Density Matrix

Table of Contents

• Introduction
• In QPanda3 Quantum Information
  • Constructing a DensityMatrix object
    • Default constructor
    • From qubit total
    • From a complex array
    • From DensityMatrix
    • From StateVector
  • Obtain internal data
    • Get numpy.ndarray
    • Get element
  • Purity
  • StateVector
  • Evolution with QCircuit
    • Evolve without updating
    • Evolve with updating
  • Boolean function
    • Valid
    • Equal

Prev Tutorial: Variational Quantum Circuit
Next Tutorial: State Vector

---

Introduction

The density matrix is a crucial concept in quantum mechanics, which is used to describe the state of a quantum system, especially when there is interaction between the system and the environment. The following is a detailed introduction to the density matrix in the quantum field:

The density matrix, also known as the density operator, is a Hermitian matrix with a trace of 1 and a squared trace of less than or equal to 1. In quantum mechanics, the density matrix is used to represent the superposition of probabilities of a system being in multiple quantum states. For the state of a quantum system, its density matrix can be expressed as \( \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|\), where \(p_i\) is the probability that the system is in the state \(|\psi_i\rangle\), satisfying \(p_i \geq 0\) and \(\sum_i\) \(p_i=1 \) , which is the normalization condition.

In QPanda3 Quantum Information

QPanda3 uses DensityMatrix in Quantum Informaton to abstract and simulate density matrix

Constructing a DensityMatrix object

Here is API doc

Default constructor

The default is to construct a density matrix for a quantum system with only one qubit and a current state of all 0s

For internal tensor product operations, the qbit index with the larger value is used as the left operand. For example, when performing a tensor product operation on a quantum system consisting of \(q[0]\), \(q[1]\), and \(q[2]\), the result corresponds to \(\left | q[2] \right \rangle \otimes \left | q[1] \right \rangle \otimes \left | q[0] \right \rangle \)

Python

from pyqpanda3.quantum_info import DensityMatrix
# Construct a DensityMatrix object
dm = DensityMatrix()
# Print result
print(dm)

Output

{
{
{(1,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},
}
}

From qubit total

Specify the total number of qubits in the quantum system, and generate a density matrix where the state of each quantum is currently 0

For internal tensor product operations, the qbit index with the larger value is used as the left operand. For example, when performing a tensor product operation on a quantum system consisting of \(q[0]\), \(q[1]\), and \(q[2]\), the result corresponds to \(\left | q[2] \right \rangle \otimes \left | q[1] \right \rangle \otimes \left | q[0] \right \rangle \)

Python

from pyqpanda3.quantum_info import DensityMatrix
# Construct a DensityMatrix object
dm = DensityMatrix(3)
# Print result
print(dm)

Output

{
{
{(1,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
}

From a complex array

Construct a density matrix based on the input two-dimensional complex array.If the matrix corresponding to the two-dimensional complex array does not satisfy the condition of trace 1 and positive definiteness, an exception is thrown. If the number of elements in a given complex array is not a power of 2, it will be automatically expanded to a power of 2. This expansion operation will expand the number of elements as little as possible

For internal tensor product operations, the qbit index with the larger value is used as the left operand. For example, when performing a tensor product operation on a quantum system consisting of \(q[0]\), \(q[1]\), and \(q[2]\), the result corresponds to \(\left | q[2] \right \rangle \otimes \left | q[1] \right \rangle \otimes \left | q[0] \right \rangle \)

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
# Prepare a complex array
data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
# Construct a DensityMatrix object
dm = DensityMatrix(data)
# Print result
print(dm)

Output

{
{
{(0.180564,0)},{(0.122156,0.0689359)},{(-0.0533788,-0.0607727)},{(-0.0616472,-0.00112235)},
}
{
{(0.122156,-0.0689359)},{(0.386702,0)},{(-0.251278,-0.091095)},{(0.110335,0.0401682)},
}
{
{(-0.0533788,0.0607727)},{(-0.251278,0.091095)},{(0.260187,0)},{(-0.14254,0.0202721)},
}
{
{(-0.0616472,0.00112235)},{(0.110335,-0.0401682)},{(-0.14254,-0.0202721)},{(0.172547,0)},
}
}

From DensityMatrix

Construct a new DensityMatrix object based on an existing DensityMatrix object

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
# Prepare a complex array
data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
# Construct a DensityMatrix object
dm = DensityMatrix(data)
# Construct another DensityMatrix object
dm2 = DensityMatrix(dm)
# Print result
print(dm2)

Output

{
{
{(0.180564,0)},{(0.122156,0.0689359)},{(-0.0533788,-0.0607727)},{(-0.0616472,-0.00112235)},
}
{
{(0.122156,-0.0689359)},{(0.386702,0)},{(-0.251278,-0.091095)},{(0.110335,0.0401682)},
}
{
{(-0.0533788,0.0607727)},{(-0.251278,0.091095)},{(0.260187,0)},{(-0.14254,0.0202721)},
}
{
{(-0.0616472,0.00112235)},{(0.110335,-0.0401682)},{(-0.14254,-0.0202721)},{(0.172547,0)},
}
}

