Test sympy.physics.quantum.pauli¶

In [1]:

from sympy import *
init_printing()

In [2]:

from sympy.physics.quantum import *
from sympy.physics.quantum.spin import *

In [9]:

from sympy.physics.quantum.pauli import *
from sympy.physics.quantum.boson import *
from sympy.physics.quantum.density import *
from sympy.physics.quantum.operatorordering import *

In [10]:

s_ops = sx, sy, sz, sp, sm = SigmaX(), SigmaY(), SigmaZ(), SigmaPlus(), SigmaMinus()

#s_ops = sx, sy, sz, sp, sm = Jx, Jy, Jz, Jplus, Jminus  # does not work well, many expressions unevaluated

In [11]:

sx1, sy1, sz1 = SigmaX(1), SigmaY(1), SigmaZ(1)
sx2, sy2, sz2 = SigmaX(2), SigmaY(2), SigmaZ(2)

In [12]:

qsimplify_pauli(sx * sy)

Out[12]:

$$i {\sigma_z}$$

In [13]:

qsimplify_pauli(sx1 * sy1 * sx2 * sy2)

Out[13]:

$$- {\sigma_z^{(1)}} {\sigma_z^{(2)}}$$

In [15]:

HTML table¶

In [12]:

from IPython.display import HTML

In [13]:

def print_table(data):
    t_table = "<table>\n%s\n</table>"
    t_row = "<tr>%s</tr>"
    t_col = "<td>%s</td>"
    table_code = t_table % "".join([t_row % "".join([t_col % ("$%s$" % latex(col))
                                                     for col in row])
                                    for row in data])
    return HTML(table_code)

Matrix representations¶

In [14]:

data = [s_ops, [represent(s) for s in s_ops]]

print_table(data)

Out[14]:

${\sigma_x}$	${\sigma_y}$	${\sigma_z}$	${\sigma_+}$	${\sigma_-}$
$\left[\begin{matrix}0 & 1\\1 & 0\end{matrix}\right]$	$\left[\begin{matrix}0 & - i\\i & 0\end{matrix}\right]$	$\left[\begin{matrix}1 & 0\\0 & -1\end{matrix}\right]$	$\left[\begin{matrix}0 & 1\\0 & 0\end{matrix}\right]$	$\left[\begin{matrix}0 & 0\\1 & 0\end{matrix}\right]$

Commutators¶

In [15]:

data = [[r"{\rm Expression}", r"{\rm Evaluated\;value}", r"{\rm Numerical\;test}"]] + \
       [[Commutator(s1, s2),
         Commutator(s1, s2).doit(),
         represent(Commutator(s1, s2)) == represent(Commutator(s1, s2).doit())]
        for s1 in s_ops for s2 in s_ops if s1 != s2]

print_table(data)

Out[15]:

${\rm Expression}$	${\rm Evaluated\;value}$	${\rm Numerical\;test}$
$\left[{\sigma_x},{\sigma_y}\right]$	$2 i {\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_x},{\sigma_z}\right]$	$- 2 i {\sigma_y}$	$\mathrm{True}$
$- \left[{\sigma_+},{\sigma_x}\right]$	$- {\sigma_z}$	$\mathrm{True}$
$- \left[{\sigma_-},{\sigma_x}\right]$	${\sigma_z}$	$\mathrm{True}$
$- \left[{\sigma_x},{\sigma_y}\right]$	$- 2 i {\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_y},{\sigma_z}\right]$	$2 i {\sigma_x}$	$\mathrm{True}$
$- \left[{\sigma_+},{\sigma_y}\right]$	$- i {\sigma_z}$	$\mathrm{True}$
$- \left[{\sigma_-},{\sigma_y}\right]$	$- i {\sigma_z}$	$\mathrm{True}$
$- \left[{\sigma_x},{\sigma_z}\right]$	$2 i {\sigma_y}$	$\mathrm{True}$
$- \left[{\sigma_y},{\sigma_z}\right]$	$- 2 i {\sigma_x}$	$\mathrm{True}$
$- \left[{\sigma_+},{\sigma_z}\right]$	$2 {\sigma_+}$	$\mathrm{True}$
$- \left[{\sigma_-},{\sigma_z}\right]$	$- 2 {\sigma_-}$	$\mathrm{True}$
$\left[{\sigma_+},{\sigma_x}\right]$	${\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_+},{\sigma_y}\right]$	$i {\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_+},{\sigma_z}\right]$	$- 2 {\sigma_+}$	$\mathrm{True}$
$- \left[{\sigma_-},{\sigma_+}\right]$	${\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_-},{\sigma_x}\right]$	$- {\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_-},{\sigma_y}\right]$	$i {\sigma_z}$	$\mathrm{True}$
$\left[{\sigma_-},{\sigma_z}\right]$	$2 {\sigma_-}$	$\mathrm{True}$
$\left[{\sigma_-},{\sigma_+}\right]$	$- {\sigma_z}$	$\mathrm{True}$

