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# DensityMatrix

class DensityMatrix(data, dims=None)

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Bases: qiskit.quantum_info.states.quantum_state.QuantumState, qiskit.quantum_info.operators.mixins.tolerances.TolerancesMixin

DensityMatrix class

Initialize a density matrix object.

Parameters

• or (data (np.ndarray or list or matrix_like or QuantumCircuit) – qiskit.circuit.Instruction): A statevector, quantum instruction or an object with a to_operator or to_matrix method from which the density matrix can be constructed. If a vector the density matrix is constructed as the projector of that vector. If a quantum instruction, the density matrix is constructed by assuming all qubits are initialized in the zero state.
• dims (int or tuple or list) – Optional. The subsystem dimension of the state (See additional information).

Raises

QiskitError – if input data is not valid.

The dims kwarg can be None, an integer, or an iterable of integers.

• Iterable – the subsystem dimensions are the values in the list with the total number of subsystems given by the length of the list.
• Int or None – the leading dimension of the input matrix specifies the total dimension of the density matrix. If it is a power of two the state will be initialized as an N-qubit state. If it is not a power of two the state will have a single d-dimensional subsystem.

## Methods

### conjugate

DensityMatrix.conjugate()

Return the conjugate of the density matrix.

### copy

DensityMatrix.copy()

Make a copy of current operator.

### dims

DensityMatrix.dims(qargs=None)

Return tuple of input dimension for specified subsystems.

### draw

DensityMatrix.draw(output=None, **drawer_args)

Return a visualization of the Statevector.

repr: ASCII TextMatrix of the state’s __repr__.

text: ASCII TextMatrix that can be printed in the console.

latex: An IPython Latex object for displaying in Jupyter Notebooks.

latex_source: Raw, uncompiled ASCII source to generate array using LaTeX.

qsphere: Matplotlib figure, rendering of density matrix using plot_state_qsphere().

hinton: Matplotlib figure, rendering of density matrix using plot_state_hinton().

bloch: Matplotlib figure, rendering of density matrix using plot_bloch_multivector().

Parameters

• output (str) – Select the output method to use for drawing the state. Valid choices are repr, text, latex, latex_source, qsphere, hinton, or bloch. Default is repr. Default can be changed by adding the line state_drawer = <default> to ~/.qiskit/settings.conf under [default].
• drawer_args – Arguments to be passed directly to the relevant drawing function or constructor (TextMatrix(), array_to_latex(), plot_state_qsphere(), plot_state_hinton() or plot_bloch_multivector()). See the relevant function under qiskit.visualization for that function’s documentation.

Returns

matplotlib.Figure or str or TextMatrix or IPython.display.Latex: Drawing of the Statevector.

Raises

ValueError – when an invalid output method is selected.

### evolve

DensityMatrix.evolve(other, qargs=None)

Evolve a quantum state by an operator.

Parameters

• QuantumChannel (other (Operator or) – or Instruction or Circuit): The operator to evolve by.
• qargs (list) – a list of QuantumState subsystem positions to apply the operator on.

Returns

the output quantum state.

Return type

QuantumState

Raises

QiskitError – if the operator dimension does not match the specified QuantumState subsystem dimensions.

### expand

DensityMatrix.expand(other)

Return the tensor product state other ⊗ self.

Parameters

other (DensityMatrix) – a quantum state object.

Returns

the tensor product state other ⊗ self.

Return type

DensityMatrix

Raises

QiskitError – if other is not a quantum state.

### expectation_value

DensityMatrix.expectation_value(oper, qargs=None)

Compute the expectation value of an operator.

Parameters

• oper (Operator) – an operator to evaluate expval.
• qargs (None or list) – subsystems to apply the operator on.

Returns

the expectation value.

Return type

complex

### from_instruction

classmethod DensityMatrix.from_instruction(instruction)

Return the output density matrix of an instruction.

The statevector is initialized in the state $|{0,\ldots,0}\rangle$ of the same number of qubits as the input instruction or circuit, evolved by the input instruction, and the output statevector returned.

Parameters

instruction (qiskit.circuit.Instruction orQuantumCircuit) – instruction or circuit

Returns

the final density matrix.

Return type

DensityMatrix

Raises

QiskitError – if the instruction contains invalid instructions for density matrix simulation.

### from_int

static DensityMatrix.from_int(i, dims)

Return a computational basis state density matrix.

Parameters

• i (int) – the basis state element.
• dims (int or tuple or list) – The subsystem dimensions of the statevector (See additional information).

Returns

The computational basis state $|i\rangle\!\langle i|$.

Return type

DensityMatrix

The dims kwarg can be an integer or an iterable of integers.

