=======================
Mathematical Background
=======================

Mathematical Background of the DROPS Visualization
--------------------------------------------------

.. drops:: I1xI2y + i*I1yI2z
   :width: 80%
   :view: C170
   :aspect: 200%
   :align: center
   :caption:

It is not necessary to understand the mathematical details of the
mapping between quantum-mechanical operators and their DROPS
representation in order to use the DROPS representation to visualize
e.g. the state of a coupled spin system.  Similar to how it is not
necessary to understand the mathematical details of the mapping
between quantum-mechanical operators for two-level systems and their
vector representation via the expectation values of the spin operators
:pton:`I_x`, :pton:`I_y`, and :pton:`I_z` in order to use the
well-known vector picture to visualize the state of uncoupled spins.

Nonetheless, it may be interesting to learn more about the
mathematical background of this mapping and the connection to Wigner
representations.  In the following, we summarize some of the
underlying ideas.  For details beyond this summary, we refer to the
manuscript `Visualizing states and operators of coupled spins systems`
[GARON2015]_ and to A. Garon’s thesis `On a new visualization tool for
quantum systems and on a time-optimal control problem for quantum
gates` [GARON2014]_, and related Mathematica package [GARONG2014]_.

Wigner Functions of Single Spins
--------------------------------


.. image:: _images/page168_wigner.png


For a system consisting of just a single spin, it is always possible
to express a related operator as a linear combination of tensor
operators :math:`T_{jm}` with (in general complex) coefficients
:math:`c_{jm}`.  This is also true for spins other than 1/2, and
includes, for example, the :term:`density operator <Density Operator>`
encoding the state of the spin.

By using the natural mapping between :term:`spherical tensor operators
<Spherical Tensor Operator>` :math:`T_{jm}` and spherical harmonics
:math:`Y_{jm}`, is it possible to represent any operator :math:`A` by
a function :math:`f` on a sphere, which can have complex values and
which can be plotted as a three-dimensional shape: for a given point
:math:`s` of the unit sphere, the complex number :math:`f(s)` can be
written as :math:`r(s) e^{iφ(s)}` which can be represented by a point
with distance :math:`r(s)` from the origin and a color to represent
the phase :math:`φ(s)` (see :ref:`Color Code`).  If the operator
:math:`A` is a density operator, the function :math:`f` is called a
Wigner function.

DROPS Display and Wigner Representation of Coupled Spins
--------------------------------------------------------

For systems of coupled spins, an operator :math:`A` can be expressed
as a linear combination of :term:`tensor operators <Tensor Operator>`.
However, this basis contains multiple tensor operators with the same
rank :math:`j`, which precludes a one-to-one map to the spherical
harmonics :math:`Y_{jm}`.  The problem can be alleviated by grouping
the basis' tensor operators into discrete sets (labeled :math:`l`),
where tensor operators with rank :math:`j` appear not more than once
in each set :math:`l`.  As the tensor operators form a complete basis,
any operator :math:`A` can then also be decomposed into components
that correspond to these discrete sets :math:`l` of :term:`tensor
operators <Tensor Operator>`:

.. math::

   A = \sum_l A^{(l)}

Thus, analogously to the case of single spins, each component
:math:`A^{(l)}` can be mapped to a function :math:`f^{(l)}` on a
sphere and represented by a corresponding colored shape labeled
:math:`l`.

This is the basic idea of the :term:`DROPS` (**D**\ iscrete
**R**\ epresentation of spin **OP**\ erator\ **S**\ ) representation of operators.  The
individual shapes, :math:`l`, are called droplets.  If the operator
:math:`A` is the density operator, the corresponding set of functions
:math:`f^{(l)}` forms a generalized Wigner representation.

Mapping of Operator Components :math:`A^{(l)}` to Individual Droplets
---------------------------------------------------------------------

If an operator :math:`A` is decomposed as described above in the form
:math:`A = \sum_l A^{(l)}` then each of the operator components
:math:`A^{(l)}` can be represented by a function :math:`f^{(l)}` on a
sphere by mapping tensor operators :math:`T_{jm}` to spherical
harmonics :math:`Y_{jm}` as in the case of single spins:

.. image:: _images/page170_drops_mapping.png

(non-)Hermitian Tensor Operators and Real vs. Complex Spherical Harmonics
-------------------------------------------------------------------------

The tensor operators :math:`T_{jm}` have two forms: Hermitian and
non-Hermitian.  The [GARON2015]_ paper uses the non-Hermitian form.  A
transformation is applied to these non-Hermitian tensor operators
:math:`T_{jm}` as well as the complex spherical harmonics
:math:`Y_{jm}`, resulting in Hermitian tensor operators and `real
spherical harmonics
<https://en.wikipedia.org/wiki/Spherical_harmonics#Real_form>`_.

Using the real spherical harmonics greatly simplifies the calculation
of the spherical harmonic shapes for SpinDrops' display.  But it also
means that the tensor basis must be the Hermitian tensor basis,
deviating from the tensor basis used in the DROPS paper [GARON2015]_.

LISA Basis
----------

There is no unique assignment of tensor operators :math:`T_j` into
discrete sets :math:`l` constrained by the condition that a
:term:`tensor operators <Tensor Operator>` :math:`T_j` with rank
:math:`j` appears not more than once in each set.  Depending on the
application, different approaches to the assignment may be preferable.
For applications with distinguishable spins, the so-called LISA basis
[GARON2015]_ has a number of very favorable properties and provides an
intuitive DROPS visualization to represent NMR experiments
graphically.  It groups tensor operators based on **Li**\ nearity,
**S**\ ubsystems and **A**\ uxiliary (LISA) criteria, such as
:ref:`permutation symmetry <Spin Permutations>`.  Applying these
criteria constrains a unique grouping, which defines the tensor
operators of the basis up to their algebraic signs.  In the standard
LISA basis the signs are chosen to result in intuitive shapes
[GARON2015]_.  The LISA basis is the default basis used by SpinDrops
to draw the DROPS.

Plain-text entry of the LISA basis is described in detail in the
section :ref:`PTON for the LISA basis`.

Tensor Grouping in the LISA basis
=================================

In the standard LISA basis, the following criteria are used for the
grouping of tensor operators into discrete sets :math:`l` such that
each set contains not more than one tensor of the same rank :math:`j`.

Linearity
+++++++++

The first criterion for the grouping of tensors is the number
:math:`k` of involved spins, which is also called the linearity of an
operator.  For three-spin systems, the linearity :math:`k` can be 0,
1, 2, or 3.  Except for :math:`k=0`, the resulting groups still
contain several tensors with the same rank :math:`j`.

Subsystems
++++++++++

The second criterion is based on the subset :math:`K` of spins that
are involved in a tensor operator.  In a system consisting of three
spins, the subsets are, for linear operators :math:`(k=1)`:
:math:`\{I1\}`, :math:`\{I2\}`, and :math:`\{I3\}`, for bilinear
operators :math:`(k=2)`: :math:`\{I1,I2\}`, :math:`\{I1,I3\}`, and
:math:`\{I2,I3\}`.  In linear and bilinear subsystems, not more than
one tensor exists with the same rank :math:`j`, so these groups can
be simply labeled according to their associated spins, however, this
is not the case for the trilinear operators :math:`(k=3)`, i.e. for
trilinear tensors involving all spins :math:`\{I1, I2, I3\}`, thus the
introduction of :ref:`auxiliary criteria` and unique :math:`τ`
grouping labels.

Auxiliary Criteria
++++++++++++++++++

For :math:`k>2`, auxiliary criteria are needed in order to define
unique groupings of tensors.  In the LISA basis, after :ref:`subsystem
<subsystems>`, the next criterion for the grouping of tensors is based
on permutation symmetry.  For systems of up to five spins, this is
sufficient to define a unique and physically motivated grouping such
that all tensors within each group have different ranks :math:`j`.

Tensor Groups for 3-spin system
===============================

The :ref:`LISA grouping of tensors <Tensor Grouping in the LISA
basis>` is schematically summarized below.  For each linearity
:math:`k` and subsystem :math:`K`, the rank :math:`j` of the existing
tensor operators is indicated.  For the trilinear terms :math:`(k=3)`,
also the permutation symmetry is indicated (in terms of so-called
Young Tableaux, see [GARON2015]_, which is used as an auxiliary
criterion to define sets of tensors in which all ranks :math:`j` are
different.  The table also indicates the labels :math:`l` of the
resulting unique sets of tensor operators and of the corresponding
droplets in the DROPS representation.

+----------+----------------------+---------+-----------------+-------------------+----------------------------------+
|Linearity |Subsystem             |Rank     |Permutation      |Set Label          |:ref:`PTON <Plain Text Operator   |
|          |                      |         |Symmetry (ranks) |                   |Notation>`                        |
|:math:`k` |                      |:math:`j`|                 |:math:`l`          |                                  |
+----------+----------------------+---------+-----------------+-------------------+----------------------------------+
|0         |:math:`\emptyset`     |0        |                 |:math:`I_d`        |:code:`½E2`, :code:`½E4`,...      |
|          |                      |         |                 |                   |                                  |
+----------+----------------------+---------+-----------------+-------------------+----------------------------------+
|1         |:math:`{I_1}`         |1        |                 |:math:`I_1`        |:code:`{1}_11m`, :code:`{1}_10`,  |
|          |                      |         |                 |                   |...                               |
|          |:math:`{I_2}`         |1        |                 |:math:`I_2`        |                                  |
|          |                      |         |                 |                   |:code:`_{2}_11m`, :code:`_{2}_10`,|
|          |:math:`{I_3}`         |1        |                 |:math:`I_3`        |...                               |
|          |                      |         |                 |                   |                                  |
|          |                      |         |                 |                   |:code:`__{3}_11m`, ...            |
|          |                      |         |                 |                   |                                  |
+----------+----------------------+---------+-----------------+-------------------+----------------------------------+
|2         |:math:`{I_1,I_2}`     |2,1,0    |                 |:math:`\{I_1,I_2\}`|:code:`_{1,2}_00`, ...            |
|          |                      |         |                 |                   |                                  |
|          |:math:`{I_1,I_3}`     |2,1,0    |                 |:math:`\{I_1,I_3\}`|:code:`_{1,3}_00`, ...            |
|          |                      |         |                 |                   |                                  |
|          |:math:`{I_2,I_3}`     |2,1,0    |                 |:math:`\{I_2,I_3\}`|:code:`_{2,3}_00`,...             |
+----------+----------------------+---------+-----------------+-------------------+----------------------------------+
|3         |:math:`{I_1,I_2,I_3}` |3        |:math:`{\tau_1}` |:math:`\{\tau_1\}` |:code:`τ1_11m`, ...,              |
|          |                      |         |(3,1)            |                   |:code:`τ1_33p`                    |
|          |                      |         |                 |                   |                                  |
|          |                      |2,2      |:math:`{\tau_2}` |:math:`\{\tau_2\}` |                                  |
|          |                      |         |(2,1)            |                   |:code:`τ2_11m`, ...,              |
|          |                      |         |                 |                   |:code:`τ2_22p`                    |
|          |                      |1,1,1    |:math:`{\tau_3}` |:math:`\{\tau_3\}` |                                  |
|          |                      |         |(2,1)            |                   |                                  |
|          |                      |         |                 |                   |:code:`τ3_11m`,..., :code:`τ3_22p`|
|          |                      |0        |:math:`{\tau_4}` |:math:`\{\tau_4\}` |or :code:`\tau3_22p`              |
|          |                      |         |(0)              |                   |                                  |
|          |                      |         |                 |                   |                                  |
|          |                      |         |                 |                   |:code:`\tau4` or :code:`τ4`       |
+----------+----------------------+---------+-----------------+-------------------+----------------------------------+

Spin Permutations
=================

.. raw:: html

    <style> .orange {color:orange} </style>
    <style> .blue {color:#00BFFF} </style>

.. role:: orange


.. role:: blue


The trilinear operators corresponding to the sets :math:`τ_1`,
:math:`τ_2`, :math:`τ_3`, and :math:`τ_4` have characteristic
symmetries with respect to permutations of spin labels.  As illustrated
schematically below, the tensor operators in set :math:`τ_1` are
symmetric :orange:`(s)` with respect to any pairwise exchange of spin
labels, i.e. any exchange of spin labels leaves these operators
invariant.  Conversely, the operators in set :math:`τ_4` are
anti-symmetric :blue:`(a)` with respect to any pairwise exchange of
spin labels, i.e. any exchange of spin labels changes the sign of
these operators.  The tensor operators in set :math:`τ_2` are symmetric
:orange:`(s)` only with respect to the exchange of spin labels 1 and
2, whereas the tensor operators in set :math:`τ_3` are anti-symmetric
:blue:`(a)` with respect to the exchange of spin labels 1 and 2.

