# Quantum Entanglement in Python
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## Version
The releases of `pyqentangle` 2.x.x is incompatible with previous releases.
The releases of `pyqentangle` 3.x.x is incompatible with previous releases.
Since release 3.1.0, the support for Python 2 was decomissioned.
## Installation
This package can be installed using `pip`.
```
>>> pip install pyqentangle
```
To use it, enter
```
>>> import pyqentangle
>>> import numpy as np
```
## Schmidt Decomposition for Discrete Bipartite States
We first express the bipartite state in terms of a tensor. For example, if the state is `|01>+|10>`, then express it as
```
>>> tensor = np.array([[0., np.sqrt(0.5)], [np.sqrt(0.5), 0.]])
```
To perform the Schmidt decompostion, just enter:
```
>>> pyqentangle.schmidt_decomposition(tensor)
[(0.7071067811865476, array([ 0., -1.]), array([-1., -0.])),
(0.7071067811865476, array([-1., 0.]), array([-0., -1.]))]
```
For each tuple in the returned list, the first element is the Schmidt coefficients, the second the component for first subsystem, and the third the component for the second subsystem.
## Schmidt Decomposition for Continuous Bipartite States
We can perform Schmidt decomposition on continuous systems too. For example, define the following normalized wavefunction:
```
>>> fcn = lambda x1, x2: np.exp(-0.5 * (x1 + x2) ** 2) * np.exp(-(x1 - x2) ** 2) * np.sqrt(np.sqrt(8.) / np.pi)
```
Then perform the Schmidt decomposition,
```
>>> modes = pyqentangle.continuous_schmidt_decomposition(biwavefcn, -10., 10., -10., 10., keep=10)
```
where it describes the ranges of x1 and x2 respectively, and `keep=10` specifies only top 10 Schmidt modes are kept. Then we can read the Schmidt coefficients:
```
>>> list(map(lambda dec: dec[0], modes))
[0.9851714310094161,
0.1690286950361957,
0.02900073920775954,
0.004975740210361192,
0.0008537020544076649,
0.00014647211608480773,
2.51306421011773e-05,
4.311736522272035e-06,
7.39777032460608e-07,
1.2692567250688184e-07]
```
The second and the third elements in each tuple in the list `decompositions` are lambda functions for the modes of susbsystems A and B respectively. The Schmidt functions can be plotted:
```
>>> xarray = np.linspace(-10., 10., 100)
plt.subplot(3, 2, 1)
plt.plot(xarray, modes[0][1](xarray))
plt.subplot(3, 2, 2)
plt.plot(xarray, modes[0][2](xarray))
plt.subplot(3, 2, 3)
plt.plot(xarray, modes[1][1](xarray))
plt.subplot(3, 2, 4)
plt.plot(xarray, modes[1][2](xarray))
plt.subplot(3, 2, 5)
plt.plot(xarray, modes[2][1](xarray))
plt.subplot(3, 2, 6)
plt.plot(xarray, modes[2][2](xarray))
```
## Useful Links
* Study of Entanglement in Quantum Computers: [https://datawarrior.wordpress.com/2017/09/20/a-first-glimpse-of-rigettis-quantum-computing-cloud/](https://datawarrior.wordpress.com/2017/09/20/a-first-glimpse-of-rigettis-quantum-computing-cloud/)
* Github page: [https://github.com/stephenhky/pyqentangle](https://github.com/stephenhky/pyqentangle)
* PyPI page: [https://pypi.python.org/pypi/pyqentangle/](https://pypi.python.org/pypi/pyqentangle/)
* Documentation: [http://pyqentangle.readthedocs.io/](http://pyqentangle.readthedocs.io/)
* RQEntangle: [https://CRAN.R-project.org/package=RQEntangle](https://CRAN.R-project.org/package=RQEntangle) (corresponding R library)
## Reference
* Artur Ekert, Peter L. Knight, "Entangled quantum systems and the Schmidt decomposition", *Am. J. Phys.* 63, 415 (1995).