• Iterable – the subsystem dimensions are the values in the list with the total number of subsystems given by the length of the list.
• Int – the integer specifies the total dimension of the state. If it is a power of two the state will be initialized as an N-qubit state. If it is not a power of two the state will have a single d-dimensional subsystem.

### from_label

classmethod DensityMatrix.from_label(label)

Return a tensor product of Pauli X,Y,Z eigenstates.

LabelStatevector

| "0" | $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ | | "1" | $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ | | "+" | $\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ | | "-" | $\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$ | | "r" | $\frac{1}{2}\begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$ | | "l" | $\frac{1}{2}\begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}$ |

Parameters

label (string) – a eigenstate string ket label (see table for allowed values).

Returns

The N-qubit basis state density matrix.

Return type

Statevector

Raises

QiskitError – if the label contains invalid characters, or the length of the label is larger than an explicitly specified num_qubits.

### is_valid

DensityMatrix.is_valid(atol=None, rtol=None)

Return True if trace 1 and positive semidefinite.

### measure

DensityMatrix.measure(qargs=None)

Measure subsystems and return outcome and post-measure state.

Note that this function uses the QuantumStates internal random number generator for sampling the measurement outcome. The RNG seed can be set using the seed() method.

Parameters

qargs (list or None) – subsystems to sample measurements for, if None sample measurement of all subsystems (Default: None).

Returns

the pair (outcome, state) where outcome is the

measurement outcome string label, and state is the collapsed post-measurement state for the corresponding outcome.

Return type

tuple

### probabilities

DensityMatrix.probabilities(qargs=None, decimals=None)

Return the subsystem measurement probability vector.

Measurement probabilities are with respect to measurement in the computation (diagonal) basis.

Parameters

• qargs (None or list) – subsystems to return probabilities for, if None return for all subsystems (Default: None).
• decimals (None or int) – the number of decimal places to round values. If None no rounding is done (Default: None).

Returns

The Numpy vector array of probabilities.

Return type

np.array

Examples

Consider a 2-qubit product state $\rho=\rho_1\otimes\rho_0$ with $\rho_1=|+\rangle\!\langle+|$, $\rho_0=|0\rangle\!\langle0|$.

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('+0')

# Probabilities for measuring both qubits
probs = rho.probabilities()
print('probs: {}'.format(probs))

# Probabilities for measuring only qubit-0
probs_qubit_0 = rho.probabilities([0])
print('Qubit-0 probs: {}'.format(probs_qubit_0))

# Probabilities for measuring only qubit-1
probs_qubit_1 = rho.probabilities([1])
print('Qubit-1 probs: {}'.format(probs_qubit_1))
probs: [0.5 0.  0.5 0. ]
Qubit-0 probs: [1. 0.]
Qubit-1 probs: [0.5 0.5]

We can also permute the order of qubits in the qargs list to change the qubit position in the probabilities output

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('+0')

# Probabilities for measuring both qubits
probs = rho.probabilities([0, 1])
print('probs: {}'.format(probs))

# Probabilities for measuring both qubits
# but swapping qubits 0 and 1 in output
probs_swapped = rho.probabilities([1, 0])
print('Swapped probs: {}'.format(probs_swapped))
probs: [0.5 0.  0.5 0. ]
Swapped probs: [0.5 0.5 0.  0. ]

### probabilities_dict

DensityMatrix.probabilities_dict(qargs=None, decimals=None)

Return the subsystem measurement probability dictionary.

Measurement probabilities are with respect to measurement in the computation (diagonal) basis.

This dictionary representation uses a Ket-like notation where the dictionary keys are qudit strings for the subsystem basis vectors. If any subsystem has a dimension greater than 10 comma delimiters are inserted between integers so that subsystems can be distinguished.

Parameters

• qargs (None or list) – subsystems to return probabilities for, if None return for all subsystems (Default: None).
• decimals (None or int) – the number of decimal places to round values. If None no rounding is done (Default: None).

Returns

The measurement probabilities in dict (ket) form.

Return type

dict

### purity

DensityMatrix.purity()

Return the purity of the quantum state.

### reset

DensityMatrix.reset(qargs=None)

Reset state or subsystems to the 0-state.

Parameters

qargs (list or None) – subsystems to reset, if None all subsystems will be reset to their 0-state (Default: None).

Returns

the reset state.

Return type

DensityMatrix

If all subsystems are reset this will return the ground state on all subsystems. If only a some subsystems are reset this function will perform evolution by the reset SuperOp of the reset subsystems.

### reverse_qargs

DensityMatrix.reverse_qargs()

Return a DensityMatrix with reversed subsystem ordering.

