the transmitted qubits and his half of the ebit
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. 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 codein three different ways (Wilde and Brun 2007b).
Suppose that an entanglement-assisted quantum code encodes
informationqubits into
physical qubits with the help of ebits.- The entanglement-assisted rate assumes that entanglement shared between sender and receiver is free. Bennett et al. make this assumption when deriving the entanglement-assisted capacity of a quantum channel for sending quantum information. The entanglement-assisted rate is
for a code with the above parameters.
- The trade-off rate assumes that entanglement is not free and a rate pair determines performance. The first number in the pair is the number of noiseless qubits generated per channel use, and the second number in the pair is the number of ebits consumed per channel use. The rate pair is
for a code with the above parameters. Quantum information theorists have computed asymptotic trade-off curves that bound the rate region in which achievable rate pairs lie. The construction for an entanglement-assisted quantum block code minimizes the number 
of ebits given a fixed number 
and 
of respective information qubits and physical qubits.
- The catalytic rate assumes that bits of entanglement are built up at the expense of transmitted qubits. A noiseless quantum channel or the encoded use of noisy quantum channel are two different ways to build up entanglement between a sender and receiver. The catalytic rate of an
code is 
.
Which interpretation is most reasonable depends on the context in which we use
the code. In any case, the parameters
, , and ultimately governperformance, regardless of which definition of the rate we use to interpret
that performance.
Example of an entanglement-assisted code
We present an example of an entanglement-assisted codethat 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
is a qubit that the sender wants toencode. The encoding unitary then rotates the canonical stabilizer to the following set of globally commuting
generators:
The receiver measures the above generators upon receipt of all qubits to
detect and correct errors.
Encoding algorithm
We continue with the previous example. Wedetail 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
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...
article):
Call the matrix to the left of the vertical bar the "
matrix" and the matrix to the right of the vertical bar the
"
matrix."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
under conjugation. TheCNOT gate, the Hadamard gate, and the Phase gate generate the Clifford group.
A CNOT gate from qubit
to qubit adds column to column in the matrix and adds column to column in the matrix. A Hadamardgate on qubit
swaps column in the matrix with column in the matrix and vice versa. A phase gate on qubit adds column in the matrix to column in the matrix. Three CNOT gates implement aqubit swap operation. The effect of a swap on qubits
and is to swap columns and in both the and matrix.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
matrix is one. ACNOT, swap, Hadamard, or combinations of these operations can achieve this
result. We can have this result in our example by swapping qubits one and two.
The matrix becomes
Perform CNOTs to clear the entries in the
matrix in the top row to theright of the leftmost entry. These entries are already zero in this example so
we need not do anything. Proceed to the clear the entries in the first row of
the
matrix. Perform a phase gate to clear the leftmost entry in the firstrow of the
matrix if it is equal to one. It is equal to zero in this caseso we need not do anything. We then use Hadamards and CNOTs to clear the other
entries in the first row of the
matrix.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
matrix. Add row two to any other row that has one as the leftmost entry in its
matrix. For our example, we add row one to row four and we add row two torows three and four. The matrix becomes
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.