In
quantum information and
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
, a cluster state is a type of highly entangled state of multiple
qubits. Cluster states are generated in
lattices of qubits with
Ising type interactions. A cluster ''C'' is a connected subset of a ''d''-dimensional lattice, and a cluster state is a pure state of the qubits located on ''C''. They are different from other types of entangled states such as
GHZ states or
W states in that it is more difficult to eliminate
quantum entanglement (via
projective measurements) in the case of cluster states. Another way of thinking of cluster states is as a particular instance of
graph states, where the underlying graph is a connected subset of a ''d''-dimensional lattice. Cluster states are especially useful in the context of the
one-way quantum computer. For a comprehensible introduction to the topic see.
Formally, cluster states
are states which obey the set eigenvalue equations:
:
where
are the correlation operators
:
with
and
being
Pauli matrices,
denoting the
neighbourhood of
and
being a set of binary parameters specifying the particular instance of a cluster state.
Examples with qubits
Here are some examples of one-dimensional cluster states (''d''=1), for
, where
is the number of qubits. We take
for all
, which means the cluster state is the unique simultaneous eigenstate that has corresponding eigenvalue 1 under all correlation operators. In each example the set of correlation operators
and the corresponding cluster state is listed.
*
:
This is an EPR-pair (up to local transformations).
*
:
:
This is the GHZ-state (up to local transformations).
*
:
:
.
:This is not a GHZ-state and
can not be converted to a GHZ-state with local operations.
In all examples
is the identity operator, and tensor products are omitted. The states above can be obtained from the all zero state
by first applying a Hadamard gate to every qubit, and then a controlled-Z gate between all qubits that are adjacent to each other.
Experimental creation of cluster states
Cluster states can be realized experimentally. One way to create a cluster state is by encoding
logical qubits into the polarization of photons, one common encoding is the following:
This is not the only possible encoding, however it is one of the simplest: with this encoding entangled pairs can be created experimentally through
spontaneous parametric down-conversion. The entangled pairs that can be generated this way have the form
equivalent to the logical state
for the two choices of the phase
the two
Bell states are obtained: these are themselves two examples of two-qubits cluster states. Through the use of linear optic devices as
beam-splitters or
wave-plates these Bell states can interact and form more complex cluster states. Cluster states have been created also in
optical lattices of
cold atoms.
Entanglement criteria and Bell inequalities for cluster states
After a cluster state was created in an experiment, it is important to verify that indeed, an entangled quantum state has been created and obtain the fidelity with respect to an ideal cluster state. There are efficient conditions to detect entanglement close to cluster states, that need only the minimal two local measurement settings. Similar conditions can also be used to estimate the fidelity with respect to an ideal cluster state. Bell inequalities have also been developed for cluster states. All these entanglement conditions and Bell inequalities are based on the stabilizer formalism.
See also
*
Bell state
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a fo ...
*
Graph state
*
Optical cluster state
References
{{Quantum computing
Quantum information science