## Acknowledgement
* [Hossein Seifoory](https://ca.linkedin.com/in/hosseinseifoory?trk=public_profile_card_url)
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"description": "# Quantum Entanglement in Python\n\n[](https://circleci.com/gh/stephenhky/pyqentangle.svg)\n[](https://github.com/stephenhky/pyqentangle/releases)\n[](https://pyqentangle.readthedocs.io/en/latest/?badge=latest)\n[](https://pyup.io/repos/github/stephenhky/pyqentangle/)\n[](https://pyup.io/repos/github/stephenhky/pyqentangle/)\n[](https://pypi.org/project/pyqentangle/)\n[](https://pypi.org/project/pyqentangle/)\n\n## Version\n\nThe releases of `pyqentangle` 2.x.x is incompatible with previous releases.\n\nThe releases of `pyqentangle` 3.x.x is incompatible with previous releases.\n\nSince release 3.1.0, the support for Python 2 was decomissioned.\n\n## Installation\n\nThis package can be installed using `pip`.\n\n```\n>>> pip install pyqentangle\n```\n\nTo use it, enter\n\n```\n>>> import pyqentangle\n>>> import numpy as np\n```\n\n## Schmidt Decomposition for Discrete Bipartite States\n\nWe first express the bipartite state in terms of a tensor. For example, if the state is `|01>+|10>`, then express it as\n\n```\n>>> tensor = np.array([[0., np.sqrt(0.5)], [np.sqrt(0.5), 0.]])\n```\n\nTo perform the Schmidt decompostion, just enter:\n\n```\n>>> pyqentangle.schmidt_decomposition(tensor)\n[(0.7071067811865476, array([ 0., -1.]), array([-1., -0.])),\n (0.7071067811865476, array([-1., 0.]), array([-0., -1.]))]\n ```\n\nFor each tuple in the returned list, the first element is the Schmidt coefficients, the second the component for first subsystem, and the third the component for the second subsystem.\n\n## Schmidt Decomposition for Continuous Bipartite States\n\nWe can perform Schmidt decomposition on continuous systems too. For example, define the following normalized wavefunction:\n\n```\n>>> fcn = lambda x1, x2: np.exp(-0.5 * (x1 + x2) ** 2) * np.exp(-(x1 - x2) ** 2) * np.sqrt(np.sqrt(8.) / np.pi)\n```\n\nThen perform the Schmidt decomposition, \n\n```\n>>> modes = pyqentangle.continuous_schmidt_decomposition(biwavefcn, -10., 10., -10., 10., keep=10)\n```\n\nwhere it describes the ranges of x1 and x2 respectively, and `keep=10` specifies only top 10 Schmidt modes are kept. Then we can read the Schmidt coefficients:\n\n```\n>>> list(map(lambda dec: dec[0], modes))\n[0.9851714310094161,\n 0.1690286950361957,\n 0.02900073920775954,\n 0.004975740210361192,\n 0.0008537020544076649,\n 0.00014647211608480773,\n 2.51306421011773e-05,\n 4.311736522272035e-06,\n 7.39777032460608e-07,\n 1.2692567250688184e-07]\n```\n\nThe second and the third elements in each tuple in the list `decompositions` are lambda functions for the modes of susbsystems A and B respectively. The Schmidt functions can be plotted:\n```\n>>> xarray = np.linspace(-10., 10., 100)\n\n plt.subplot(3, 2, 1)\n plt.plot(xarray, modes[0][1](xarray))\n plt.subplot(3, 2, 2)\n plt.plot(xarray, modes[0][2](xarray))\n\n plt.subplot(3, 2, 3)\n plt.plot(xarray, modes[1][1](xarray))\n plt.subplot(3, 2, 4)\n plt.plot(xarray, modes[1][2](xarray))\n\n plt.subplot(3, 2, 5)\n plt.plot(xarray, modes[2][1](xarray))\n plt.subplot(3, 2, 6)\n plt.plot(xarray, modes[2][2](xarray))\n```\n\n\n\n\n## Useful Links\n\n* Study of Entanglement in Quantum Computers: [https://datawarrior.wordpress.com/2017/09/20/a-first-glimpse-of-rigettis-quantum-computing-cloud/](https://datawarrior.wordpress.com/2017/09/20/a-first-glimpse-of-rigettis-quantum-computing-cloud/)\n* Github page: [https://github.com/stephenhky/pyqentangle](https://github.com/stephenhky/pyqentangle)\n* PyPI page: [https://pypi.python.org/pypi/pyqentangle/](https://pypi.python.org/pypi/pyqentangle/)\n* Documentation: [http://pyqentangle.readthedocs.io/](http://pyqentangle.readthedocs.io/)\n* RQEntangle: [https://CRAN.R-project.org/package=RQEntangle](https://CRAN.R-project.org/package=RQEntangle) (corresponding R library)\n\n## Reference\n* Artur Ekert, Peter L. Knight, \"Entangled quantum systems and the Schmidt decomposition\", *Am. J. Phys.* 63, 415 (1995).\n\n## Acknowledgement\n* [Hossein Seifoory](https://ca.linkedin.com/in/hosseinseifoory?trk=public_profile_card_url)\n",
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