states.ghz
==========

.. py:module:: states.ghz

.. autoapi-nested-parse::

   GHZ (Greenberger-Horne-Zeilinger) is used to represent a maximally entangled state.

   In the GHZ state, the state of qubits are completely dependent on the state of other qubits. This state is an important
   part of quantum computing as it is commonly used in algorithms, protocols, error corrections, cryptography, etc.



Functions
---------

.. autoapisummary::

   states.ghz.ghz


Module Contents
---------------

.. py:function:: ghz(dim, num_qubits, coeff = None)

   Generate a (generalized) GHZ state :cite:`Greenberger_2007_Going`.

   Returns a :code:`num_qubits`-partite GHZ state acting on :code:`dim` local dimensions, described
   in :cite:`Greenberger_2007_Going`. For example, :code:`ghz(2, 3)` returns the standard 3-qubit GHZ state on qubits.
   The output of this function is a dense NumPy array.

   For a system of :code:`num_qubits` qubits (i.e., :code:`dim = 2`), the GHZ state can be written
   as

   .. math::
       |GHZ \rangle = \frac{1}{\sqrt{n}} \left(|0\rangle^{\otimes n} +
       |1 \rangle^{\otimes n} \right)).

   .. rubric:: Examples

   When :code:`dim = 2`, and :code:`num_qubits = 3` this produces the standard GHZ state

   .. math::
       \frac{1}{\sqrt{2}} \left( |000 \rangle + |111 \rangle \right).

   Using :code:`|toqito⟩`, we can see that this yields the proper state.

   >>> from toqito.states import ghz
   >>> ghz(2, 3)
   array([[0.70710678],
          [0.        ],
          [0.        ],
          [0.        ],
          [0.        ],
          [0.        ],
          [0.        ],
          [0.70710678]])

   As this function covers the generalized GHZ state, we can consider higher dimensions. For instance here is the GHZ
   state in :math:`\mathbb{C}^{4^{\otimes 7}}` as

   .. math::
       \frac{1}{\sqrt{30}} \left(|0000000 \rangle + 2|1111111 \rangle +
       3|2222222 \rangle + 4|3333333\rangle \right).

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: Number of qubits is not a positive integer.
   :param dim: The local dimension.
   :param num_qubits: The number of parties (qubits/qudits)
   :param coeff: (default `[1, 1, ..., 1])/sqrt(dim)`:
                 a 1-by-`dim` vector of coefficients.
   :returns: Numpy vector array as GHZ state.