From StateVector

Construct a new DensityMatrix object from an existing StateVector object

Python

import numpy as np
from pyqpanda3.quantum_info import DensityMatrix
from pyqpanda3.quantum_info import StateVector
 
def test_construct_from_stv():
    # Construct a StateVector object
    stv = StateVector(np.array([1 + 1.j, 2 + 2.j, 3 + 3.j, 4 + 4.j, 5 + 5.j, 6 + 6.j, 7 + 7.j, 8 + 8.j]))
    # Construct a DensityMatrix object
    dm = DensityMatrix(stv)
    # Print result
    print("dm:",dm)
 
if __name__ == "__main__":
    test_construct_from_stv()

Output

dm: {
{
{(0.00490196,0)},{(0.00980392,0)},{(0.0147059,0)},{(0.0196078,0)},{(0.0245098,0)},{(0.0294118,0)},{(0.0343137,0)},{(0.0392157,0)},
}
{
{(0.00980392,0)},{(0.0196078,0)},{(0.0294118,0)},{(0.0392157,0)},{(0.0490196,0)},{(0.0588235,0)},{(0.0686275,0)},{(0.0784314,0)},
}
{
{(0.0147059,0)},{(0.0294118,0)},{(0.0441176,0)},{(0.0588235,0)},{(0.0735294,0)},{(0.0882353,0)},{(0.102941,0)},{(0.117647,0)},
}
{
{(0.0196078,0)},{(0.0392157,0)},{(0.0588235,0)},{(0.0784314,0)},{(0.0980392,0)},{(0.117647,0)},{(0.137255,0)},{(0.156863,0)},
}
{
{(0.0245098,0)},{(0.0490196,0)},{(0.0735294,0)},{(0.0980392,0)},{(0.122549,0)},{(0.147059,0)},{(0.171569,0)},{(0.196078,0)},
}
{
{(0.0294118,0)},{(0.0588235,0)},{(0.0882353,0)},{(0.117647,0)},{(0.147059,0)},{(0.176471,0)},{(0.205882,0)},{(0.235294,0)},
}
{
{(0.0343137,0)},{(0.0686275,0)},{(0.102941,0)},{(0.137255,0)},{(0.171569,0)},{(0.205882,0)},{(0.240196,0)},{(0.27451,0)},
}
{
{(0.0392157,0)},{(0.0784314,0)},{(0.117647,0)},{(0.156863,0)},{(0.196078,0)},{(0.235294,0)},{(0.27451,0)},{(0.313725,0)},
}
}

Obtain internal data

Get numpy.ndarray

Get the internal data of the density matrix and return it in the form of numpy.ndarray

Here is API doc for DensityMatrix.ndarray

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
def test():
    # Prepare a complex array
    data = np.array(
        [[0.18056413 + 0.j  , 0.12215589 + 0.06893592j ,- 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j , 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j ,- 0.251278 + 0.09109501j,0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j  ,0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
 
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
    # Obtain internal data
    data = dm.ndarray()
    # Print result
    print("data:",data)
 
if __name__ == "__main__":
    test()

Output

data: [[ 0.18056413+0.j          0.12215589+0.06893592j -0.05337882-0.06077268j
  -0.06164719-0.00112235j]
 [ 0.12215589-0.06893592j  0.38670196+0.j         -0.251278  -0.09109501j
   0.11033526+0.04016816j]
 [-0.05337882+0.06077268j -0.251278  +0.09109501j  0.26018725+0.j
  -0.14253993+0.02027215j]
 [-0.06164719+0.00112235j  0.11033526-0.04016816j -0.14253993-0.02027215j
   0.17254666+0.j        ]]

Get element

Get an element of internal data by matrix row and column index

Here is API doc for DensityMatrix.dim

Here is API doc for DensityMatrix.at

Python

from pyqpanda3.quantum_info import DensityMatrix,StateVector
import numpy as np
def test_dim_at():
    # Prepare a complex array
    data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
 
    print("element in dm:")
    for i in range(dm.dim()):
        for j in range(dm.dim()): # using dim function obtain dim information
            print(dm.at(i,j)) # using at function to obtain internal data
 
if __name__ == "__main__":
    test_dim_at()

Output

element in dm:
(0.18056413+0j)
(0.12215589+0.06893592j)
(-0.05337882-0.06077268j)
(-0.06164719-0.00112235j)
(0.12215589-0.06893592j)
(0.38670196+0j)
(-0.251278-0.09109501j)
(0.11033526+0.04016816j)
(-0.05337882+0.06077268j)
(-0.251278+0.09109501j)
(0.26018725+0j)
(-0.14253993+0.02027215j)
(-0.06164719+0.00112235j)
(0.11033526-0.04016816j)
(-0.14253993-0.02027215j)
(0.17254666+0j)