For a tensor product state this is equivalent to reversing the order of tensor product subsystems. For a density matrix $\rho = \rho_{n-1} \otimes ... \otimes \rho_0$ the returned state will be $\rho_0 \otimes ... \otimes \rho_{n-1}$.

Returns

the state with reversed subsystem order.

Return type

DensityMatrix

### sample_counts

DensityMatrix.sample_counts(shots, qargs=None)

Sample a dict of qubit measurement outcomes in the computational basis.

Parameters

• shots (int) – number of samples to generate.
• qargs (None or list) – subsystems to sample measurements for, if None sample measurement of all subsystems (Default: None).

Returns

sampled counts dictionary.

Return type

Counts

This function samples measurement outcomes using the measure probabilities() for the current state and qargs. It does not actually implement the measurement so the current state is not modified.

The seed for random number generator used for sampling can be set to a fixed value by using the stats seed() method.

### sample_memory

DensityMatrix.sample_memory(shots, qargs=None)

Sample a list of qubit measurement outcomes in the computational basis.

Parameters

• shots (int) – number of samples to generate.
• qargs (None or list) – subsystems to sample measurements for, if None sample measurement of all subsystems (Default: None).

Returns

list of sampled counts if the order sampled.

Return type

np.array

This function samples measurement outcomes using the measure probabilities() for the current state and qargs. It does not actually implement the measurement so the current state is not modified.

The seed for random number generator used for sampling can be set to a fixed value by using the stats seed() method.

### seed

DensityMatrix.seed(value=None)

Set the seed for the quantum state RNG.

### tensor

DensityMatrix.tensor(other)

Return the tensor product state self ⊗ other.

Parameters

other (DensityMatrix) – a quantum state object.

Returns

the tensor product operator self ⊗ other.

Return type

DensityMatrix

Raises

QiskitError – if other is not a quantum state.

### to_dict

DensityMatrix.to_dict(decimals=None)

Convert the density matrix to dictionary form.

This dictionary representation uses a Ket-like notation where the dictionary keys are qudit strings for the subsystem basis vectors. If any subsystem has a dimension greater than 10 comma delimiters are inserted between integers so that subsystems can be distinguished.

Parameters

decimals (None or int) – the number of decimal places to round values. If None no rounding is done (Default: None).

Returns

the dictionary form of the DensityMatrix.

Return type

dict

Examples

The ket-form of a 2-qubit density matrix $rho = |-\rangle\!\langle -|\otimes |0\rangle\!\langle 0|$

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('-0')
print(rho.to_dict())
{
'00|00': (0.4999999999999999+0j),
'10|00': (-0.4999999999999999-0j),
'00|10': (-0.4999999999999999+0j),
'10|10': (0.4999999999999999+0j)
}

For non-qubit subsystems the integer range can go from 0 to 9. For example in a qutrit system

import numpy as np
from qiskit.quantum_info import DensityMatrix

mat = np.zeros((9, 9))
mat[0, 0] = 0.25
mat[3, 3] = 0.25
mat[6, 6] = 0.25
mat[-1, -1] = 0.25
rho = DensityMatrix(mat, dims=(3, 3))
print(rho.to_dict())
{'00|00': (0.25+0j), '10|10': (0.25+0j), '20|20': (0.25+0j), '22|22': (0.25+0j)}

For large subsystem dimensions delimiters are required. The following example is for a 20-dimensional system consisting of a qubit and 10-dimensional qudit.

import numpy as np
from qiskit.quantum_info import DensityMatrix

mat = np.zeros((2 * 10, 2 * 10))
mat[0, 0] = 0.5
mat[-1, -1] = 0.5
rho = DensityMatrix(mat, dims=(2, 10))
print(rho.to_dict())
{'00|00': (0.5+0j), '91|91': (0.5+0j)}

### to_operator

DensityMatrix.to_operator()

Convert to Operator

### to_statevector

DensityMatrix.to_statevector(atol=None, rtol=None)

Return a statevector from a pure density matrix.

Parameters

• atol (float) – Absolute tolerance for checking operation validity.
• rtol (float) – Relative tolerance for checking operation validity.

Returns

The pure density matrix’s corresponding statevector.

Corresponds to the eigenvector of the only non-zero eigenvalue.

Return type

Statevector

Raises

QiskitError – if the state is not pure.

### trace

DensityMatrix.trace()

Return the trace of the density matrix.

## Attributes

### atol

Default value: 1e-08

Return data.

### num_qubits

Return the number of qubits if a N-qubit state or None otherwise.

### rtol

Default value: 1e-05

Return settings.