Anticommutators¶

In [16]:

def elim_zero_matrix(e):
    return 0 if e == Matrix([[0, 0], [0, 0]]) else e

In [17]:

I_op = IdentityOperator(2)

data = [[r"{\rm Expression}", r"{\rm Evaluated\;value}", r"{\rm Numerical\;test}"]] + \
       [[AntiCommutator(s1, s2),
         AntiCommutator(s1, s2).doit(),
         elim_zero_matrix(represent(AntiCommutator(s1, s2))) == represent(I_op * AntiCommutator(s1, s2).doit())]
        for s1 in s_ops for s2 in s_ops if s1 != s2]

print_table(data)

Out[17]:

${\rm Expression}$	${\rm Evaluated\;value}$	${\rm Numerical\;test}$
$\left\{{\sigma_x},{\sigma_y}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_x},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_x}\right\}$	$1$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_x}\right\}$	$1$	$\mathrm{True}$
$\left\{{\sigma_x},{\sigma_y}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_y},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_y}\right\}$	$i$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_y}\right\}$	$- i$	$\mathrm{True}$
$\left\{{\sigma_x},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_y},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_x}\right\}$	$1$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_y}\right\}$	$i$	$\mathrm{True}$
$\left\{{\sigma_+},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_+}\right\}$	$1$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_x}\right\}$	$1$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_y}\right\}$	$- i$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_z}\right\}$	$0$	$\mathrm{True}$
$\left\{{\sigma_-},{\sigma_+}\right\}$	$1$	$\mathrm{True}$

Adjoint¶

In [18]:

Dagger(sx), Dagger(sy), Dagger(sz)

Out[18]:

$$\begin{pmatrix}{\sigma_x}, & {\sigma_y}, & {\sigma_z}\end{pmatrix}$$

In [19]:

Dagger(sm), Dagger(sp)

Out[19]:

$$\begin{pmatrix}{\sigma_+}, & {\sigma_-}\end{pmatrix}$$

In [20]:

sm, sp

Out[20]:

$$\begin{pmatrix}{\sigma_-}, & {\sigma_+}\end{pmatrix}$$

Operator products¶

In [26]:

s = [sx, sy, sz, sm, sp]

data = [[s2] + [(qsimplify_pauli(s1 * s2), represent((qsimplify_pauli(s1 * s2) * I_op).expand()) == represent(s1) * represent(s2))
         for s1 in s] for s2 in s]

print_table(data)

Out[26]:

