In [1]:

import qutip
from qutip.qinstrument import (
    QInstrument,
    basis_measurement,
    pauli_measurement
)

Basics of instruments¶

In [2]:

H = qutip.operations.hadamard_transform()

In [3]:

ket_plus = H * qutip.basis(2, 0)

In [4]:

z_instrument = basis_measurement(2)
z_instrument

Out[4]:

QInstrument id=23190ff7e80 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0,) Seq(1,)
}

In [5]:

z_instrument.sample(ket_plus)

Out[5]:

(Seq(0,),
 Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True
 Qobj data =
 [[1. 0.]
  [0. 0.]])

In [6]:

(H * z_instrument)(ket_plus)

Out[6]:

{Seq(0,): Outcome(probability=0.5000000000000002, output_state=Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True
 Qobj data =
 [[0.5 0.5]
  [0.5 0.5]]),
 Seq(1,): Outcome(probability=0.5000000000000002, output_state=Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True
 Qobj data =
 [[ 0.5 -0.5]
  [-0.5  0.5]])}

Combining instruments¶

In [7]:

z_instrument * z_instrument

Out[7]:

QInstrument id=23190ff9d00 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0, 0) Seq(1, 1)
}

In [8]:

qutip.tensor([z_instrument] * 2)

Out[8]:

QInstrument id=23190ff9460 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(Par(0, 0),) Seq(Par(0, 1),) Seq(Par(1, 0),) Seq(Par(1, 1),)
}

In [9]:

qutip.tensor(z_instrument.with_finite_visibility(0.95) ** 2, z_instrument)

Out[9]:

QInstrument id=23190ff3880 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(Par(Seq(0, 0), 0),) Seq(Par(Seq(0, 0), 1),) Seq(Par(Seq(1, 0), 0),) Seq(Par(Seq(1, 0), 1),) Seq(Par(Seq(0, 1), 0),) Seq(Par(Seq(0, 1), 1),) Seq(Par(Seq(1, 1), 0),) Seq(Par(Seq(1, 1), 1),)
}

Eliminating impossible outcomes¶

In [10]:

z_instrument * z_instrument

Out[10]:

QInstrument id=23190ff3d90 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0, 0) Seq(1, 1)
}

In [11]:

z_instrument.with_finite_visibility(0.95) ** 2

Out[11]:

QInstrument id=23190fc7550 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0, 0) Seq(1, 0) Seq(0, 1) Seq(1, 1)
}

Arbitrary dims¶

In [12]:

basis_measurement(3)

Out[12]:

QInstrument id=23190fc79d0 {
    dims [[[3], [3]], [[3], [3]]]
    outcomes Seq(0,) Seq(1,) Seq(2,)
}

Incomplete instruments¶

In [13]:

qutip.QInstrument(qutip.projection(4, 0, 0)).complete()

Out[13]:

QInstrument id=23190fc7b50 {
    dims [[[4], [4]], [[4], [4]]]
    outcomes Seq() Seq('⊥',)
}

In [14]:

qutip.QInstrument(qutip.projection(4, 0, 0)).complete().reindex()

Out[14]:

QInstrument id=23190fc7eb0 {
    dims [[[4], [4]], [[4], [4]]]
    outcomes Seq(0,) Seq(1,)
}

Arbitrary outcome labels¶

In [15]:

pauli_measurement("ZZ") * pauli_measurement("XX")

Out[15]:

QInstrument id=23190fbc9d0 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(+XX, +ZZ) Seq(-XX, +ZZ) Seq(+XX, -ZZ) Seq(-XX, -ZZ)
}

In [16]:

(pauli_measurement("ZZ") * pauli_measurement("XX")) ** 2

Out[16]:

QInstrument id=23190fbccd0 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(+XX, +ZZ, +XX, +ZZ) Seq(-XX, +ZZ, -XX, +ZZ) Seq(+XX, -ZZ, +XX, -ZZ) Seq(-XX, -ZZ, -XX, -ZZ)
}

In [17]:

(
    pauli_measurement("ZZ") * pauli_measurement("XX")
).with_finite_visibility(0.95) ** 2

Out[17]:

QInstrument id=23190fb1ac0 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(+XX, +ZZ, +XX, +ZZ) Seq(-XX, +ZZ, +XX, +ZZ) Seq(+XX, -ZZ, +XX, +ZZ) Seq(-XX, -ZZ, +XX, +ZZ) Seq(+XX, +ZZ, -XX, +ZZ) Seq(-XX, +ZZ, -XX, +ZZ) Seq(+XX, -ZZ, -XX, +ZZ) Seq(-XX, -ZZ, -XX, +ZZ) Seq(+XX, +ZZ, +XX, -ZZ) Seq(-XX, +ZZ, +XX, -ZZ) Seq(+XX, -ZZ, +XX, -ZZ) Seq(-XX, -ZZ, +XX, -ZZ) Seq(+XX, +ZZ, -XX, -ZZ) Seq(-XX, +ZZ, -XX, -ZZ) Seq(+XX, -ZZ, -XX, -ZZ) Seq(-XX, -ZZ, -XX, -ZZ)
}

When using custom output labels, you can specify sequences of outcomes with , and parallel outcomes with ;. Integers in strings are automatically converted, but other strings are preserved.

