Entanglement-assisted stabilizer formalism

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

Π

n

{\displaystyle \Pi ^{n}}

over

n

{\displaystyle n}

qubits.

The sender can make clever use of her shared

ebits so that the global stabilizer is Abelian and thus forms a valid

quantum error-correcting code.

Definition

We review the construction of an entanglement-assisted code (Brun et al. 2006). Suppose that

there is a nonabelian subgroup

S

⊂

Π

n

{\displaystyle {\mathcal {S}}\subset \Pi ^{n}}

of size

n

−

k

=

2

c

+

s

{\displaystyle n-k=2c+s}

.

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

{

Z

¯

1

,

…

,

Z

¯

s

+

c

,

X

¯

s

+

1

,

…

,

X

¯

s

+

c

}

{\displaystyle \left\{{\bar {Z}}_{1},\ldots ,{\bar {Z}}_{s+c},{\bar {X}}_{s+1},\ldots ,{\bar {X}}_{s+c}\right\}}

for

S

{\displaystyle {\mathcal {S}}}

with the following commutation relations:

[

Z

¯

i

,

Z

¯

j

]

=

0

∀

i

,

j

,

{\displaystyle \left[{\bar {Z}}_{i},{\bar {Z}}_{j}\right]=0\ \ \ \ \ \forall i,j,}

[

X

¯

i

,

X

¯

j

]

=

0

∀

i

,

j

,

{\displaystyle \left[{\bar {X}}_{i},{\bar {X}}_{j}\right]=0\ \ \ \ \ \forall i,j,}

[

X

¯

i

,

Z

¯

j

]

=

0

∀

i

≠

j

,

{\displaystyle \left[{\bar {X}}_{i},{\bar {Z}}_{j}\right]=0\ \ \ \ \ \forall i\neq j,}

{

X

¯

i

,

Z

¯

i

}

=

0

∀

i

.