ecterrab

A Complete Guide for Tensor computation...

Posted: ecterrab 14837 Product: Maple

January 23 2019

7 1

A Complete Guide for performing Tensors computations using Physics

This is an old request, a complete guide for using Physics to perform tensor computations. This guide, shown below with Sections closed, is linked at the end of this post as a pdf file with all the sections open, and also as a Maple worksheet that allows for reproducing its contents. Most of the computations shown are reproducible in Maple 2018.2.1, and a significant part also in previous releases, but to reproduce everything you need to have the Maplesoft Physics Updates version 283 or higher installed. Feedback one how to improve this presentation is welcome.

Physics is a package developed by Maplesoft, an integral part of the Maple system. In addition to its commands for Quantum Mechanics, Classical Field Theory and General Relativity, Physics includes 5 other subpackages, three of them also related to General Relativity: Tetrads , ThreePlusOne and NumericalRelativity (work in progress), plus one to compute with Vectors and another related to the Standard Model (this one too work in progress).

The presentation is organized as follows. Section I is complete regarding the functionality provided with the Physics package for computing with tensors in Classical and Quantum Mechanics (so including Euclidean spaces), Electrodynamics and Special Relativity. The material of section I is also relevant in General Relativity, for which section II is all devoted to curved spacetimes. (The sub-section on the Newman-Penrose formalism needs to be filled with more material and a new section devoted to the EnergyMomentum tensor is appropriate. I will complete these two things as time permits.) Section III is about transformations of coordinates, relevant in general.

For an alphabetical list of the Physics commands with a brief one-line description and a link to the corresponding help page see Physics: Brief description of each command .

I. Spacetime and tensors in Physics

This section contains all what is necessary for working with tensors in Classical and Quantum Mechanics, Electrodynamics and Special Relativity. This material is also relevant for computing with tensors in General Relativity, for which there is a dedicated Section II. Curved spacetimes .

	Default metric and signature, coordinate systems

Tensors, their definition, symmetries and operations

Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of General Relativity. In addition, one of the strengths of Physics is that you can define tensors, in natural ways, by indicating a matrix or array with its components, or indicating any generic tensorial expression involving other tensors.

In Maple, tensor indices are letters, as when computing with paper and pencil, lowercase or upper case, latin or greek, entered using indexation, as in , and are displayed as subscripts as in . Contravariant indices are entered preceding the letter with ~, as in , and are displayed as superscripts as in . You can work with two or more kinds of indices at the same time, e.g., spacetime and space indices.

To input greek letters, you can spell them, as in mu for , or simpler: use the shortcuts for entering Greek characters . Right-click your input and choose Convert To → 2-D Math input to give to your input spelled tensorial expression a textbook high quality typesetting.

Not every indexed object or function is, however, automatically a tensor. You first need to define it as such using the Define command. You can do that in two ways:

1.	Passing the tensor being defined, say , possibly indicating symmetries and/or antisymmetries for its indices.

2.	Passing a tensorial equation where the left-hand side is the tensor being defined as in 1. and the right-hand side is a tensorial expression - or an Array or Matrix - such that the components of the tensor being defined are equal to the components of the tensorial expression.

After defining a tensor - say or - you can perform the following operations on algebraic expressions involving them

•	Automatic formatting of repeated indices, one covariant the other contravariant

•	Automatic handling of collisions of repeated indices in products of tensors

•	Simplify products using Einstein's sum rule for repeated indices.

•	SumOverRepeatedIndices of the tensorial expression.

•	Use TensorArray to compute the expression's components

•	TransformCoordinates .

If you define a tensor using a tensorial equation, in addition to the items above you can:

•	Get each tensor component by indexing, say as in or

•	Get all the covariant and contravariant components by respectively using the shortcut notation and .

•	Use any of the special indexing keywords valid for the pre-defined tensors of Physics; they are: definition, nonzero, and in the case of tensors of 2 indices also trace, and determinant.