In [18]:

QInstrument({
    '0, 1, 2; 4, ⊥': qutip.projection(4, 0, 0)
})

Out[18]:

QInstrument id=23190fb1f70 {
    dims [[[4], [4]], [[4], [4]]]
    outcomes Seq(Par(Seq(0, 1, 2), Seq(4, '⊥')),)
}

For more complicated labels, Seq and Par can also be used directly. For example, the string '0, 1, 2; 4, ⊥' can also be written more explicitly as Seq(Par(Seq(0, 1, 2), Seq(4, '⊥')),).

In [19]:

from qutip.qinstrument import Seq, Par

In [20]:

QInstrument({
    Seq(Par(Seq(0, 1, 2), Seq(4, '⊥')),): qutip.projection(4, 0, 0),
    Seq(Par(Seq(0, 1, 2), Seq(Par('A', 'B'), 'C')),): qutip.projection(4, 1, 1)
})

Out[20]:

QInstrument id=2319106e490 {
    dims [[[4], [4]], [[4], [4]]]
    outcomes Seq(Par(Seq(0, 1, 2), Seq(4, '⊥')),) Seq(Par(Seq(0, 1, 2), Seq(Par('A', 'B'), 'C')),)
}

Nonselective processes¶

In [21]:

ins = (
    pauli_measurement("ZZ") * pauli_measurement("XX")
).with_finite_visibility(0.95) ** 2

In [22]:

ins.nonselective_process

Out[22]:

Quantum object: dims=[[[2, 2], [2, 2]], [[2, 2], [2, 2]]], shape=(16, 16), type='super', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}0.501 & 0 & 0 & -2.082\times10^{ -17 } & 0 & \cdots & 0 & -2.082\times10^{ -17 } & 0 & 0 & 0.499\\0 & 0.003 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.003 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0\\-2.082\times10^{ -17 } & 0 & 0 & 0.501 & 0 & \cdots & 0 & 0.499 & 0 & 0 & -2.082\times10^{ -17 }\\0 & 0 & 0 & 0 & 0.003 & \cdots & 0 & 0 & 0 & 0 & 0\\\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots\\0 & 0 & 0 & 0 & 0 & \cdots & 0.003 & 0 & 0 & 0 & 0\\-2.082\times10^{ -17 } & 0 & 0 & 0.499 & 0 & \cdots & 0 & 0.501 & 0 & 0 & -2.082\times10^{ -17 }\\0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0.003 & 0 & 0\\0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0.003 & 0\\0.499 & 0 & 0 & -2.082\times10^{ -17 } & 0 & \cdots & 0 & -2.082\times10^{ -17 } & 0 & 0 & 0.501\end{array}\right)\end{equation*}

Checking CP/TP/HP¶

In [23]:

ins.iscptp

Out[23]:

True

In [24]:

ins.ishp

Out[24]:

True

In [25]:

ins.istp

Out[25]:

True

In [26]:

ins.iscp

Out[26]:

True

Operator overloading¶

In [27]:

z_instrument & z_instrument.with_finite_visibility(0.95)

Out[27]:

QInstrument id=2319108d730 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(Par(0, 0),) Seq(Par(0, 1),) Seq(Par(1, 0),) Seq(Par(1, 1),)
}

In [28]:

z_instrument ^ 3

Out[28]:

QInstrument id=23190fc7df0 {
    dims [[[2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2]]]
    outcomes Seq(Par(0, 0, 0),) Seq(Par(0, 0, 1),) Seq(Par(0, 1, 0),) Seq(Par(0, 1, 1),) Seq(Par(1, 0, 0),) Seq(Par(1, 0, 1),) Seq(Par(1, 1, 0),) Seq(Par(1, 1, 1),)
}

Conditional processes¶

In [29]:

reset = z_instrument.if_({
    1: qutip.to_super(qutip.sigmax())
})
reset

Out[29]:

QInstrument id=2319108d4c0 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0,) Seq(1,)
}

In [30]:

reset.nonselective_process

Out[30]:

