Properties

Functions for calculating single-state properties like normalisation and purity. More...

Functions
qreal	calcPurity (Qureg qureg)
qreal	calcTotalProb (Qureg qureg)

Detailed Description

Functions for calculating single-state properties like normalisation and purity.

Function Documentation

calcPurity()

qreal calcPurity

(

Qureg

qureg

)

Calculates the purity of qureg, which is a measure of its mixedness.

Formulae

Let \(N\) be the number of qubits in qureg.

• When qureg is a density matrix \( \dmrho \) (as expected), this function returns

  \[ \tr{ \dmrho^2 } = \sum\limits_{i,j} \left| \dmrho_{ij} \right|^2 \]

  where \( \dmrho_{ij} \) is the \((i,j)\)-th element of \( \dmrho \).

  A purity of 1 indicates that the matrix is pure and can be expressed as

  \[ \dmrho \equiv \ketbra{\phi}{\phi} \]

  where \( \ket{\phi} \) is some pure state expressible as a statevector.

  In contrast, a purity less than 1 indicates the matrix is mixed and can be understood as a convex combination of multiple (at least two) pure states. That is,

  \[ \dmrho \equiv \sum\limits_n p_n \ketbra{\phi}{\phi}_n, \]

  where \(p_n \in [0,1]\) and sum to 1 whenever \(\dmrho\) is a valid and correctly normalised density matrix. Mixedness can result, for example, from Decoherence.

  The minimum purity of an \(N\)-qubit density matrix is \( 1/2^N \), which is admitted only by the maximally-mixed state \( \dmrho = \hat{\id} / 2^N \).

• When qureg is a statevector \( \svpsi \), this function returns

  \[ \tr{ \ketbra{\psi}{\psi} \; \ketbra{\psi}{\psi} } = \left( \sum\limits_i |\psi_i|^2 \right)^2 \]

  where \(\psi_i\) is the \(i\)-th amplitude of \(\svpsi\). This is always 1 for any valid statevector, and is otherwise equivalent to the output of calcTotalProb(), squared.

Constraints

• The output of this function is only a reliable measure of purity when qureg is correctly normalised. For example, an invalid density matrix can return a purity of 1, such as the \(N\)-qubit maximally-mixed state scaled by factor \( 2^N \). Note that the function calcTotalProb() alone cannot be used to validate validity since it only consults diagonal elements, whereas the purity is informed by all elements.

Equivalences

• When qureg is a valid density matrix (specifically, Hermitian), this function is faster than, but mathematically equivalent to, calling calcInnerProduct() and passing qureg twice.

  qcomp out = calcInnerProduct(qureg, qureg);
  qreal pur = real(out); // im=0

• When qureg is a statevector, this function returns the output of calcTotalProb(), squared.

Example

Qureg qureg = createDensityQureg(5);
initRandomPureState(qureg);
 
// = 1
qreal purity1 = calcPurity(qureg);
reportScalar("purity1", purity1);
 
mixTwoQubitDepolarising(qureg, 0, 1, 0.5);
 
// < 1
qreal purity2 = calcPurity(qureg);
reportScalar("purity2", purity2);

Parameters

[in]

qureg

the reference state, which is unchanged.

Returns

The purity of qureg.

Exceptions

error

if qureg is uninitialised.

Attention

This function's input validation has not yet been unit tested, so erroneous usage may produce unexpected output. Please use with caution!

See also

• calcFidelity()
• calcTotalProb()

Author

Tyson Jones

Definition at line 293 of file calculations.cpp.

                              {
    validate_quregFields(qureg, __func__);
 
    // assuming Hermiticity, Tr(rho^2) = sum_ij |rho_ij|^2,
    // and Tr(|x><x|x><x|) = (sum_i |x_i|^2)^2
    qreal prob = localiser_statevec_calcTotalProb(qureg);
    return (qureg.isDensityMatrix)? prob : prob * prob;
}

Referenced by TEST_CASE(), TEST_CASE(), TEST_CASE(), and TEST_CASE().

calcTotalProb()

qreal calcTotalProb

(

Qureg

qureg

)

Calculates the probability normalisation of the given qureg. This is the probability of the qureg being in any outcome state, which is expected to equal 1.

Formulae

Let \(N\) be the number of qubits in qureg.

• When qureg is a statevector \( \svpsi \) with \(i\)-th amplitude \(\psi_i\), this function returns

  \[ \sum\limits_{i=0}^{2^N-1} |\psi_i|^2. \]

• When qureg is a density matrix \( \dmrho \) with \(i\)-th diagonal element \( \dmrho_{ii} \), this function returns

  \[ \sum\limits_{i=0}^{2^N-1} \re{ \rho_{ii} } \]

Constraints

• As above, only the real components of the diagonal elements of a density matrix are consulted; these are the only amplitudes consulted by functions which calculate probabilities in the computational basis. As such, this function gives no indication of the general validity of density matrices, such as whether they are Hermitian, whether the diagonals are real, and whether the off-diagoanl elements are valid.

Equivalences

• This function is faster than, but mathematically equivalent to, summing the outputs of other functions which calculate probabilitie across all possible outcomes.

  // choice is arbitrary
  int qubit = 0;
   
  qreal totalProb = (
      calcProbOfQubitOutcome(qureg, qubit, 0) + 
      calcProbOfQubitOutcome(qureg, qubit, 1));

Example

Qureg qureg = createDensityQureg(5);
initRandomMixedState(qureg, 1<<5);
 
// differs from 1 by numerical error
qreal totalProb = calcTotalProb(qureg);

Parameters

[in]

qureg

the reference state, which is unchanged.

Returns

The probability normalisation of qureg.

Exceptions

error

if qureg is uninitialised.

Attention

This function's input validation has not yet been unit tested, so erroneous usage may produce unexpected output. Please use with caution!

See also

• calcPurity()
• calcProbsOfAllMultiQubitOutcomes()

Author

Tyson Jones

Definition at line 278 of file calculations.cpp.

                                 {
    validate_quregFields(qureg, __func__);
 
    return (qureg.isDensityMatrix)?
        localiser_densmatr_calcTotalProb(qureg): 
        localiser_statevec_calcTotalProb(qureg);
}

Referenced by setQuregToRenormalized(), TEST_CASE(), TEST_CASE(), TEST_CASE(), and TEST_CASE().