•	No need to specify the tensor dependency for differentiation purposes - it is inferred automatically from its definition.

•	Redefine any particular tensor component using Library:-RedefineTensorComponent

•	Minimizing the number of independent tensor components using Library:-MinimizeTensorComponent

•	Compute the number of independent tensor components - relevant for tensors with several indices and different symmetries - using Library:-NumberOfTensorComponents .

The first two sections illustrate these two ways of defining a tensor and the features described. The next sections present the existing functionality of the Physics package to compute with tensors.

	Defining a tensor passing the tensor itself

	Defining a tensor passing a tensorial equation

	Automatic formatting of repeated tensor indices and handling of their collisions in products

	Tensor symmetries

	Substituting tensors and tensor indices

	Simplifying tensorial expressions

	SumOverRepeatedIndices

	Visualizing tensor components - Library:-TensorComponents and TensorArray

	Modifying tensor components - Library:-RedefineTensorComponent

	Enhancing the display of tensorial expressions involving tensor functions and derivatives using CompactDisplay

	The LeviCivita tensor and KroneckerDelta

	The 3D space metric and decomposing 4D tensors into their 3D space part and the rest

	Total differentials, the d_[mu] and dAlembertian operators

	Tensorial differential operators in algebraic expressions

	Inert tensors

	Functional differentiation of tensorial expressions with respect to tensor functions

	The Pauli matrices and the spacetime Psigma[mu] 4-vector

	The Dirac matrices and the spacetime Dgamma[mu] 4-vector

	Quantum not-commutative operators using tensor notation

II. Curved spacetimes

Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of general relativity, respectively entered and displayed as: Einstein[mu,nu] = , Ricci[mu,nu] = , Riemann[alpha, beta, mu, nu] = , Weyl[alpha, beta, mu, nu], = , and the Christoffel symbols Christoffel[alpha, mu, nu] = and Christoffel[~alpha, mu, nu] = respectively of first and second kinds. The Tetrads and ThreePlusOne subpackages have other predefined related tensors. This section is thus all about computing with tensors in General Relativity.

	Loading metrics from the database of solutions to Einstein's equations

	Setting the spacetime metric indicating the line element or a Matrix

	Covariant differentiation: the D_[mu] operator and the Christoffel symbols

	The Einstein, Ricci, Riemann and Weyl tensors of General Relativity

	A conversion network for the tensors of General Relativity

	Tetrads and the local system of references - the Newman-Penrose formalism

	The ThreePlusOne package and the 3+1 splitting of Einstein's equations

	III. Transformations of coordinates

	See Also
	Physics , Conventions used in the Physics package , Physics examples , Physics Updates

Download Tensors_-_A_Complete_Guide.mw, or the pdf version with sections open: Tensors_-_A_Complete_Guide.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Overview of the Physics Updates...

Posted: ecterrab 14837 Product: Maple

January 11 2019

8 0

Overview of the Physics Updates

One of the problems pointed out several times about the Physics package documentation is that the information is scattered. There are the help pages for each Physics command, then there is that page on Physics conventions, one other with Examples in different areas of physics, one "What's new in Physics" page at each release with illustrations only shown there. Then there are a number of Mapleprimes post describing the Physics project and showing how to use the package to tackle different problems. We seldomly find the information we are looking for fast enough.

This post thus organizes and presents all those elusive links in one place. All the hyperlinks below are alive from within a Maple worksheet. A link to this page is also appearing in all the Physics help pages in the future Maple release. Comments on practical ways to improve this presentation of information are welcome.

Description

As part of its commitment to providing the best possible environment for algebraic computations in Physics, Maplesoft launched, during 2014, a Maple Physics: Research and Development website. That enabled users to ask questions, provide feedback and download updated versions of the Physics package, around the clock.

The "Physics Updates" include improvements, fixes, and the latest new developments, in the areas of Physics, Differential Equations and Mathematical Functions. Since Maple 2018, you can install/uninstall the "Physics Updates" directly from the MapleCloud .