Quantum object: dims=[[[2], [2]], [[2], [2]]], shape=(4, 4), type='super', isherm=False\begin{equation*}\left(\begin{array}{*{11}c}1 & 0 & 0 & 1\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{array}\right)\end{equation*}

In [31]:

z_instrument.if_({
    1: qutip.QInstrument(qutip.to_super(qutip.sigmax()))
})

Out[31]:

QInstrument id=2319108d490 {
    dims [[[2], [2]], [[2], [2]]]
    outcomes Seq(0,) Seq(1,)
}

In [32]:

V = (
    (qutip.qeye(2) & qutip.qip.operations.cnot()) *
    (qutip.qeye(2) & qutip.qip.operations.hadamard_transform() & qutip.qeye(2)) *
    (qutip.qeye(2) & qutip.basis(2, 0) & qutip.basis(2, 0))
)
    
prepare_bell = qutip.sprepost(V, V.dag())

In [33]:

prepare_bell

Out[33]:

Quantum object: dims=[[[2, 2, 2], [2, 2, 2]], [[2], [2]]], shape=(64, 4), type='super', isherm=False\begin{equation*}\left(\begin{array}{*{11}c}0.500 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0.500 & 0 & 0 & 0\\0 & 0.500 & 0 & 0\\\vdots & \vdots & \vdots & \vdots\\0 & 0 & 0.500 & 0\\0 & 0 & 0 & 0.500\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0.500\end{array}\right)\end{equation*}

In [34]:

prepare_bell(qutip.basis(2, 0)).ptrace([0])

Out[34]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}1 & 0\\0 & 0\end{array}\right)\end{equation*}

In [35]:

prepare_bell(qutip.basis(2, 0)).ptrace([1, 2])

Out[35]:

Quantum object: dims=[[2, 2], [2, 2]], shape=(4, 4), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}0.500 & 0 & 0 & 0.500\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0.500 & 0 & 0 & 0.500\end{array}\right)\end{equation*}

In [36]:

bell_measurement = pauli_measurement("ZZ") * pauli_measurement("XX")

In [37]:

bell_measurement

Out[37]:

QInstrument id=2319108d6a0 {
    dims [[[2, 2], [2, 2]], [[2, 2], [2, 2]]]
    outcomes Seq(+XX, +ZZ) Seq(-XX, +ZZ) Seq(+XX, -ZZ) Seq(-XX, -ZZ)
}

In [38]:

teleport_wo_correction = (
    (bell_measurement & qutip.to_super(qutip.qeye(2))) *
    prepare_bell
)
teleport_wo_correction

Out[38]:

QInstrument id=231910999d0 {
    dims [[[2, 2, 2], [2, 2, 2]], [[2], [2]]]
    outcomes Seq(+XX, +ZZ) Seq(-XX, +ZZ) Seq(+XX, -ZZ) Seq(-XX, -ZZ)
}

In [39]:

teleport = teleport_wo_correction.if_({
    qutip.Seq("XX", "ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.qeye(2)),
    qutip.Seq("-XX", "ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmaz()),
    qutip.Seq("XX", "-ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmax()),
    qutip.Seq("-XX", "-ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmay()),
})

In [40]:

teleport.nonselective_process(qutip.basis(2, 0)).ptrace([2])

Out[40]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}1 & 0\\0 & 0\end{array}\right)\end{equation*}

In [41]:

teleport.nonselective_process(qutip.basis(2, 1)).ptrace([2])

Out[41]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}0 & 0\\0 & 1\end{array}\right)\end{equation*}

In [42]:

teleport.nonselective_process(qutip.qip.operations.hadamard_transform() * qutip.basis(2, 0)).ptrace([2])

Out[42]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}0.500 & 0.500\\0.500 & 0.500\end{array}\right)\end{equation*}

In [43]:

teleport_wo_correction = (
    (bell_measurement.with_finite_visibility(0.95) & qutip.to_super(qutip.qeye(2))) *
    prepare_bell
)

In [44]:

teleport = teleport_wo_correction.if_({
    qutip.Seq("XX", "ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.qeye(2)),
    qutip.Seq("-XX", "ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmaz()),
    qutip.Seq("XX", "-ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmax()),
    qutip.Seq("-XX", "-ZZ"): qutip.to_super((qutip.qeye(2) ^ 2) & qutip.sigmay()),
})

In [45]:

teleport.nonselective_process(qutip.qip.operations.hadamard_transform() * qutip.basis(2, 0)).ptrace([2])

Out[45]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', isherm=True\begin{equation*}\left(\begin{array}{*{11}c}0.500 & 0.475\\0.475 & 0.500\end{array}\right)\end{equation*}

In [ ]: