/homological-tools-4QM-octave

Primary LanguageMATLABGNU General Public License v2.0GPL-2.0

Octave Cohomological Toolbox for the Quantum Mechanic

License: GPLv2 or later

License: GPL v2

Description

This is a set of Octave functions built to compute the cohomology of the cochain complexes introduced in the paper 'Homological Tools for the Quantum Mechanic' (arXiv:1901.02011).

For software that can compute ranks and explicit bases of cohomology components in Mathematica, see: https://github.com/tmainiero/homological-tools-4QM-mathematica.

How to Download

Git

git clone https://github.com/tmainiero/homological-tools-4QM-octave.git

From the Github web interface

Click that fancy green "Clone or download" button on the top right! Then "Download ZIP".

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From the Github web interface:

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  2. Locate the "Raw" button (On the top right at the time of writing) and right click.
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Descriptions of Basic Functions

To compute a list of ranks of cohomology groups

  1. gnscohomrk(psi,N,dims)

  2. ecohomrk(psi,N,dims)

Computes the dimensions of the GNS (case 1) or commutant (case 2) cohomology vector spaces of the state psi (pure or mixed) of an N-partite system. The output is a dimension vector of the form [H^0, H^1,..., H^(N-1)]. where H^(i) is the rank of the ith GNS/commutant complex. This is equivalently the list of coefficients of the associated Poincare polynomial.

  • psi can be either:

    1. A row vector representing a pure state, or a density state: e.g. psi = [1,0,0,0]
    2. A density state (a self-adjoint square matrix), e.g. psi = [1,0,0,0;0,0,0,0;0,0,0,0;0,0,0,0,0].

  • N is the number of tensor factors/primitive subsystems (the "partiteness").

  • dims is an optional argument giving the dimension vector: the list of dimensions of Hilbert spaces at the tensor factors. It must be a vector of length N. If this argument is left empty, it will be assumed all subsystems are dimension 2.

The inputs psi, N, and dims must be sensible, i.e. psi must be able to be represented as an honest N-partite (density) state on primitive subsystems with dimension vector [d_1,...d_N]: i.e. if dims is empty: it must be written as either an 1 x 2^(N) (row) vector or a 2^(N) x 2^(N) square matrix; if dims=[d_1, ..., d_N], then it must be a length d_1d_2...d_N row vector or a d_1d_2...d_N square matrix.

Basic State Manipulation

ket(basis,dims)

Ket vector given the standard computational basis on a multipartite system: outputs the vector |basis(1)> \otimes |basis(2)> \otimes \cdots \otimes |basis(n)> = |basis(1) ... basis(N)>. Output is a row vector in Octave.

  • dims is an optional row vector specifying the Hilbert space dimension of each primitive subsystem. If this argument is left empty it is assumed that there are N (= length of basis) subsystems of dimension 2.

  • basis is a list of non-negative integers [b_1,b_2,...b_N], where b_i runs from 0 to dims(i) - 1. Here b_k represents the standard basis element |b_i> of \mathbb{C}^(dims(i)) with a 1 in the (d_k - 1)^(th) position, and zeros elsewhere.

ghz(N,dims)

The (unnormalized) GHZ density state given by (|00....0> + |11...1>)(<00...0| + <11..1|) for an N-partite qubit system. (This is an unnormalized density state associated to the Bell state |00> + |11> when N=2).

-N is the number of primitive subsystems.

-dims is an optional dimension vector of length N of the ambient system: assumed to be a list of 2's when empty.

wst(N,dims)

The (unnormalized) W density state given by (|0....01> + |0...10> + ...+ |1...0>)(<|0....01| + <0...10| + ...+ <1...0|) for an N-partite qubit system. (This is an unnormalized density state associated to the Bell state |01> + |10> when N=2).

-N is the number of primitive subsystems.

-dims is an optional dimension vector of length N of the ambient system: assumed to be a list of 2's when empty.

tensor(a,b,...)

Returns the kronecker product of its arguments. To produce the state psi = |01> for instance, we would write psi = tensor([0,1],[1,0]).

This function is taken from Toby Cubitt.

Acknowledgements:

Basic linear algebraic operations: partialtrace.m, tensor.m, and syspermute.m are taken from Toby Cubitt's Matlab software (c.f. http://www.dr-qubit.org/matlab.html) licensed under GPLv2.