Maplesoft incorporated the results of this accelerated exchange with people around the world into the successive versions of Maple. Below there are two sections

•	The Updates of Physics, as an organized collection of links per Maple release, where you can find a description with examples of the subjects developed in the Physics package, from 2012 till 2019.

•	The Mapleprimes Physics posts, containing the most important posts describing the Physics project and showing the use of the package to tackle problems in General Relativity and Quantum Mechanics.

The update of Physics in Maple 2018 and back to Maple 16 (2012)

•	Physics Updates during 2018

a.	Tensor product of Quantum States using Dirac's Bra-Ket Notation

b.	Coherent States in Quantum Mechanics

c.	The Zassenhaus formula and the algebra of the Pauli matrices

d.	Multivariable Taylor series of expressions involving anticommutative (Grassmannian) variables

e.	New SortProducts command

f.	A Complete Guide for Tensor computations using Physics

•	Physics Maple 2018 updates

g.	Automatic handling of collision of tensor indices in products

h.	User defined algebraic differential operators

i.	The Physics:-Cactus package for Numerical Relativity

j.	Automatic setting of the EnergyMomentumTensor for metrics of the database of solutions to Einstein's equations

k.	Minimize the number of tensor components according to its symmetries, relabel, redefine or count the number of independent tensor components

l.	New functionality and display for inert names and inert tensors

m.	Automatic setting of Dirac, Paul and Gell-Mann algebras

n.	Simplification of products of Dirac matrices

o.	New Physics:-Library commands to perform matrix operations in expressions involving spinors with omitted indices

p.	Miscellaneous improvements

•	Physics Maple 2017 updates

q.	General Relativity: classification of solutions to Einstein's equations and the Tetrads package

r.	The 3D metric and the ThreePlusOne (3 + 1) new Physics subpackage

s.	Tensors in Special and General Relativity

t.	The StandardModel new Physics subpackage

•	Physics Maple 2016 updates

u.	Completion of the Database of Solutions to Einstein's Equations

v.	Operatorial Algebraic Expressions Involving the Differential Operators d_[mu], D_[mu] and Nabla

w.	Factorization of Expressions Involving Noncommutative Operators

x.	Tensors in Special and General Relativity

y.	Vectors Package

z.	New Physics:-Library commands

aa.	Redesigned Functionality and Miscellaneous

•	Physics Maple 2015 updates

ab.	Simplification

ac.

Tensors

ad.	Tetrads in General Relativity

ae.	More Metrics in the Database of Solutions to Einstein's Equations

af.	Commutators, AntiCommutators, and Dirac notation in quantum mechanics

ag.	New Assume command and new enhanced Mode: automaticsimplification

ah.	Vectors Package

ai.	New Physics:-Library commands

aj.	Miscellaneous

•	Physics Maple 18 updates

ak.	Simplification

al.	4-Vectors, Substituting Tensors

am.	Functional Differentiation

an.	More Metrics in the Database of Solutions to Einstein's Equations

ao.	Commutators, AntiCommutators

ap.	Expand and Combine

aq.	New Enhanced Modes in Physics Setup

ar.

Dagger

as.	Vectors Package

at.	New Physics:-Library commands

au.	Miscellaneous

•	Physics Maple 17 updates

av.	Tensors and Relativity: ExteriorDerivative, Geodesics, KillingVectors, LieDerivative, LieBracket, Antisymmetrize and Symmetrize

aw.	Dirac matrices, commutators, anticommutators, and algebras

ax.	Vector Analysis

ay.	A new Library of programming commands for Physics

•	Physics Maple 16 updates

az.	Tensors in Special and General Relativity: contravariant indices and new commands for all the General Relativity tensors

ba.	New commands for working with expressions involving anticommutative variables and functions: Gtaylor, ToFieldComponents, ToSuperfields

