Inverse Participation Ratio (quanguru.QuantumToolbox.IPR)#

Contains functions to calculate delocalisation measure (Inverse participation ratio, shortly IPR) in various cases.

Functions#

iprKet(basis, ket)

Calculates inverse participation ratio 1/(\sum_{i}|c_{i,k}|^{4}) of a ket |k\rangle = \sum_{i}c_{i,k}|i\rangle in a given basis \{|i\rangle\}.

iprKetNB(ket)

Calculates the IPR 1/\sum_{i}|c_{i,k}|^{4} of a ket |k\rangle := \begin{bmatrix} c_{1,k} \\ \vdots \\ c_{i,k} \\ \vdots \\c_{\mathcal{D},k} \end{bmatrix}_{\mathcal{D}\times 1} by using each entry c_{i,k} as a complex amplitude.

Function Name

Docstrings

Examples

Unit Tests

Tutorials

iprKet

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iprKetNB

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iprKet(basis: List[Union[scipy.sparse._base.spmatrix, numpy.ndarray]], ket: Union[scipy.sparse._base.spmatrix, numpy.ndarray]) → float[source]#

Calculates inverse participation ratio 1/(\sum_{i}|c_{i,k}|^{4}) of a ket |k\rangle = \sum_{i}c_{i,k}|i\rangle in a given basis \{|i\rangle\}. The complex probability amplitudes satisfy \sum_{i}|c_{i,k}|^{2} = 1, therefore IPR = 1 is perfectly localised, and IPR = 1/\mathcal{D} is uniformly localised in \mathcal{D} dimensional space.

Parameters

  • basis (matrixList) – a ket state

  • ket (Matrix) – a complete basis

Returns

inverse participation ratio

Return type

float

Examples

>>> completeBasis = completeBasis(dimension=2)
>>> state0 = normalise(0.2*basis(2, 0) + 0.8*basis(2,1))
>>> iprKet(completeBasis, state0)
1.1245136186770428
>>> state1 = normalise(0.5*basis(2, 0) + 0.5*basis(2,1))
>>> iprKet(completeBasis, state1)
2.000000000000001
>>> state2 = basis(2,1)
>>> iprKet(completeBasis, state2)
1.0
iprKetNB(ket: Union[scipy.sparse._base.spmatrix, numpy.ndarray]) → float[source]#

Calculates the IPR 1/\sum_{i}|c_{i,k}|^{4} of a ket |k\rangle := \begin{bmatrix} c_{1,k} \\ \vdots \\ c_{i,k} \\ \vdots \\c_{\mathcal{D},k} \end{bmatrix}_{\mathcal{D}\times 1} by using each entry c_{i,k} as a complex amplitude.

Parameters

ket (Matrix) – a ket state

Returns

inverse participation ratio

Return type

float

Examples

>>> state0 = normalise(0.2*basis(2, 0) + 0.8*basis(2,1))
>>> iprKetNB(state0)
1.1245136186770428
>>> state1 = normalise(0.5*basis(2, 0) + 0.5*basis(2,1))
>>> iprKetNB(state1)
2.000000000000001
>>> state2 = basis(2,1)
>>> iprKetNB(state2)
1.0
>>> state3 = basis(2,0)
>>> iprKetNB(state3)
1.0

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