${\sigma_x}$	$\begin{pmatrix}1, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- i {\sigma_z}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}i {\sigma_y}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}\frac{1}{2} \left(1 - {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}\frac{1}{2} \left(1 + {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$
${\sigma_y}$	$\begin{pmatrix}i {\sigma_z}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}1, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- i {\sigma_x}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- \frac{i}{2} \left(1 - {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}\frac{i}{2} \left(1 + {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$
${\sigma_z}$	$\begin{pmatrix}- i {\sigma_y}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}i {\sigma_x}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}1, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}{\sigma_-}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- {\sigma_+}, & \mathrm{True}\end{pmatrix}$
${\sigma_-}$	$\begin{pmatrix}\frac{1}{2} + \frac{{\sigma_z}}{2}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- \frac{i}{2} \left(1 + {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}- {\sigma_-}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}0, & \mathrm{False}\end{pmatrix}$	$\begin{pmatrix}\frac{1}{2} \left(1 + {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$
${\sigma_+}$	$\begin{pmatrix}\frac{1}{2} - \frac{{\sigma_z}}{2}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}\frac{i}{2} \left(1 - {\sigma_z}\right), & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}{\sigma_+}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}\frac{1}{2} - \frac{{\sigma_z}}{2}, & \mathrm{True}\end{pmatrix}$	$\begin{pmatrix}0, & \mathrm{False}\end{pmatrix}$

In [27]:

represent(sm) * represent(sm)

Out[27]:

$$\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]$$

Powers¶

In [28]:

[sx ** n for n in range(10)]

Out[28]:

$$\begin{bmatrix}1, & {\sigma_x}, & 1, & {\sigma_x}, & 1, & {\sigma_x}, & 1, & {\sigma_x}, & 1, & {\sigma_x}\end{bmatrix}$$

In [29]:

[sy ** n for n in range(10)]

Out[29]:

$$\begin{bmatrix}1, & {\sigma_y}, & 1, & {\sigma_y}, & 1, & {\sigma_y}, & 1, & {\sigma_y}, & 1, & {\sigma_y}\end{bmatrix}$$

In [30]:

[sz ** n for n in range(10)]

Out[30]:

$$\begin{bmatrix}1, & {\sigma_z}, & 1, & {\sigma_z}, & 1, & {\sigma_z}, & 1, & {\sigma_z}, & 1, & {\sigma_z}\end{bmatrix}$$

In [31]:

[sm ** n for n in range(10)]

Out[31]:

$$\begin{bmatrix}1, & {\sigma_-}, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0\end{bmatrix}$$

In [32]:

[sp ** n for n in range(10)]

Out[32]:

$$\begin{bmatrix}1, & {\sigma_+}, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0\end{bmatrix}$$

States¶

In [33]:

up = SigmaZKet(0)
down = SigmaZKet(1)

ket = (up + down)/sqrt(2)

In [34]:

ket, qapply(sx * ket), qapply(sy * ket), qapply(sz * ket)

Out[34]:

$$\begin{pmatrix}\frac{\sqrt{2}}{2} \left({\left|0\right\rangle } + {\left|1\right\rangle }\right), & \frac{\sqrt{2}}{2} {\left|0\right\rangle } + \frac{\sqrt{2}}{2} {\left|1\right\rangle }, & - \frac{\sqrt{2} i}{2} {\left|0\right\rangle } + \frac{\sqrt{2} i}{2} {\left|1\right\rangle }, & \frac{\sqrt{2}}{2} {\left|0\right\rangle } - \frac{\sqrt{2}}{2} {\left|1\right\rangle }\end{pmatrix}$$

In [35]:

ket, qapply(sm * ket), qapply(sp * ket)

Out[35]:

$$\begin{pmatrix}\frac{\sqrt{2}}{2} \left({\left|0\right\rangle } + {\left|1\right\rangle }\right), & \frac{\sqrt{2}}{2} {\left|1\right\rangle }, & \frac{\sqrt{2}}{2} {\left|0\right\rangle }\end{pmatrix}$$

In [36]:

represent(qapply(sx * ket)) == represent(sx) * represent(ket)

Out[36]:

True

In [37]:

represent(qapply(sy * ket)) == represent(sy) * represent(ket)

Out[37]:

True

In [38]:

represent(qapply(sz * ket)) == represent(sz) * represent(ket)

Out[38]:

True

In [39]:

represent(qapply(sm * ket)) == represent(sm) * represent(ket)

Out[39]:

True

In [40]:

represent(qapply(sp * ket)) == represent(sp) * represent(ket)

Out[40]:

True

Applications¶

Baker–Campbell–Hausdorff formula¶

In [41]:

from sympy_quantum_extra import *

---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-41-b89e5cd86f97> in <module>()
----> 1 from sympy_quantum_extra import *

