quick_start.nb

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Quick Start Guide for Mathematica Homological Toolkit for the Quantum Mechanic\
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The package BasicStable can be run independently, it provides a list of \
useful quantum mechanical operations.\
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Assuming this notebook is in the same folder with the package BasicStable, we \
begin by running the package by evaluating the cell below.\
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Cell["Kets, Density states, GHZ states, and W-states", "Subsection",
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One can create basic ket vectors for qubit systems, i.e. |0> is given by\
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We can work with qudit systems by specifying a dimension vector: i.e. the \
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The function pureToDensity helps us convert any (normalized) pure state into \
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The function tensor provides useful syntax for taking tensor products of \
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The GHZ state, W-state and their generalizations are implemented via the \
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These functions make sense for <=2 partite systems; for instance, ghzPure[2] \
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We can work with different ambient dimensions: the 3-partite GHZ state \
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We can embed this states into qudit systems by listing off the dimensions of \
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Cell["\<\
Permutations, Partial Traces, and Tensoring by identity operators\
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If we feel like permuting around tensor factors we can do so, the following \
takes in a pure state |01> , and applies the permutation (2,1) (taking the \
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We can also work with density states/matrices and change up the tensor factor \
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We can take partial traces: partialTrace[rho, systr, dimvect]  takes the \
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\[CloseCurlyQuote].  Better yet, let\[CloseCurlyQuote]s just call its \
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Instead of specifying the systems we want to trace over, we can also specify \
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We can also extend operators by tensoring by the identity at arbitrary \
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Cell["Let\[CloseCurlyQuote]s make up some matrices", "Text",
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I want to form 1_{2} \\otimes rr, i.e. tensor with a 1 in the first position \
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I want rr and mm to live on the first and third tensor factors of a 4 - \
partite system\
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We can compute the state index of a multipartite density state\
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The syntax treats stateIndex as a function of four parameters\
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We can also compute euler characteristics of associated GNS and commutant \
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Cell["\<\
The package StateHomologyStable contains all cohomological operations and \
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The syntax is easiest when dealing with qubits only: the first argument \
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The result is a list of dimensions, the first element of the list (read left \
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Instead of a list, we can output the Poincar\[EAcute] polynomial directly\
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If we are dealing with qudit states instead of qubit states; when all Hilbert \
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dimensions of the Hilbert spaces at the tensor factors\
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If we don\[CloseCurlyQuote]t want the Hilbert spaces at each tensor factor to \
have the same dimension, we can specify the list of dimensions as the second \
argument (the number of tensor factors is not necessary to specify as it is \
read off from the length of the list of dimensions).\
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Cell["\<\
One can even compute cohomologies of a coarsening of a multipartite state \
with respect to some partition; the partition is given as the third argument, \
after a listof the dimensions of each tensor factor (before the partition).\
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Cell["Here are some other fun polynomials to look at", "Text",
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Unfortunately, the brute force algorithm used to compute these quantities \
causes the computation time grow exponentially in the number of tensor \
factors.  You might find on a personal computer, (depending on the state \
under consideration and your computer) that above 10 or so tensor factors you\
\[CloseCurlyQuote]re just bringing the universe closer to heat death.\
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Using similar syntax to the above, we can also extract explicit \
generators/bases of cohomology components.  The following gives the \
generators of the degree 0 component of GNS cohomology of the tripartite GHZ \
state, presented as a a list, each of whose elements themselves are lists of \
of assignments of operators to each tensor factor\
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The following gives the generators of the degree 0 component of commutant \
cohomology of the tripartite GHZ state\
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