bb.	Vector Analysis: geometrical coordinates with funcional dependency

Mapleprimes Physics posts

1.	The Physics project at Maplesoft

2.	Mini-Course: Computer Algebra for Physicists

3.	A Complete Guide for Tensor computations using Physics

4.	Perimeter Institute-2015, Computer Algebra in Theoretical Physics (I)

5.	IOP-2016, Computer Algebra in Theoretical Physics (II)

6.	ACA-2017, Computer Algebra in Theoretical Physics (III)

•	General Relativity

7.	General Relativity using Computer Algebra

8.	Exact solutions to Einstein's equations

9.	Classification of solutions to Einstein's equations and the ThreePlusOne (3 + 1) package

10.	Tetrads and Weyl scalars in canonical form

11.	Equivalence problem in General Relativity

12.	Automatic handling of collision of tensor indices in products

13.	Minimize the number of tensor components according to its symmetries

•	Quantum Mechanics

14.	Quantum Commutation Rules Basics

15.	Quantum Mechanics: Schrödinger vs Heisenberg picture

16.	Quantization of the Lorentz Force

17.	Magnetic traps in cold-atom physics

18.	The hidden SO(4) symmetry of the hydrogen atom

19.	(I) Ground state of a quantum system of identical boson particles

20.	(II) The Gross-Pitaevskii equation and Bogoliubov spectrum

21.	(III) The Landau criterion for Superfluidity

22.	Simplification of products of Dirac matrices

23.	Algebra of Dirac matrices with an identity matrix on the right-hand side

24.	Factorization with non-commutative variables

25.	Tensor Products of Quantum State Spaces

26.	Coherent States in Quantum Mechanics

27.	The Zassenhaus formula and the Pauli matrices

•	Physics package generic functionality

28.	Automatic simplification and a new Assume (as in "extended assuming")

29.	Wirtinger derivatives and multi-index summation

	See Also
	Conventions used in the Physics package , Physics , Physics examples , A Complete Guide for Tensor computations using Physics

Download Physics-Updates.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

The Zassenhaus formula and the Pauli mat...

Posted: ecterrab 14837 Product: Maple

January 11 2019

6 0

The Zassenhaus formula and the algebra of the Pauli matrices

Edgardo S. Cheb-Terrab¹ and Bryan C. Sanctuary²

(1) Maplesoft

(2) Department of Chemistry, McGill University, Montreal, Quebec, Canada

The implementation of the Pauli matrices and their algebra were reviewed during 2018, including the algebraic manipulation of nested commutators, resulting in faster computations using simpler and more flexible input. As it frequently happens, improvements of this type suddenly transform research problems presented in the literature as untractable in practice, into tractable.

As an illustration, we tackle below the derivation of the coefficients entering the Zassenhaus formula shown in section 4 of [1] for the Pauli matrices up to order 10 (results in the literature go up to order 5). The computation presented can be reused to compute these coefficients up to any desired higher-order (hardware limitations may apply). A number of examples which exploit this formula and its dual, the Baker-Campbell-Hausdorff formula, occur in connection with the Weyl prescription for converting a classical function to a quantum operator (see sec. 5 of [1]), as well as when solving the eigenvalue problem for classes of mathematical-physics partial differential equations [2].
To reproduce the results below - a worksheet with this contents is linked at the end - you need to have your Maple 2018.2.1 updated with the Maplesoft Physics Updates version 280 or higher.

References

[1] R.M. Wilcox, "Exponential Operators and Parameter Differentiation in Quantum Physics", Journal of Mathematical Physics, V.8, 4, (1967.

[2] S. Steinberg, "Applications of the lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations", Journal of Differential Equations, V.26, 3, 1977.

[3] K. Huang, "Statistical Mechanics", John Wiley & Sons, Inc. 1963, p217, Eq.(10.60).

Formulation of the problem

The Zassenhaus formula expresses as an infinite product of exponential operators involving nested commutators of increasing complexity

=

Given , and their commutator , if and commute with , for and the Zassenhaus formula reduces to the product of the first three exponentials above. The interest here is in the general case, when and , and the goal is to compute the Zassenhaus coefficients in terms of , for arbitrary finite n. Following [1], in that general case, differentiating the Zassenhaus formula with respect to and multiplying from the right by one obtains

This is an intricate formula, which however (see eq.(4.20) of [1]) can be represented in abstract form as

$"0=((∑)(lambda^n)/(n!) {A^n,B})+2 lambda ((∑) (∑)(lambda^(n+m))/(n! m!) {A^m,B^n,C[2]})+3 lambda^2 ((∑) (∑) (∑)(lambda^(n+m+k))/(n! m! k!) {A^k,B^m,(C[2])^n,C[3]})+ ..."$

from where an equation to be solved for each is obtained by equating to 0 the coefficient of . In this formula, the repeated commutator bracket is defined inductively in terms of the standard commutator by

${B, A^0} = B, {B, A^(n+1)} = %Commutator(A, {A^n, B^n})$

${C[j], B^n, A^0} = {C[j], B^n}, {C[j], A^m, B^n} = %Commutator(A, {A^`-`(m, 1), B^n, C[j]^k})$

and higher-order repeated-commutator brackets are similarly defined. For example, taking the coefficient of and and respectively solving each of them for and one obtains

This method is used in [3] to treat quantum deviations from the classical limit of the partition function for both a Bose-Einstein and Fermi-Dirac gas. The complexity of the computation of grows rapidly and in the literature only the coefficients up to have been published. Taking advantage of developments in the Physics package during 2018, below we show the computation up to and provide a compact approach to compute them up to arbitrary finite order.

Computing up to

Set the signature of spacetime such that its space part is equal to +++ and use lowercaselatin letters to represent space indices. Set also , and to represent quantum operators

>

(1)

To illustrate the computation up to , a convenient example, where the commutator algebra is closed, consists of taking and as Pauli Matrices which, multiplied by the imaginary unit, form a basis for the group, which in turn exponentiate to the relevant Special Unitary Group . The algebra for the Pauli matrices involves a commutator and an anticommutator

>

(2)

Assign now and to two Pauli matrices, for instance

>

(3)

>

(4)

Next, to extract the coefficient of from

$"0=((∑)(lambda^n)/(n!) {A^n,B})+2 lambda ((∑) (∑)(lambda^(n+m))/(n! m!) {A^m,B^n,C[2]})+3 lambda^2 ((∑) (∑) (∑)(lambda^(n+m+k))/(n! m! k!) {A^k,B^m,(C[2])^n,C[3]})+..."$

to solve it for we note that each term has a factor multiplying a sum, so we only need to take into account the first terms (sums) and in each sum replace by the corresponding . For example, given to compute we only need to compute these first three terms:

$0 = Sum(lambda^n*{B, A^n}/factorial(n), n = 1 .. 2)+2*lambda*(Sum(Sum(lambda^(n+m)*{C[2], A^m, B^n}/(factorial(n)*factorial(m)), n = 0 .. 1), m = 0 .. 1))+3*lambda^2*(Sum(Sum(Sum(lambda^(n+m+k)*{C[3], A^k, B^m, C[2]^n}/(factorial(n)*factorial(m)*factorial(k)), n = 0 .. 0), m = 0 .. 0), k = 0 .. 0))$

then solving for one gets .

Also, since to compute we only need the coefficient of , it is not necessary to compute all the terms of each multiple-sum. One way of restricting the multiple-sums to only one power of consists of using multi-index summation, available in the Physics package (see Physics:-Library:-Add ). For that purpose, redefine sum to extend its functionality with multi-index summation

>

(5)

Now we can represent the same computation of without multiple sums and without computing unnecessary terms as

$0 = Sum(lambda^n*{B, A^n}/factorial(n), n = 1)+2*lambda*(Sum(lambda^(n+m)*{C[2], A^m, B^n}/(factorial(n)*factorial(m)), n+m = 1))+3*lambda^2*(Sum(lambda^(n+m+k)*{C[3], A^k, B^m, C[2]^n}/(factorial(n)*factorial(m)*factorial(k)), n+m+k = 0))$