/Users/rob/Dropbox/projects/sympy-quantum-notebooks/sympy_quantum_extra.py in <module>()
      8 from sympy.physics.quantum.fermion import *
      9 from sympy.physics.quantum.operatorordering import *
---> 10 from sympy.physics.quantum.expectation import Expectation
     11 
     12 from sympy.physics.quantum.pauli import (SigmaX, SigmaY, SigmaZ, SigmaMinus,

ImportError: No module named 'sympy.physics.quantum.expectation'

In [42]:

rc = recursive_commutator(sy, sz, 5).doit().expand()

rc

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-42-3ba084e7083e> in <module>()
----> 1 rc = recursive_commutator(sy, sz, 5).doit().expand()
      2 
      3 rc

NameError: name 'recursive_commutator' is not defined

In [31]:

A, B = Operator("A"), Operator("B")

Commutator(A, B).doit().doit()
# Add.doit() - > Mul.doit() -> ...

Out[31]:

$$A B - B A$$

In [32]:

Commutator(sx, sy).doit()
# sx * sy - sy * sx

Out[32]:

$$2 i {\sigma_z}$$

In [33]:

A = Matrix([[1, 2], [3, 4]])
B = Matrix([[1, 2], [3, 4]])
A * B, MatMul(A, B)

Out[33]:

$$\left ( \left[\begin{matrix}7 & 10\\15 & 22\end{matrix}\right], \quad \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right] \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]\right )$$

In [34]:

1 * 3

Out[34]:

$$3$$

In [35]:

IdentityOperator(2).is_commutative

Out[35]:

False

In [36]:

qsimplify(qsimplify(rc))

Out[36]:

$$32 i {\sigma_x}$$

In [37]:

represent(sx * sx)

Out[37]:

$$1$$

In [38]:

represent(sx) ** 2 == Integer(1) * represent(IdentityOperator(2))

Out[38]:

True

In [38]:

Test: $e^{i\theta/2 \sigma_y}\sigma_xe^{-i\theta/2 \sigma_y} = \cos(\theta)\sigma_x + \sin(\theta)\sigma_z$¶

In [39]:

theta = symbols("theta")

In [40]:

U = exp((I * theta / 2)* sy)

In [41]:

unitary_transformation(U, sx)

Out[41]:

$$\sin{\left (\theta \right )} {\sigma_z} + \cos{\left (\theta \right )} {\sigma_x}$$

Test: $e^{i\theta/2 \sigma_y}\sigma_ze^{-i\theta/2 \sigma_y} = \cos(\theta)\sigma_z - \sin(\theta)\sigma_x$¶

In [42]:

unitary_transformation(U, sz)

Out[42]:

$$- \sin{\left (\theta \right )} {\sigma_x} + \cos{\left (\theta \right )} {\sigma_z}$$

Calculate $e^{i\theta/2 \sigma_y}\sigma_-e^{-i\theta/2 \sigma_y}$¶

In [43]:

unitary_transformation(U, sm)

Out[43]:

$$\left(\frac{1}{2} \cos{\left (\theta \right )} - \frac{1}{2}\right) {\sigma_x} + \frac{{\sigma_z}}{2} \sin{\left (\theta \right )} + {\sigma_-}$$

Calculate $e^{i\theta/2 \sigma_y}\sigma_+e^{-i\theta/2 \sigma_y}$¶

In [44]:

unitary_transformation(U, sp)

Out[44]:

$$\left(\frac{1}{2} \cos{\left (\theta \right )} - \frac{1}{2}\right) {\sigma_x} + \frac{{\sigma_z}}{2} \sin{\left (\theta \right )} + {\sigma_+}$$

Versions¶

In [7]:

%reload_ext version_information

%version_information sympy

Out[7]:

Software	Version
Python	3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]
IPython	2.3.0
OS	Darwin 13.4.0 x86_64 i386 64bit
sympy	0.7.5-git
Thu Oct 09 13:14:39 2014 JST

In [ ]: