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
The theory of quantum error correction plays a prominent role in the practical realization and engineering of
quantum computing and quantum communication devices. The first quantum
error-correcting codes are strikingly similar to classical bloc ...
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 operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s.
The sender's
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s do not necessarily have to form an
Abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
subgroup of the
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\begin ...
over
qubits.
The sender can make clever use of her shared
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s so that the global stabilizer is Abelian and thus forms a valid
quantum error-correcting code
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that ...
.
Definition
We review the construction of an entanglement-assisted code (Brun ''et al.'' 2006). Suppose that
there is a
nonabelian subgroup of size
.
Application of the fundamental theorem of
symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
(Lemma 1 in the first external reference)
states that there exists a minimal set of independent generators
for
with the following
commutation
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
relations:
:
:
:
:
The decomposition of
into the above minimal generating set
determines that the code requires
ancilla qubits and
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s. The code
requires an
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
for every
anticommuting
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
pair in the minimal generating set.
The simple reason for this requirement is that an
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
is a simultaneous
-
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
of the
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s
. The second
qubit
in the
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
transforms the
anticommuting
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
pair
into a
commuting
Commuting is periodically recurring travel between one's place of residence and place of work or study, where the traveler, referred to as a commuter, leaves the boundary of their home community. By extension, it can sometimes be any regul ...
pair
. The above decomposition also
minimizes the number of
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s required for the code---it is an optimal decomposition.
We can partition the
nonabelian group into two
subgroups: 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 EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s:
:
.
Entanglement-assisted stabilizer code error correction conditions
The two subgroups
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
,
:
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
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s. The unencoded state is a simultaneous +1-
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
of
the following
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s:
:
The
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s to the right of the vertical bars indicate the receiver's half
of the shared
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s. The encoding unitary transforms the unencoded
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s
to the following encoded
Pauli operator
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
s:
:
The sender transmits all of her
qubits over the noisy
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. An example of classical information i ...
. The
receiver then possesses the transmitted qubits and his half of the
ebit EBIT, Ebit or ebit may refer to:
*EBIT, or Earnings before interest and taxes, in finance
*EBIT, or Electron beam ion trap, in physics
*An ebit (quantum state), a two-party quantum state with quantum entanglement
Quantum entanglement is the ph ...
s. He
measures the above encoded operators to diagnose the error. The last step is
to correct 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
information
qubits 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
In the theory of quantum communication, the entanglement-assisted classical capacity of a quantum channel is the highest rate at which classical information can be transmitted from a sender to receiver when they share an unlimited amount of noise ...
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