Finally, we need a computational representation for the repeated commutator bracket

${B, A^0} = B, {B, A^(n+1)} = %Commutator(A, {A^n, B^n})$

One way of representing this commutator bracket operation is defining a procedure, say F, with a cache to avoid recomputing lower order nested commutators, as follows

>

(6)

>

For example,

>

(7)

>

(8)

>

(9)

We can set now the value of

>

(10)

and enter the formula that involves only multi-index summation

>

H := sum(lambda^n*F(A, B, n)/factorial(n), n = 2)+2*lambda*(sum(lambda^(n+m)*F(A, F(B, C[2], n), m)/(factorial(n)*factorial(m)), n+m = 1))+3*lambda^2*(sum(lambda^(n+m+k)*F(A, F(B, F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = 0))

(1/2)*lambda^2*%Commutator(Physics:-Psigma[1], %Commutator(Physics:-Psigma[1], Physics:-Psigma[3]))+2*lambda*(lambda*%Commutator(Physics:-Psigma[1], I*Physics:-Psigma[2])+lambda*%Commutator(Physics:-Psigma[3], I*Physics:-Psigma[2]))+3*lambda^2*C[3]

(11)

from where we compute by solving for it the coefficient of , and since due to the mulit-index summation this expression already contains as a factor,

>

C[3] = (2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

(12)

In order to generalize the formula for H for higher powers of , the right-hand side of the multi-index summation limit can be expressed in terms of an abstract N, and H transformed into a mapping:

>

H := unapply(sum(lambda^n*F(A, B, n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(A, F(B, C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))+3*lambda^2*(sum(lambda^(n+m+k)*F(A, F(B, F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = N-2)), N)

(13)

Now we have

>

(14)

>

(15)

The following is already equal to (11)

>

(1/2)*lambda^2*%Commutator(Physics:-Psigma[1], %Commutator(Physics:-Psigma[1], Physics:-Psigma[3]))+2*lambda*(lambda*%Commutator(Physics:-Psigma[1], I*Physics:-Psigma[2])+lambda*%Commutator(Physics:-Psigma[3], I*Physics:-Psigma[2]))+3*lambda^2*C[3]

(16)

In this way, we can reproduce the results published in the literature for the coefficients of Zassenhaus formula up to by adding two more multi-index sums to (13). Unassign first

>

We compute now up to in one go

>

for j to 4 do C[j+1] := Simplify(solve(H(j), C[j+1])) end do

(2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

-(8/9)*Physics:-Psigma[1]-(158/45)*Physics:-Psigma[3]-((16/3)*I)*Physics:-Psigma[2]

(17)

The nested-commutator expression solved in the last step for is

>

(18)

With everything understood, we want now to extend these results generalizing them into an approach to compute an arbitrarily large coefficient , then use that generalization to compute all the Zassenhaus coefficients up to . To type the formula for H for higher powers of is however prone to typographical mistakes. The following is a program, using the Maple programming language , that produces these formulas for an arbitrary integer power of :

Formula := proc(A, B, C, Q)

This Formula program uses a sequence of summation indices with as much indices as the order of the coefficient we want to compute, in this case we need 10 of them

>

(19)

To avoid interference of the results computed in the loop (17), unassign again

>

Now the formulas typed by hand, used lines above to compute each of , and , are respectively constructed by the computer

>

sum(lambda^n*F(Physics:-Psigma[1], Physics:-Psigma[3], n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))

(20)

>

sum(lambda^n*F(Physics:-Psigma[1], Physics:-Psigma[3], n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))+3*lambda^2*(sum(lambda^(n+m+k)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = N-2))

(21)

>

(22)

Construct then the formula for and make it be a mapping with respect to N, as done for after (16)

>

(23)

Compute now the coefficients of the Zassenhaus formula up to all in one go

>

for j to 9 do C[j+1] := Simplify(solve(H(j), C[j+1])) end do

(2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

-(8/9)*Physics:-Psigma[1]-(158/45)*Physics:-Psigma[3]-((16/3)*I)*Physics:-Psigma[2]

(1030/81)*Physics:-Psigma[1]-(8/81)*Physics:-Psigma[3]+((1078/405)*I)*Physics:-Psigma[2]

((11792/243)*I)*Physics:-Psigma[2]+(358576/42525)*Physics:-Psigma[1]+(12952/135)*Physics:-Psigma[3]

(87277417/492075)*Physics:-Psigma[1]+(833718196/820125)*Physics:-Psigma[3]+((35837299048/17222625)*I)*Physics:-Psigma[2]

-((449018539801088/104627446875)*I)*Physics:-Psigma[2]-(263697596812424/996451875)*Physics:-Psigma[1]+(84178036928794306/2197176384375)*Physics:-Psigma[3]

(3226624781090887605597040906/21022858292748046875)*Physics:-Psigma[1]+(200495118165066770268119656/200217698026171875)*Physics:-Psigma[3]+((2185211616689851230363020476/4204571658549609375)*I)*Physics:-Psigma[2]

(24)

Notes: with the material above you can compute higher order values of . For that you need:

1.	Unassign as done above in two opportunities, to avoid interference of the results just computed.

2.	Indicate more summation indices in the sequence in (19), as many as the maximum value of n in .

3.	Have in mind that the growth in size and complexity is significant, with each taking significantly more time than the computation of all the previous ones.

4.	Re-execute the input line (23) and the loop (24).

>

Download The_Zassenhause_formula_and_the_Pauli_Matrices.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Coherent States in Quantum Mechanics...

Posted: ecterrab 14837 Product: Maple

January 05 2019

5 0

Coherent States in Quantum Mechanics

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

Coherent states are among the most relevant representations for the state of a quantum system. These states, that form an overcomplete basis, minimize the quantum uncertainty between position x and momentum p (they satisfy the Heisenberg uncertainty principle with equality and their expectation values satisfy the classical equations of motion). Coherent states are widely used in quantum optics and quantum mechanics in general; they also mathematically characterize the concept of Planck cells. Part of this development is present in Maple 2018.2.1. To reproduce what you see below, however, you need a more recent version, as the one distributed within the Maplesoft Physics Updates (version 276 or higher). A worksheet with this contents is linked at the end of this post.

Definition and the basics

>

Set a quantum operator and corresponding annihilation / creation operators

>

(1.1)

>

(1.2)

>

(1.3)

In what follows, on the left-hand sides the product operator used is `*`, which properly represents, but does not perform the attachment of Bras Kets and operators. On the right-hand sides the product operator is `.`, that performs the attachments. Since the introduction of Physics in the Maple system, we have that

>

(1.4)

>

(1.5)

>

(1.6)

New development during 2018: coherent states, the eigenstates of the annihilation operator , with all of their properties, are now understood as such by the system

>

(1.7)

is an eigenket of but not of

>

(1.8)

The norm of these states is equal to 1

>

(1.9)

These states, however, are not orthonormal as the occupation number states are, and since is not Hermitian, its eigenvalues are not real but complex numbers. Instead of (1.6) , we now have

>

%Bracket(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, alpha), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, beta)) = exp(-(1/2)*abs(alpha)^2-(1/2)*abs(beta)^2+conjugate(alpha)*beta)

(1.10)

At ,

>

(1.11)

Their scalar product with the occupation number states , using the inert %Bracket on the left-hand side and the active Bracket on the other side:

>

%Bracket(Physics:-Bra(A, n), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(alpha)^2)*alpha^n/factorial(n)^(1/2)

(1.12)

The expansion of coherent states into occupation number states, first representing the product operation using `*`, then performing the attachments replacing `*` by `.`

