pyqentangle 4.0.6
Quantum Entanglement in Python
License: MIT
Requires Python: >=3.7
Keywords: quantum physics, quantum information, Schmidt decomposition, entanglement

# 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))
```

![alt](https://github.com/stephenhky/pyqentangle/raw/master/fig/three_harmonic_modes.png)


## 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)