Boole-Bell-Type Inequalities in \mathematica
Boole-Bell-Type Inequalities in Mathematica
Stefan Filipp and Karl Svozil
Institut für Theoretische Physik,
Technische Universität Wien
Wiedner Hauptstraß e 8-10/136,
A-1040 Vienna, Austria
e-mail: svozil@tuwien.ac.at
Abstract
Classical Pitowsky correlation polytopes with particular emphasis on the Minkowski-Weyl representation theorem are reviewed.
A numerical study of the generation of optimal Boole-Bell type inequalities
for arbitrary experimental setups is presented.
These inequalities represent the faces of
classical Pitowsky correlation polytopes
[1,2], which are based on
the ``conditions of possible experience''
stated by George Boole in the 19th century[3].
We introduce CddIF, a Mathematica package created as an interface between
Mathematica and the cdd program
by Komei Fukuda[4], which represents a highly efficient
method to solve the hull problem
for general classical correlation polytopes.
Contents
1 Boole-Bell Type Inequalities
1.1 Simple urn model
1.2 Geometrical interpretation
1.3 Minkowski-Weyl representation theorem
1.4 From vertices to inequalities
1.5 From inequalities to vertices
1.6 Quantum mechanical context
2 Installation
2.1 Mathematica
2.2 cdd
2.3 Installation on windows-platforms
3 Description Of Functions
3.1 CddFormat
Description:
Example:
3.2 ToCddExtFile
Description:
Example:
3.3 TruthTable
Description:
Example:
3.4 RunCdd
Description:
Example:
3.5 ShowVRep
Description:
Example:
3.6 ConvToHRep
Description:
Example:
3.7 ReadInHRep
Description:
Example:
3.8 GetInequFromHRep
Description:
Example:
3.9 InequToRead
Description:
Example:
3.10 GetViolInequalities
Description:
Example:
3.11 PlotInequalities
Description:
Example:
3.12 ContPlotInequalities
Description:
Example:
3.13 Cdd
Description:
Example:
4 Examples
4.1 Three particles and two measurement directions
4.2 Violations of inequalities
4.3 Graphical representation
4.4 Two particles and three measurement directions
4.5 Violations of inequalities
4.6 Graphical representation
1 Boole-Bell Type Inequalities And Their Geometric Representation
In the middle of the 19th century the English mathematician George Boole
formulated a theory of "conditions of possible experience"
[5,3,6,1,7].
These conditions are related to relative frequencies of
logically connected
events and are expressed by certain equations or inequalities.
More recently, similar equations for a particular setup which are relevant in the
quantum mechanical context have been discussed by Clauser and Horne and others
[8,9,10].
Pitowsky has given a geometrical interpretation in terms of correlation polytopes
[11,1,2,7].
1.1 Simple urn model
Consider an urn containing some balls of different colors and styles:
Each ball can be described by two properties,
let us say ÿellow" and "wooden",
so each ball can have the property ÿellow" or the property "wooden",
but it can also have both - ÿellow and wooden".
The state of the urn can be given by the probabilities to draw a ball
with one of these properties: p1 is the proportion of yellow balls in the urn, p2
the proportion of wooden ones and p12 denotes
the proportion of yellow and wooden balls.
If there are enough balls in the urn these proportions are in fact the probabilities to get a ball with the special property by drawing. Clearly the inequalities
are fulfilled by the proportions and so p1, p2 and p12 can be seen as probabilities of two events and their joint event only if these inequalities are satisfied. Simply by taking some appropriate numbers (p1 = 0.6, p2=0.72 and p12=0.32) we can see, that equations (1) are not sufficient. If we take the probability to draw a ball which is either yellow or wooden (p1 + p2 - p12) into consideration, a new inequality can be found that is not satisfied by the numbers chosen:
It can be shown that the inequalities (1) and (2)
are necessary and sufficient for the numbers p1, p2 and p12 to represent
probabilities of two events and their joint [1].
1.2 Geometrical interpretation
Itamar Pitowsky
[11,1,2,7] has suggested a geometric interpretation.
Let us take the set of all numbers (p1, p2, p12)
satisfying the inequalities stated above as a
set of vectors in a three-dimensional real space.
This procedure yields a closed convex polytope with
vertices (0,0,0), (1,0,0), (0,1,0) and (1,1,1) (cf. Figure 1).
The extreme points (vertices) can be interpreted as follows:
(0,0,0) is a case where no yellow and no wooden balls are in the urn at all,
(1,0,0) is representing the configuration that all balls are yellow and no one is wooden.
(0,1,0) is representing the configuration that all balls are wooden and no one is yellow.
(1,1,1) is a case with only yellow and at the same time wooden balls.
Figure
Figure 1: Polytope associtated with the urn model
1.3 Minkowski-Weyl representation theorem
The Minkowski-Weyl representation theorem (e.g., [12]) states that compact convex sets
are ``spanned'' by their extreme points; and furthermore that the
representation of this polytope by the inequalities corresponding to the planes of their
faces is an equivalent one.
Stated differently, every convex polytope in an Euclidean space has a dual description:
either as the convex hull of its vertices (V-representation),
or as the intersection of a finite number of half-spaces,
each one given by a linear inequality (H-representation)
This equivalence is known as the Weyl-Minkowski theorem.
The problem to obtain all inequalities from the vertices of a convex
polytope is known as the hull problem. One solution strategy
is the Double Description Method [13] which we shall use but not review here.
1.4 From vertices to inequalities
For the above simple urn model, the inequalities are rather intuitive
and can be easily obtained by guessing.
This is impossible in the general case involving more events
and more joint probabilities thereof.
In order to obtain the relevant inequalities-Boole's
``conditions of possible experience''-we have to find a hopefully
constructive way to derive them.
Recall that a vector is an element of the polytope if and only if
it can be represented as a certain bounded convex combination,
i.e., a bounded linear span, of the vertices.
More precisely, let us denote the convex hull conv(K) of a finite set of points
K = {x1,¼,xn} Î Rd by
|
conv (K) = |
ì
í
î
|
l1 xi+¼+ln xn |
ê
ê
|
n ³ 1, li ³ 0, |
n
å
i = 1
|
li = 1 |
ü
ý
þ
|
. |
| (3) |
In the probabilistic context, the coefficients li are
interpreted as the probability that the event represented
by the extreme point xi occurs, whereby
K represents the complete set of all atoms of a Boolean algebra.
The geometric interpretation of K is the set of all extreme
points of the correlation polytope.
In summary, the connection between the convex hull of
the extreme points of a correlation polytope
and the inequalities representing its faces is guaranteed
by the Minkowski-Weyl representation theorem.
A constructive solution of the corresponding hull problem exists
(but is NP-hard [2]).
For the special urn model introduced above this means
that any three numbers (p1, p2 and p12) must fulfill an equation
dictated by Kolmogorov's probability axioms [14]:
|
(p1, p2,p12) = l1 (0,0,0) + l2 (0,1,0) + l3 (1,0,0) + l4 (1,1,1) = (l2 + l4, l3 + l4, l4). |
| (4) |
It is important here to realize that these logical possibilities are exhaustive,
That is, by definition,
there cannot be any other classical case which is not already
included in the above possibilities (0,0,0),(1,0,0),(0,1,0),(1,1,1).
the third ``component bit'' of the vector is a function of the first components.
Actually, the function is a multiplication, since we are dealing with
the classical logical ``and'' operation here.
(If one or more cases are omitted, the corresponding polytope would not be
optimal; i.e., it would be embedded in the optimal one.)
Therefore, any ``state'' of a physical system represented by a probability distribution
has to satisfy the constraint
The extreme cases li = 1, lj = 0 for i Î {1,2,3,4} and j ¹ i
just correspond to
the vertices spanning the classical correlation polytope as the convex sum (3).
A generalization to arbitrary configurations is straightforward.
To solve the hull problem for more general cases, an efficient algorithm has to be used.
There are some algorithms to solve this problem,
but they run in exponential time in the number of events,
thus it can be solved only for small enough cases to get a solution in conceivable time.
1.5 From inequalities to vertices
Conversely, a vector is an element of the convex polytope if and only
if its coordinates satisfy a set of linear inequalities which represent the
supporting hyper-planes of that polytope.
The problem to find the extreme points (vertices) of the polytope
from the set of inequalities is dual to the hull problem considered above.
1.6 Quantum mechanical context
In the quantum mechanical case the elementary irreducible
events are clicks in particle detectors
and the probabilities
have to be calculated using the formalism of quantum mechanics.
Figure
Figure 2: Experimental setting to test the violation of Boole - Bell type inequalities
As an example we take a source that produces pairs of spin-[1/2]
particles in a singlet-state
( |yñ = [1/(Ö2)](|¯ñ-|¯ñ)).
The particles fly apart along the z axis and after the particles have separated,
measurements on spin components along one out of two directions are made.
If, for simplicity, the measurements are made in the x-y plane perpendicular
to the trajectory of the particles, the direction of the measurement can be given by angles measured from the vertical x axis (a1 and a2 on the one side, b1 and b2 on the other side). On each side the measurement angle is chosen randomly for each pair of incoming particles and each measurement can yield two results - in [((h/2p) )/2] units: ``+1'' for spin up and ``-1''
for spin down (cf. Figure 2).
Deploying this configuration we get probabilities to find a particle measured
along the axis specified by the angles a1, a2, b1 and b2
either in spin up or in spin down state - denoted as pa1, pa2, pb1, pb2
- and we also take the joint event of finding a particle on one side at the
angle a1 (a2)
in a specific spin state and the other particle on the other side
along the vector b1 (b2) in a specific spin state,
denoted as pa1b1, pa2b1, pa1b2 and pa2b2. T
o construct the convex polytope to this experiment
we build up a truth table of all possible events using a ``1'' as ``spin up is detected along the specific axis'' and a ``0'' as ``spin down is detected along the specific axis'' (table 1).
| a1 | a2 | b1 | b1 | a1 b1 | a1 b2 | a2 b1 | a2 b2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 1: Truth table for four propositions
The rows of this table are then identified with the vertices of the convex polytope.
By using the Minkowski-Weyl theorem and by
solving the hull problem,
the vertices determine
the hyper-planes confining the polytope,
i.e. the inequalities which the probabilities have to satisfy.
As a result the following inequalities are gained:
|
|
|
0 £ paibi £ pai £ 1, 0 £ paibi £ pbi £ 1 |
| |
| |
|
|
| (6) |
|
|
|
-1 £ pa1b1 + pa 1b2 + pa2b2 - pa2b1 - pa1 - pb2 £ 0 |
|
|
-1 £ pa2b1 + pa2b2 + pa1b2 - pa1b1 - pa2 - pb2 £ 0 |
|
|
-1 £ pa1b2 + pa1b1 + pa2b1 - pa2b2 - pa1 - pb1 £ 0 |
|
|
-1 £ pa2b2 + pa2b1 + pa1b1 - pa1b2 - pa2 - pb1 £ 0 |
|
|
|
| (7) |
The last four inequalities are known as Clauser-Horne inequalities. As noticed above the probabilities have to be seen in a quantum mechanical context. If we consider the singlet state of spin-[1/2] particles |yñ = [1/(Ö2)](|¯ñ-|¯ñ) it is well known that the probability to find the particles both either in spin up or in spin down states is given by P(q) = P¯¯(q) = [1/2]sin2(q/2) - where q is the angle between the measurement directions. The single event probability is clearly pi = [1/2]. By choosing
|
a1 = - |
p
3
|
a2 = b1 = |
p
3
|
b2 = |
p
3
|
|
| (8) |
as measurement directions, we get for p = (pa1,pa2,pb1,pb2,pa1b1,pa2b1,pa1b2,pa2b2):
|
p = ( |
1
2
|
, |
1
2
|
, |
1
2
|
, |
1
2
|
, |
3
8
|
, |
3
8
|
,0, |
3
8
|
) |
| (9) |
and one of the inequalities found in (7) is violated:
|
pa1b1 + pa1b2 + pa2b2 - pa2b1 - pa1 - pb2 = |
3
8
|
+ |
3
8
|
+ |
3
8
|
- 0 - |
1
2
|
- |
1
2
|
= |
1
8
|
> 0 |
| (10) |
The generalization is straightforward.
Violations of certain inequalities involving classical probabilisties-Boole's
``conditions of possible experience'' [3]-also appear in higher dimensions in
configurations containing more particles and/or more measurement directions.
We shall consider more examples below.
2 Installation
2.1 Mathematica
All functions described in the following section can be found in
the Mathematica-package cddif.m.
In general this package has to be loaded into the
current Mathematica-kernel by the command
<<'path to cddif.m'/cddif.m, short
description and usage of the functions is available by entering ?'<function>'.
To guarantee a proper run of all functions it is necessary (and hopefully sufficient)
that cdd is located in any directory listed in the PATH-variable (usually /bin, /usr/bin, /usr/local/bin, ...)1 or in the current working directory, which can be shown by evaluating Directory[] or changed using the function SetDirectory [ directory_String] . If you like to avoid the frequent use of this function you can append this command to the package-file cddif.m before the line End[] so that on each loading of the package the directory is set automatically to your personal working directory.
cdd is a C++ (ANSI C) implementation of the Double Description Method
[13] for generating all vertices (i.e. extreme points)
and extreme rays of a general convex polyhedron given by a system of
linear inequalities and the dual problem of generating a system of
linear inequalities given all vertices by Komei Fukuda[4].
At this point I only refer to the documentation of the program for
the installation of the cdd - package, in particular to the file cdd.readme
included in the package and to the online documentation at
http://www.ifor.math.ethz.ch/~afukuda/cddman/cddman.html. cdd is available for free and you can download it from the homepage of
Komei Fukuda[4] (http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html), where you can also find a manual to the usage of cdd, especially descriptions to the format of the input- and output-files and to options that can be passed to cdd.
2.3 Installation on windows-platforms
cdd is available to compile on UNIX-like systems, to run cdd on Windows - systems you can either try to compile it in Windows2
or look somewhere else, if anybody has already done this work .
Currently a version runnable on Windows platforms of cdd can be downloaded from
http://tph.tuwien.ac.at/~svozil/cdd/cdd.exe.
(a different compilation is at http://www.wis.kuleuven.ac.be/wis/algebra/kathleen/files/cdd061.exe).
On Windows-systems Mathematica must be able to find cdd in a directory listed in the PATH-variable or in the current working directory, too. To set the PATH-variable in Windows 2000/NT go to the ``control panel'' and click on the ``system properties'', then click ``advanced'' and there is a place where the variable PATH is specified. Here you can add your path to cdd (separated by a semicolon). In WindowsME you need to go execute ``msconfig'' to get to the System Configuration Utility - in ``Environment'' you can edit the PATH-variable and for Windows98/95 you must edit the file ``autoexec.bat'' to get the path set.
Finally it can be necessary to rename the cdd - executable file (e.g. from cdd061.exe) to cdd.exe. The CddIF-Package uses cdd as default command to run cdd, using the function SetCddCmd [ cmd_String] you can change this behavior. Like already stated above you can also add this commandline to the package-file cddif.m just before the End[]-statement to change the default command automatically when loading the package.
3 Description Of Functions
In this section all functions of the CddIF - package are listed. For each function the syntax including the necessary parameters (if parameters are optional, it has an `` opt: '' as prefix), a description and an example is given.
3.1 CddFormat
CddFormat [ vertices, opt: options]
- vertices (List):
- List of m vertices in n dimensions of the form
{{x11,x12,...,x1n},{x21,...,x2n},...,{xm1,xm2,...,xmn}}
- options (List):
- Options to cdd (e. g. adjacency, nondegenerate, minindex,...) - see documentation to cdd (http://www.ifor.math.ethz.ch/ fukuda/cddman/cddman.html)
Description: A list of vertices, which can be determined for example by TruthTable [..., IncludeVars®False] , are converted to a format recognized by cdd. Additionally options to cdd can be declared.
Example:
- In[]=
- CddFormat [ {{1,0,0},{0,1,0},{1,1,1}}]
- Out[]=
- {V-representation begin,{3, 4, integer},
{1, 1, 0, 0}, {1, 0, 1, 0}, {1,1, 1, 1},
end}
3.2 ToCddExtFile
ToCddExtFile [ file, vertices, opt: options] or
ToCddExtFile [ file, particles, measurements, opt: options]
- file (String):
- Filename for output of H-representation (``.ext''-suffix is automatically appended)
- vertices (List):
- List of m vertices in n dimensions of the form
{{x11,x12,...,x1n},{x21,...,x2n},...,{xm1,xm2,...,xmn}}
- options (List):
- Options to cdd (e. g. adjacency, nondegenerate, minindex,...) - see documentation to cdd (http://www.ifor.math.ethz.ch/ fukuda/cddman/cddman.html)
- particles (Integer):
- Number of particles
- measurements (Integer):
- Number of possible measurements to each particle (equivalent to number of detection angles)
Description: Creates a file with ``.ext''-extension that contains the data of the given configuration to use in cdd. Eighter a list of vertices of the considered correlation polytop or the number of particles used and the possible measurements to each can be handed over. In the latter case the list of vertices is generated automatically.
Example:
- In[]=
- ToCddExtFile [ ``test'',2,3]
- Out[]=
- test.ext
writes the file ``test.ext'' to the current working directory, containing the vertices of the 2-particles 3-measurements configuration.
3.3 TruthTable
TruthTable [ particles, measurements, opt: options]
- particles (Integer):
- Number of particles
- measurements (Integer):
- Number of possible measurements to each particle (equivalent to number of detection angles)
- opt: options:
- The only possible option is IncludeVars ® True/False. If IncludeVars ® True is defined, the function includes a list of variables belonging to the given configuration as titles of the columns and output will be made in MatrixForm, otherwise a list containing all vertices is returned. Default is IncludeVars ® True.
Description: Creates a truth table of the given configuration, containing all vertices of the corresponding correlation polytopes. For generating this table all possible single events are rated eighter 0 or 1 (i. e. true or false) and the joint events are evaluated using the logical AND operation.
Example:
- In[]=
- TruthTable [ 2,2]
- Out[]=
-
3.4 RunCdd
RunCdd [ file]
- file (String):
- File handed over to cdd as command parameter (automatically extended with ``ext.''-suffix.
Description: RunCdd [...] executes cdd (with file as parameter) and returns the corresponding H-representation, which can be used in various other functions like PlotInequalities [... ] or GetViolInequalities [... ] .
Using this function you have to pay attention to the potentially long runtime in calculating the faces (i. e. the inequalities) of the correlation polytope. It can be more beneficial to use ToCddExtFile [ file,...] to create a ``.ext''-file, followed by executing cdd outside of Mathematica (eventually on a faster computer) to convert the date to H-representation stored in an ``.ine''-file. Afterwards you can read in this file utilizing ReadInHRep [ file]
Example:
- In[]=
- RunCdd [ ``test'']
- Out[]=
- {{H-representation},{begin},{684,16,real},{2,0,-2,....},....,{end}},
whereas 2-particles 3-measurement configuration is taken into consideration here.
3.5 ShowVRep
ShowVRep [ particles, measurements]
- particles (Integer):
- Number of particles
- measurements (Integer):
- Number of possible measurements to each particle (equivalent to number of detection angles)
Description: Shows the V-representation of a given configuration.
Example:
- In[]=
- ShowVRep [ 2,2]
- Out[]=
- {V-representation
begin,
{16, 9, integer},
{1, 0, 0, 0, 0, 0, 0, 0,0},{1, 1, 0, 0, 0, 0, 0, 0, 0},
{1, 0, 1, 0, 0, 0, 0, 0, 0},{1, 1, 1, 0, 0, 0, 0, 0, 0},
{1, 0, 0, 1, 0, 0, 0, 0, 0},{1, 1, 0, 1, 0, 1, 0, 0, 0},
{1, 0, 1, 1, 0, 0, 0, 1, 0},{1, 1, 1, 1, 0, 1, 0, 1, 0},
{1, 0, 0, 0, 1, 0, 0, 0, 0},{1, 1, 0, 0, 1, 0, 1, 0, 0},
{1, 0, 1, 0, 1, 0, 0, 0, 1},{1, 1, 1, 0, 1, 0, 1, 0, 1},
{1, 0, 0, 1, 1, 0, 0, 0, 0},{1, 1, 0, 1, 1, 1, 1, 0, 0},
{1, 0, 1, 1, 1, 0, 0, 1, 1},{1, 1, 1, 1, 1, 1, 1, 1, 1},
end}
3.6 ConvToHRep
ConvToHRep [ particles, measurements, opt: file, opt: options] or
ConvToHRep [ vertices, opt: file, opt: options]
- particles (Integer):
- Number of particles
- measurements (Integer):
- Number of possible measurements to each particle (equivalent to number of detection angles)
- vertices (List):
- List of m vertices in n dimensions of the form
{{x11,x12,...,x1n},{x21,...,x2n},...,{xm1,xm2,...,xmn}}
- file (String):
- Filename that is used for the conversion from a ``.ext''-file to a ``.ine''-file which is equivalent to a conversion from V-representation to H-representation. Default is ``tmp''.
- options (List):
- Options to cdd (e. g. adjacency, nondegenerate, minindex,...) - see documentation to cdd(http://www.ifor.math.ethz.ch/ fukuda/cddman/cddman.html)
Description: This function converts a given configuration (n particles, m Measurements) or a given list of vertices from V-representation into a H-representation. Like above3.4 the potentially long calculation time has to be taken into consideration, depending on the complexity of the problem.
Example:
- In[]=
- ConvToHRep [ 2,3,''2_3'']
- Out[]=
- {{H-representation},{begin},{684, 16,real},{2, 0,...}...,{end}},
wheras in this case the files ``2_3.ext'' (created by Mathematica containing the data for cdd) and ``2_3.ine'' (created by cdd as result of the calculation) are generated in the current working directory.
3.7 ReadInHRep
ReadInHRep [ file]
- file (String):
- ''.ine'' - file containing the H-representation which is to be read in.
Description: Reads the H-representation from a given ``.ine''-file for further use in various functions like GetViolInequalities [... ] or PlotInequalities [... ] .
Example:
- In[]=
- ReadInHRep [ ``2_3'']
- Out[]=
- {{H-representation},{begin},{684, 16,real},{2, 0,...}...,{end}}
3.8 GetInequFromHRep
GetInequFromHRep [ hrep]
- hrep (List):
- H-representation yielded for example from the function ReadInHRep [... ] or ConvToHRep [... ] .
Description: Returns the inequalities from a given H-representation as a list. To make the list more readable you can apply InequToRead [... ] on it.
Example:
- In[]=
- GetInequFromHRep [ConvToHRep [2,1] ]
- Out[]=
- {{a1 - a1b1 + b1, 1}, {-a1 + a1b1, 0}, {a1b1 - b1, 0}, {-a1b1, 0}}
3.9 InequToRead
InequToRead [ inequalities]
- inequalites (List):
- List of inequalities yielded from GetInequFromHRep [... ]
Description: Makes the list of inequalities yielded from GetInequFromHRep [... ] more readable.
Example:
- In[]=
- InequToRead [GetInequFromHRep [ConvToHRep [2,1] ] ]
- Out[]=
-
3.10 GetViolInequalities
GetViolInequalities [ hrep, angles, functions, inequ-nr, violation] or
GetViolInequalities [ file, angles, functions, inequ-nr, violation, opt: options]
- hrep (List):
- H-representation yielded for example from the function ReadInHRep [... ] or ConvToHRep [... ] .
- file (String):
- File containing the demanded H-representation in cdd format.
- angles (List):
- List of measurement angles for each particle, whereas the dimension of the list must represent the configuration. If you choose the configuration ``2 particles - 2 measurements'' the list must have the dimension [2,2], in this case the particles a and b are measured each along two axis given by the angles a1, a2, b1 and b2, so this parameter has the form {{a1 , a2} , {b1 , b2}}.
- functions (Symbol):
- Functions to calculate the quantum mechanical probability of the events. Considering for example two spin-[1/2] particles in a singlett state, the probability to find the particles both either in spin ``up'' or both in spin ``down'' states is given by P(q,f) = P¯¯(q,f) = [1/2]sin2[[((q- f))/2]], where q and f are the measurement angles of the particles.
In defining these functions you have to notice, that for all possible events (single events, two-particle-events,...) an apropriate function definition has to exist, each taking a list as parameter (e.g. Prob[{x_,y_}]=[1/2]Sin[(x - y)/2]2).
- inequ-nr (List):
- Range of rows in H-representation used for checking violated inequalities. Specifying this can be useful, if many inequalities have to be evaluated. The form of the parameter is {min,max} oder All.
- violation (Real):
- Only inequalities are printed out, that are violated more than this parameter. Default value is 0.
- opt: options:
- Options can be PrintOut ® True/False, which specifies, if the inequalities are printed out during evaluation or not.
Description: GetViolInequalities [... ] calculates the discrepancy of inequalities using the given probability functions and therefore the quantum mechanical violation for a distinct adjustment (i. e. special angles) of the detectors and returns all violated inequalities.
Example:
- In[]=
- GetViolInequalities [ConvToHRep [2,2] ,{{-[(p)/6],0},{0,[(p)/6]}},Prob]
- Out[]=
- (-a1b1+a1b2+a2-a2b1-a2b2+b1 £ 1 [9/8])
The probability for the single event of measuring one particle in spin ``up'' or spin ``down'' at any angle is given by Prob[{x_}]=[1/2] and the probability of the joint event has been calculated by Prob[{x_,y_}]=[1/2]Sin[(x - y)/2]2.
3.11 PlotInequalities
PlotInequalities [ hrep, range, angles, functions, opt: options] or
PlotInequalities [ file, range, angles, functions, inequ-nr, violation, opt: options]
- hrep (List):
- H-representation yielded for example from the function ReadInHRep [... ] or ConvToHRep [... ] .
- file (String):
- File containing the demanded H-representation in cdd format.
- range (List):
- Parameter specifying the range for the variable x, which is plotted on the horizontal axis. It has the form {x , xmin , xmax}(see Mathematica function Plot [... ] )
- angles (List):
- List of measurement angles for each particle, whereas the dimension of the list must represent the configuration. If you choose the configuration ``2 particles - 2 measurements'' the list must have the dimension [2,2], in this case the particles a and b are measured each along two axis given by the angles a1, a2, b1 and b2, so this parameter has the form {{a1 , a2} , {b1 , b2}}.
- functions (Symbol):
- Functions to calculate the quantum mechanical probability of the events. Considering for example two spin-[1/2] particles in a singlett state, the probability to find the particles both either in spin ``up'' or both in spin ``down'' states is given by P(q,f) = P¯¯(q,f) = [1/2]sin2[[((q- f))/2]], where q and f are the measurement angles of the particles.
In defining these functions you have to notice, that for all possible events (single events, two-particle-events,...) an apropriate function definition has to exist, each taking a list as parameter (e.g. Prob[{x_,y_}]=[1/2]Sin[(x - y)/2]2).
- inequ-nr (List):
- Range of rows in H-representation used for checking violated inequalities. Specifying this can be useful, if many inequalities have to be evaluated. The form of the parameter is {min,max} oder All.
- violation (Real):
- Only inequalities are printed out, that are violated more than this parameter. Default value is 0.
- opt: options:
- Options for the Mathematica- function Plot [... ] can be handed over.
Description: This function yields a plot of (violated) inequalities, whereas the function plotted is f(x) = p(x)-b derived from the inequalites of the form p(x) £ b (p(x) is a linear combination of functions to calculate the probabilities of (joint) events, dependent on one variable). Consequently the degree of violation is represented as a positive value of f(x).
Take for example the case ``2 particles - 2 measurement directions'', where the inequality
|
-pa1b1+pa1b2+pa2-pa2b1-pa2b2+pb1 £ 1 |
|
appears. The probability for the single event of measuring one particle in spin ``up'' or spin ``down'' at any angle is given by
|
pa1(x) = pa2(x) = pb1(x) = pb2(x) = |
1
2
|
|
|
and the probability of the joint event can be calculated by
|
pa1b1(x,y) = pa2b1(x,y) = pa1b1(x,y) = pa2b2(x,y) = |
1
2
|
sin[(x - y)/2]2. |
|
If we define the measurement angles by
|
a1 = - |
p
3
|
+x a2 = b1 = 0 b2 = 2*p |
|
the inequality can be written as
|
1+ |
1
2
|
sin |
1
2
|
(- |
p
3
|
-x)²- |
sinx²
2
|
- |
1
2
|
sin |
1
2
|
(- |
p
3
|
+x)² £ 1 |
|
The left side is dependent on x (p(x) = 1+[1/2]sin[1/2](-[(p)/3]-x)²-[sinx²/2]-[1/2]sin[1/2](-[(p)/3]+x)²) and b = 1.
The function to be plotted is f(x) = p(x)-b, thus
|
f(x) = 1+ |
1
2
|
sin |
1
2
|
(- |
p
3
|
-x)²- |
sinx²
2
|
- |
1
2
|
sin |
1
2
|
(- |
p
3
|
+x)² - 1 |
|
Example:
- In[]=
- PlotInequalities [ConvToHRep [2,2] ,{x,0,p},{{-[(p)/3]+x,0},{0,2 x}},Prob]
- Out[]=
-
Figure
The functions to calculate the probabilities have been defined as Prob[{x_}]:=[1/2] for a single event and Prob[{x_,y_}]:=[1/2]Sin[(x - y)/2]2 for two-particle events.
3.12 ContPlotInequalities
ContPlotInequalities [ hrep, rangex, rangey, angles, func, ineq-nr, violation, opt: options] or
ContPlotInequalities [ file, rangex, rangey, angles, func, ineq-nr, violation, opt: options]
- hrep (List):
- H-representation yielded for example from the function ReadInHRep [... ] or ConvToHRep [... ] .
- rangex (List):
- Parameter specifying the range for the variable x, which is plotted on the horizontal axis. It has the form {x , xmin , xmax}(see Mathematica function Plot [... ] )
- rangex (List):
- Parameter specifying the range for the variable y, which is plotted on the vertical axis. It has the form {y , ymin , ymax}(see Mathematica function Plot [... ] )
- angles (List):
- List of measurement angles for each particle, whereas the dimension of the list must represent the configuration. If you choose the configuration ``2 particles - 2 measurements'' the list must have the dimension [2,2], in this case the particles a and b are measured each along two axis given by the angles a1, a2, b1 and b2, so this parameter has the form {{a1 , a2} , {b1 , b2}}.
- functions (Symbol):
- Functions to calculate the quantum mechanical probability of the events. Considering for example two spin-[1/2] particles in a singlett state, the probability to find the particles both either in spin ``up'' or both in spin ``down'' states is given by P(q,f) = P¯¯(q,f) = [1/2]sin2[[((q- f))/2]], where q and f are the measurement angles of the particles.
In defining these functions you have to notice, that for all possible events (single events, two-particle-events,...) an apropriate function definition has to exist, each taking a list as parameter (e.g. Prob[{x_,y_}]=[1/2]Sin[(x - y)/2]2).
- inequ-nr (List):
- Range of rows in H-representation used for checking violated inequalities. Specifying this can be useful, if many inequalities have to be evaluated. The form of the parameter is {min,max} oder All.
- violation (Real):
- Only inequalities are printed out, that are violated more than this parameter. Default value is 0.
- opt: options:
- Options for the Mathematica-function Plot [... ] can be handed over.
Description: Like the function PlotInequalities [... ] ContPlotInequalities [... ] yields a graphical representation of the violation of Boole-Bell type inequalities, but in this case the functions are dependant on two variables: f(x,y) = p(x,y)-b derived from p(x,y) £ b, where p(x,y) is a linear combination of functions to calculate the probability for single or joint events. Like in the description of the PlotInequalities [... ] -function we take the ``2 particles - 2 measurement directions'', the only difference is the selection of the measurement angles:
|
a1 = x a2 = b1 = 0 b2 = y |
|
Thus the function f(x,y) is now given by
|
f(x,y) = 1- |
1
2
|
sin |
x
2
|
²+ |
1
2
|
sin |
x-y
2
|
²- |
1
2
|
sin |
y
2
|
² - 1 |
|
and can be plotted as contour plots. A higher level of violation is represented by a darker contour layer.
Example:
- In[]=
- ContPlotInequalities [ConvToHRep [2,2] ,{x,0,p},{y,0,p},{{x,0},{0,y}},Prob,All]
- Out[]=
-
Figure Figure
The functions to calculate the probabilities have been defined as Prob[{x_}]:=[1/2] for a single event and Prob[{x_,y_}]:=[1/2]Sin[(x - y)/2]2 for two-particle events.
Cdd [ values, opt: command]
- values (List):
- Data handed over to cdd in an input-file. Each element of the list must be a string and represents a row in the output-file.
- opt: command (String):
- Command to be executed on the generated output-file (default is ``cdd'' - the default value can be changed utilizing the function SetCddCmd [ newcommand_String] ). Here you can specify for example ``cddf'' or ``cddr''.
Description: Simple Interface to run cdd in Mathematica. The current version cannot distinguish, whether cdd has produced correct output or not, so you have to pay attention while using this function.
Example:
- In[]=
- Cdd [ {``H-Representation'',``begin'',``6 4 real'',``2 -1 0 0'',``2 0 -1 0'',``-1 1 0 0'',``-1 0 1 0'',``-1 0 0 1'',``4 -1 -1 0'',``end''}]
- Out[]=
-
{{``*'', ``cdd:'', ``Double'', ``Description'', ``Method'', ``C-Code:Version'', ``0.61b'',
``(November'', 29, ``1997)''}, {``*'', ``Copyright'', ``(C)'', 1996, ``Komei'',
``Fukuda,'', ``fukuda@ifor.math.ethz.ch''}, {``*Input'', ``File:tmp.ine'', ``('', 6,
``x'', ``4)''}, {``*HyperplaneOrder:'', ``LexMin''}, {``*Degeneracy'',
``preknowledge'', ``for'', ``computation:'', ``None'', ``(possible'',
``degeneracy)''}, {``*Vertex/Ray'', ``enumeration'', ``is'',
``chosen.''}, {``*Computation'', ``completed'', ``at'', ``Iteration'',
6.}, {``*Computation'', ``starts'', ``at'', ``Thu'', ``Mar'', 22, ``18:48:36'',
2001}, {``*'', ``terminates'', ``at'', ``Thu'', ``Mar'', 22, ``18:48:36'',
2001}, {``*Total'', ``processor'', ``time'', ``='', 0, ``seconds''}, {``*'', ``='', 0,
``hour'', 0, ``min'', 0, ``sec''}, {``*FINAL'', ``RESULT:''}, {``*Number'', ``of'',
``Vertices'', ``='', 4, ``Rays'', ``='', 1}, {``V-representation''}, {``begin''}, {5,
4, ``real''}, {1, 2, 1, 1}, {1, 1, 1, 1}, {1, 1, 2, 1}, {1, 2, 2, 1}, {0, 0,
0, 1}, {``end''}, {``hull''}}
4 Examples
These two examples and are originally solved in a paper by Pitowsky and Svozil [15].
The associated Mathematica - notebooks are ``3_2.nb''
(three particles - 2 measurement directions) and ``2_3.nb''
(two particles - three measurement directions)
4.1 Three particles and two measurement directions
In this configuration three particles (a, b and c) are measured in detectors which can be switched between two angles each. Consequently there are six different propositions for single particle events: a1, a2, b1, b2, c1, c2, supposing that a1 is the detection (i. e. the click in a counter) of the particle a in the detector set along the axis specified by the first angle for particle a, b2 the detection of particle b at the second angle for this particle, and so on ... (cf. Figure 3).
Figure
Figure 3: Setting for "2 particles - 3 angles"
If we also take two and three particle events into account (for example the event a2c1 means a click in the counter for particle a at the second angle AND a click in the counter for particle c at the first angle), there are 26 different events:
a1, a2, b1, b2, c1, c2, a1b1, a1b2, a2b1, a2b2, a1c1, a1c2, a2c1, a2c2, b1c1, b1c2, b2c1, b2, c2,a1b1c1, a1b1c2, a1b2c1, a1b2c2, a2b1c1, a2b1c2, a2b2c1, a2b2c2
4.2 Violations of inequalities
The truth table for this configuration can be obtained utilizing the function TruthTable [ 3,2] 3, executing ConvToHRep [3,2] yielded the appropriate H-representation, but this would last quite long, due to the complexity of the correlation polytope for this setting (there are 53856 hyper-planes limiting the polytope). Because of this fact trying to read in the H-representation created by cdd (using ReadInHRep [... ] ) could also result in memory resource problems by Mathematica.
To avoid this symptoms it is suggested to export the list of vertices and apply cdd outside of Mathematica to the file containing the list of vertices (V-representation). This can be done by invoking ToCddExtFile [ ``3_2'',3,2] , which creates a file ``3_2.ext''. This file can be handed over to cdd as parameter to get the file ``3_2.ine'' comprising the H-representation of the correlation polytope (Command: ``cdd 3_2.ext'').
Now the search for violated inequalities can begin using the function GetViolInequalities [ ... ] :
May be accepted that the functions to calculate the quantum probabilities of the (joint) events (Prob) have been defined by Prob[{x_}]:=[1/2], Prob[{x_,y_}]:=[1/4] and Prob[{x_,y_,z_}]:=[1/8]*(1-Sin[x+y+z]), where x, y and z are the angles used for detection of each particle,
GetViolInequalities [``3_2.ine'',{{0,[(p)/2]},{0,[(p)/2]},{0,[(p)/2]}},Prob,All,0.4]
yields:
| {{-3 a1+2 a1b1+a1b1c1-4 a1b1c2+3 a1b2-3 a1b2c1-
|
|
| a1b2c2+a1c1+3 a1c2+2 a2b1-2 a2b1c1-a2b1c2-
|
|
| 2 a2b2+a2b2c1+3 a2b2c2+a2c1-a2c2-2 b1+
|
|
| b1c1+2 b1c2+b2c1-2 b2c2-c1 £ 0,0.5}, |
|
| {-2 a1+2 a1b1+a1b1c1-4 a1b1c2+2 a1b2-2 a1b2c1- |
|
| a1b2c2+a1c1+2 a1c2+3 a2b1-3 a2b1c1-a2b1c2-
|
|
| 2 a2b2+a2b2c1+3 a2b2c2+a2c1-2 a2c2-3 b1+
|
|
b1c1+3 b1c2+b2c1-b2c2-c1 £ 0,0.5},
|
|
| {-2 a1+a1b1+a1b1c1-4 a1b1c2+2 a1b2-3 a1b2c1-
|
|
| a1b2c2+2 a1c1+2 a1c2+2 a2b1-3 a2b1c1- |
|
| a2b1c2-a2b2+a2b2c1+2 a2b2c2+a2c1-a2c2- |
|
| 2 b1+2 b1c1+2 b1c2+b2c1-b2c2-2 c1 £ 0,0.5},
|
|
| {-2 a1+2 a1b1+a1b1c1-4 a1b1c2+2 a1b2-3 a1b2c1- |
|
| a1b2c2+3 a1c2+2 a2b1-3 a2b1c1-a2b1c2- |
|
| 2 a2b2+a2b2c1+2 a2b2c2+a2c1-2 b1+2 b1c1+
|
|
| 2 b1c2+b2c1-b2c2-c1-c2 £ 0,0.5},
|
|
| {-a1+a1b1+a1b1c1-4 a1b1c2+a1b2-3 a1b2c1- |
|
| a1b2c2+a1c1+2 a1c2+a2+a2b1-2 a2b1c1- |
|
| a2b1c2-2 a2b2+a2b2c1+3 a2b2c2-a2c2-b1+ |
|
| b1c1+2 b1c2+b2+b2c1-2 b2c2-c1 £ 1,1.5},
|
|
| {-a1+2 a1b1-a1b1c1-3 a1b1c2+a1b2-4 a1b2c1+
|
|
| a1b2c2+3 a1c1+a2+a2b1-2 a2b1c1-a2b1c2- |
|
| 2 a2b2+a2b2c1+3 a2b2c2-a2c2-2 b1+2 b1c1+
|
|
| 2 b1c2+b2+2 b2c1-3 b2c2-2 c1+c2 £ 1,1.5},
|
|
| {-2 a1+2 a1b1-a1b1c1-3 a1b1c2+a1b2-2 a1b2c1-
|
|
| a1b2c2+2 a1c1+2 a1c2+a2+a2b1-4 a2b1c1+
|
|
| a2b1c2-2 a2b2+a2b2c1+3 a2b2c2+2 a2c1-3 a2c2-
|
|
| b1+3 b1c1+b2-b2c2-2 c1+c2 £ 1,1.5}
|
|
| {...}...}
|
4.3 Graphical representation
Using PlotInequalities [... ] a graph can be created showing the violation of inequalities dependent on one variable. Defining the probability functions as above, executing
PlotInequalities [ ``3_2.ine'',{x,0,p},{{0,x},{0,x},{0,x},Prob,{10000,20000},0.4]
yields the following plot (cf. Figure 4), whereas the corresponding H-representation has to be stored in the file ``3_2.ine'', {10000,20000} indicates the range of row numbers taken for calculating the graph and 0.4 is the minimal degree of violation to include the inequality in the graph:
Figure
Figure 4: PlotInequalities[``3_2.ine'',{x,0,p},{{0,x},{0,x},{0,x},Prob,{10000,20000},0.4]
To display inequalities dependent on two variables the function ContPlotInequalities [... ] is provided. This function shows the violation as a contour plot, a more violated set of detection angles results in a darker region in the plot.
ContPlotInequalities [ ``3_2.ine'',{x,0,p},{x,0,p},{{0,x},{0,y},{x,y}},Prob,{10000,20000},0.4]
returns ContourGraphics-objects, which can be displayed for example by executing
Show [GraphicsArray [Partition [ cont, 3[[ {1,2},All]]] ] ] // TableForm
(see figure 5).
Figure Figure
Figure Figure
Figure 5: ContPlotInequalities[``3_2.ine'',{x,0,p},{x,0,p},{{0,x},{0,y},{x,y}},Prob,{10000,20000},0.4]
Show[GraphicsArray[Partition[cont,3][[{1,2},All]] // TableForm
4.4 Two particles and three measurement directions
In the case of two particles (a and b) with three properties (whereas the properties are three different angles of the detectors for each particle denoted by a1, a2, a3, b1, b2, b3 - see figure 6) 15 different events can be found:
{a1, a2, a3, b1, b2, b3, c1, c2, c3, a1b1, a1b2, a1b3, a2b1, a2b2, a2b3, a3b1, a3b2, a3b3}
Figure
Figure 6: Setting for "2 particles - 3 angles"
4.5 Violations of inequalities
Using TruthTable [2,3] all vertices of the corresponding correlation polytope can be found - we get a dimension of 15 and 64 vertices as result (table 2).
| a1 | a2 | a3 | b1 | b2 | b3 | a1b1 | a1b2 | a1b3 | a2b1 | a2b2 | a2b3 | a3b1 | a3b2 | a3b3 |
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0cm20ptTruth table for 6 propositions*
|
Executing hrep=ConvToHRep [2,3] a H-representation of the polytope described by the truth table above can be created. This results in 684 hyper-planes respectively 684 inequalities from the 64 vertices limiting the polytope:
| {{H-representation}, {begin},{684, 16, real}, |
|
| {2, 0, -2, 1, -1, 0, -1, 1, -1, 0, 1, 1, 1, -1, -1, 1}, |
|
| {2, -2, 0, 1, -1, 0, -1, 1, 1, 1, 1, -1, 0, -1, -1, 1}, |
|
| {...}, |
|
| {1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, |
|
| {end},{Konfiguration,2,3}}
|
All inequalities can be displayed by
GetInequFromHRep [hrep] // InequToRead
The result of this operation is
| -a1b1 + a1b2 + 2 a2 - a2b1 - a2b2 - a2b3 - a3 + a3b1 + a3b2 - a3b3 + b1 + b3 £ 2 |
|
| 2a1 - a1b1 - a1b2 - a1b3 - a2b1 + a2b2 - a3 + a3b1 + a3b2 - a3b3 + b1 + b3 £ 2 |
|
| a1 - a1b1 - a1b3 - a2b1 + a2b2 + a2b3 - a3 + a3b1 + a3b2 - a3b3 + b1 - b2 £ 1 |
|
| ... |
|
| ... |
|
| a1- a1b1 + b1 £ 1
|
Using
GetViolInequalities [hrep,{{0,[(2p)/3],[(4p)/3]},{{0,[(2p)/3],[(4p)/3]}},Prob,All] // TableForm
all inequalities can be displayed that are violated at the specific angles a1 = b1 = 0, a2 = b2 = [(2p)/3] and a3 = b3 = [(4p)/3] taking the functions Prob[{x_}]:=[1/2] and Prob[{x_ ,y_}]:=Sin[[(x-y)/2]]/2 to calculate the quantum probabilities, which is equivalent to the probability to find two spin-[1/2] particles in a singlet state ( |yñ = [1/(Ö2)](|¯ñ- |¯ñ)) both either in spin ``up'' or both in spin ``down''.
|
|
|
-a1 - a1b1 + a1b2 + a1b3 - a2 + a2b1 + a2b3 + a3b1 + a3b2 - a3b3 - b1 - b2 £ 0 |
| |
| |
|
-a1 - a1b1 + a1b2 + a1b3 + a2b1 - a2b2 + a2b3 - a3 + a3b1 + a3b2 - b1 - b3 £ 0 |
| |
|
|
|
-a1 + a1b2 + a1b3 - a2 + a2b1 - a2b2 + a2b3 + a3b1 + a3b2 - a3b3 - b1 - b2 £ 0 |
| |
|
|
-a1b1 + a1b2 + a1b3 - a2 + a2b1 - a2b2 + a2b3 - a3 + a3b1 + a3b2 - b2 - b3 £ 0 |
| |
|
|
-a1 + a1b2 + a1b3 + a2b1 - a2b2 + a2b3 - a3 + a3b1 + a3b2 - a3b3 - b1 - b3 £ 0 |
| |
|
|
-a1b1 + a1b2 + a1b3 - a2 + a2b1 + a2b3 - a3 + a3b1 + a3b2 - a3b3 - b2 - b3 £ 0 |
| |
|
|
-a1 + a1b2 + a1b3 - a2b2 + a2b3 - b3 £ 0 |
| |
|
|
-a1b1 + a1b3 - a2 + a2b1 + a2b3 - b3 £ 0 |
| |
|
|
-a1b1 + a1b2 - a3 + a3b1 + a3b2 - b2 £ 0 |
| |
|
|
-a1 + a1b2 + a1b3 + a3b2 - a3b3 - b2 £ 0 |
| |
|
|
a2b1 - a2b2 - a3 + a3b1 + a3b2 - b1 £ 0 |
| |
|
|
-a2 + a2b1 + a2b3 + a3b1 - a3b3 - b1 £ 0 |
| |
|
4.6 Graphical representation
Like in the configuration ``three particles - two angles'' described above a graphical representation can be generated either dependent on one or dependent on two variables.
On the one hand side by using
PlotInequalities [hrep,{x,0,p},{{0,x,[(4p)/3]},{{0,x,[(4p)/3]}},Prob]
we get a plot of all violated inequalities (cf. Figure 7),
Figure
Figure 7: PlotInequalities[hrep,{x,0,p},{{0,x,[(4p)/3]},{{0,x,[(4p)/3]}},Prob]
on the other hand side contour plots of all inequalities violated for example more than 0.2 can be generated by executing
cont=ContPlotInequalities [hrep,{x,0,p},{y,0,p},{{0,x,y},{{0,x,y}},Prob,All,0.2]
To display the outcome of the calculation entering Show /@ cont results in ContourPlots of the following form (cf. Figure 8):
Figure Figure
Figure Figure
Figure 8: ContPlotInequalities[hrep,{x,0,p},{y,0,p},{{0,x,y},{{0,x,y}},Prob,All,0.2]
Show /@ %
References
- [1]
-
Itamar Pitowsky.
Quantum Probability-Quantum Logic.
Springer, Berlin, 1989.
- [2]
-
Itamar Pitowsky.
Correlation polytopes their geometry and complexity.
Mathematical Programming, 50:395-414, 1991.
- [3]
-
George Boole.
On the theory of probabilities.
Philosophical Transactions of the Royal Society of London,
152:225-252, 1862.
- [4]
-
Komei Fukuda.
cdd program.
2000.
- [5]
-
George Boole.
An investigation of the laws of thought.
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- [6]
-
Theodore Hailperin.
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foundations of mathematics ; 85).
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Itamar Pitowsky.
George Boole's `conditions od possible experience' and the
quantum puzzle.
Brit. J. Phil. Sci., 45:95-125, 1994.
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-
Itamar Pitowsky.
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Conceptions of the Universe. Kluwer, Dordrecht, 1989.
- [12]
-
Günter M. Ziegler.
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- [13]
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T.S. Motzkin, H. Raiffa, G.L. Thompson, and R.M. Thrall.
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University Press, New Jersey, Princeton, 1953.
- [14]
-
A. N. Kolmogorov.
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English translation in [16].
- [15]
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Itamar Pitowsky and Karl Svozil.
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-
A. N. Kolmogorov.
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Footnotes:
1for setting environment variables look at the manual of set or env
2the manual says: ``The program cdd.c is written in ANSI C, and thus it should run on personal computers without any changes if one uses a compiler supporting ANSI standard.''
3due to lack of space not listed here, but it can be found in the Mathematica- notebook 3_2.nb
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