Purity

Obtain the purity of the density matrix

Here is API doc for DensityMatrix.purity

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
def test_purity():
    # Prepare a complex array
    data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
    print("dm:", dm)
    # Obtain the purity of the density matrix
    purity = dm.purity()
    # Print result
    print("purity:",purity)
 
if __name__ == "__main__":
    test_purity()

Output

dm: {
{
{(0.180564,0)},{(0.122156,0.0689359)},{(-0.0533788,-0.0607727)},{(-0.0616472,-0.00112235)},
}
{
{(0.122156,-0.0689359)},{(0.386702,0)},{(-0.251278,-0.091095)},{(0.110335,0.0401682)},
}
{
{(-0.0533788,0.0607727)},{(-0.251278,0.091095)},{(0.260187,0)},{(-0.14254,0.0202721)},
}
{
{(-0.0616472,0.00112235)},{(0.110335,-0.0401682)},{(-0.14254,-0.0202721)},{(0.172547,0)},
}
}
 
purity: (0.5515582694688418+0j)

StateVector

Obtain the corresponding state vector

Here is API doc for DensityMatrix.to_statevector

Python

from pyqpanda3.quantum_info import DensityMatrix
def test_to_statevector():
    # Construct a DensityMatrix object
    dm = DensityMatrix(3)
    # Obtain the corresponding state vector
    stv = dm.to_statevector()
    # Print result
    print("stv:",stv)
 
if __name__ == "__main__":
    test_to_statevector()

Output

stv: [(1+0j), 0j, 0j, 0j, 0j, 0j, 0j, 0j]

Evolution with QCircuit

Use quantum circuit QCircuit to evolve

Evolve without updating

Use the quantum circuit QCircuit to evolve the density matrix without updating the internal data of the original DensityMatrix object; The result of evolution is returned as a new DensityMatrix object

Here is API doc for DensityMatrix.evolve

Python

from pyqpanda3.quantum_info import DensityMatrix
from pyqpanda3.core import *
def test_evolve():
    # Construct a DensityMatrix object
    dm = DensityMatrix(3)
    # Construct a QCircuit object
    cir = QCircuit()
    cir << X(0)<<SWAP(1,2)<<CP(1,0,3.14)
    # evolve with qcircuit
    dm2 = dm.evolve(cir)
    # Print result
    print("dm2:",dm2)
 
if __name__ == "__main__":
    test_evolve()

Output

dm2: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(1,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
}

Evolve with updating

Use the quantum circuit QCircuit to evolve the density matrix and update the internal data of the original DensityMatrix object

Here is API doc for DensityMatrix.update_by_evolve

Python

from pyqpanda3.quantum_info import DensityMatrix,StateVector
from pyqpanda3.core import *
def test_update_by_evolve():
    # Construct a DensityMatrix object
    dm = DensityMatrix(3)
    # Construct a QCircuit object
    cir = QCircuit()
    cir << X(0) << SWAP(1, 2) << CP(1, 0, 3.14)
    # evolve with qcircuit
    dm.update_by_evolve(cir)
    # Print result
    print("dm:", dm)
 
if __name__ == "__main__":
    test_update_by_evolve()

Output

dm: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(1,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
}

Boolean function

Valid

Verify whether the internal data of the density matrix is legal The term "legal" here refers to whether it meets the requirements of being semi-positive definite and having a trace of 1

Here is API doc for DensityMatrix.is_valid

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
def test_is_valid():
    # Prepare a complex array
    data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
    # Print result
    print("is_valid:",dm.is_valid())
 
if __name__ == "__main__":
    test_is_valid()

Output

is_valid: True

Equal

Determine whether the internal data of two DensityMatrix are the same

Here is API doc

Python

from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
def test_eq():
    # Prepare a complex array
    data = np.array(
        [[0.18056413 + 0.j, 0.12215589 + 0.06893592j, - 0.05337882 - 0.06077268j, - 0.06164719 - 0.00112235j],
         [0.12215589 - 0.06893592j, 0.38670196 + 0.j, - 0.251278 - 0.09109501j, 0.11033526 + 0.04016816j],
         [-0.05337882 + 0.06077268j, - 0.251278 + 0.09109501j, 0.26018725 + 0.j, - 0.14253993 + 0.02027215j],
         [-0.06164719 + 0.00112235j, 0.11033526 - 0.04016816j, - 0.14253993 - 0.02027215j, 0.17254666 + 0.j]]
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
    # Construct another DensityMatrix object
    dm2 = dm
    # Print result
    print("dm==dm2:",dm==dm2)
 
if __name__ == "__main__":
    test_eq()

Output

dm==dm2: True