.. image:: _images/page174_lisa_symmetry.png


More Spin Permutations
======================

The symmetry of the droplets :math:`τ_1`, :math:`τ_2`, :math:`τ_3`,
and :math:`τ_4` with respect to permutations of spin labels is
illustrated for the operator :pton:`4I1xI2yI3z` (a). Permuting spins 1
and 2 results in the operator 
(b) :pton:`4I1yI2xI3z` and as expected, the droplets :math:`τ_1` and
:math:`τ_2`, which are symmetric under the (1,2) permutation, are
unchanged.  In contrast, the droplets :math:`τ_3` and :math:`τ_4`,
which are anti-symmetric with regard to this operation, change sign
(resulting in inverted colors).  The (1,3) permutation results in the
operator :pton:`4I3xI2yI1z` => :pton:`4I1zI2yI3x` (c) and the (2,3)
permutation results in the operator :pton:`4I1xI3yI2z` =>
:pton:`4I1xI2zI3y` (d). As discussed :ref:`above <Spin Permutations>`,
droplet :math:`τ_1` is also symmetric with respect to these
permutations and is identical to case (a). Droplet :math:`τ_4` is
anti-symmetric with respect to these permutations, resulting in a
color change from red to green.

.. table::
   :class: no-cap-num

   +--------------------+--------------------+--------------------+--------------------+
   |(a) :math:`()`      |(b) :math:`(1,2)`   |(c) :math:`(1,3)`   |(d) :math:`(2,3)`   |
   |                    |                    |                    |                    |
   |:drop:`/4%75^Cq120  |:drop:`/4%75^Cq120  |:drop:`/4%75^Cq120  |:drop:`/4%75^Cq120  |
   |4I1xI2yI3z`         |4I1yI2xI3z`         |4I1zI2yI3x`         |4I1xI2zI3y`         |
   +--------------------+--------------------+--------------------+--------------------+

Coherence Order
---------------

An operator :math:`A` has a well-defined coherence order :math:`p` if
a rotation around the z axis by an arbitrary angle α reproduces the
operator :math:`A` up to an additional phase factor :math:`exp(-ipα)`,
i.e. if


.. math::

   A
   \xrightarrow[ \text{around z Axis} ]
   {\text{rotation by angle }\alpha}
   e^{-ip\alpha} A


Similarly, a droplet representing a function :math:`f(l)` corresponds
to an operator term :math:`A(l)` with well-defined coherence order
:math:`p`, if a rotation around the z axis by an arbitrary angle
:math:`α` reproduces the droplet up to an additional phase factor
:math:`exp(-ipα)`.  For simplicity, in the following we drop the
superscript “:math:`(l)`”.  For example, let us consider the rotation
of the following droplet with :math:`p=+1` by :math:`π/2` (i.e. 90°)
around the z axis:

.. table::
   :class: no-cap-num nocol norow

   +-----------------------------+---------------------------+-------------------------------------+
   |:drop:`/3"" I1m`             |:math:`\xrightarrow[{\text{|:drop:`/3"" exp(-i*pi/2)*I1m`        |
   |                             |around z Axis}}]{          |                                     |
   |                             |\text{rotation by angle }  |                                     |
   |                             |\alpha=\frac{\pi}{2}}`     |                                     |
   |                             |                           |                                     |
   +-----------------------------+---------------------------+-------------------------------------+
   |:math:`f(s)=r(s)e^{i\phi(s)}`|                           |:math:`f'(s)=e^{ip\alpha}f(s) =      |
   |                             |                           |r(s)e^{i\phi'(s)}`                   |
   |                             |                           |                                     |
   |                             |                           |with :math:`φ'(s) = φ(s) - pα =      |
   |                             |                           |φ(s)-π/2`                            |
   +-----------------------------+---------------------------+-------------------------------------+

For a droplet with coherence order :math:`p`, the corresponding
function :math:`f(s) = r(s) e^{iφ(s)}` is transformed by a z rotation
with an angle α to :math:`f’(s) = r(s) e^{iφ’(s)}` with :math:`φ’(s) =
φ(s) - pα`.

To further illustrate this point, consider again :math:`p=+1` and
:math:`α=π/2`:

.. list-table::
   :class: no-cap-num nocol norow

   * - .. drops:: I1m
          :width: 40%
          :caption:
     - .. math::

         \xrightarrow[\text{ around z Axis}]{ \text{rotation by angle }                                       
         \alpha=\frac{\pi}{2}}
     - .. drops:: exp(-i*pi/2)*I1m
          :width: 40%
          :caption:
   * - :math:`\uparrow`
     - 
     - :math:`\uparrow`


After the rotation, the shape of the droplet is the same but the color
(representing the phase of the function :math:`f`) has changed.
Notice that the color of the point on the bottom of the drop (most
negative in the y direction) has changed from yellow
(:ref:`corresponding to <Color Wheel>` :math:`φ = π/2`) to red
(corresponding to :math:`φ’=0`), which is expected from the general
formula given above:

.. math::

   φ’(s) & = φ(s) - pα \\
         & = π/2 - (+1) (π/2) \\
         & = 0