>

Sum(Physics:-`*`(Physics:-Ket(A, n), Physics:-Bra(A, n)), n = 0 .. infinity)

(1.13)

>

Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Physics:-`*`(Sum(Physics:-`*`(Physics:-Ket(A, n), Physics:-Bra(A, n)), n = 0 .. infinity), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha))

(1.14)

>

Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity)

(1.15)

Hide now the ket label. When in doubt, input show to see the Kets with their labels explicitly shown

>

(1.16)

Define eigenkets of the annihilation operator, with two different eigenvalues for experimentation

>

(1.17)

>

(1.18)

Because the properties of coherent states are now known to the system, the following computations proceed automatically. The left-hand sides use the `*`, while the right-hand sides use the `.`

>

(1.19)

>

(1.20)

>

(1.21)

>

(1.22)

Properties of Coherent states

The mean value of the occupation number N

The occupation number operator N is given by

>

(2.1.1)

>

(2.1.2)

>

(2.1.3)

N is diagonal in the basis of the Fock (occupation number) space

>

(2.1.4)

•	The mean value of N in a coherent state

>

(2.1.5)

>

(2.1.6)

The mean value of

>

(2.1.7)

>

(2.1.8)

The standard deviation for a state

>

(2.1.9)

In conclusion, a coherent state has a finite spreading . Coherent states are good approximations for the states of a laser, where the laser intensity I is proportional to the mean value of the photon number, I f , and so the intensity fluctuation, .

•	The mean value of the occupation number N in an occupation number state

>

(2.1.10)

>

(2.1.11)

The mean value of the occupation number N in a state is thus n itself, as expected since represents a (Fock space) state of n (quase-) particles. Accordingly,

>

(2.1.12)

>

(2.1.13)

The standard deviation for a state , is thus

>

(2.1.14)

That is, in a Fock state, , there is no intensity fluctuation.

The specific properties of coherent states implemented can be derived explicitly departing from the projection of into the basis of occupation number states and the definition of as the operator that annihilates the vacuum

>

Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Physics:-`*`(Sum(Physics:-`*`(Physics:-Ket(A, n), Physics:-Bra(A, n)), n = 0 .. infinity), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha))

(2.2.1)

>

Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity)

(2.2.2)

To derive from the formula above, start multiplying by

>

Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))

(2.2.3)

In view of , discard the first term of the sum

>

Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 1 .. infinity))

(2.2.4)

Change variables ; in the result rename

>

Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Sum(exp(-(1/2)*abs(alpha)^2)*Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(A, n+1))*alpha^(n+1)/(factorial(n)^(1/2)*(n+1)^(1/2)), n = 0 .. infinity)

(2.2.5)

Activate the product by replacing, in the right-hand side, the product operator `*` by `.`

>

Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Sum(exp(-(1/2)*abs(alpha)^2)*Physics:-Ket(A, n)*alpha^(n+1)/factorial(n)^(1/2), n = 0 .. infinity)

(2.2.6)

By inspection the right-hand side of (2.2.6) is equal to times the right-hand side of (2.2.2)

>

alpha*Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)-Physics:-`*`(`#msup(mi("a"),mo("&uminus0;"))`, Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = alpha*(Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))-(Sum(exp(-(1/2)*abs(alpha)^2)*Physics:-Ket(A, n)*alpha^(n+1)/factorial(n)^(1/2), n = 0 .. infinity))

(2.2.7)

>

(2.2.8)

•	Overview of the coherent states distribution

Consider the projection of over an occupation number state

>

%Bracket(Physics:-Bra(A, n), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(alpha)^2)*alpha^n/factorial(n)^(1/2)

(2.2.9)

An overview of the distribution of coherent states for a sample of values of n and is thus as follows

>

The distribution can be explored for ranges of values of n and using Explore

>

	=

	=

`<|>`(beta, alpha) = exp(conjugate(beta)*alpha-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2)

The identity in the title can be derived departing again from the the projection of a coherent stateinto the basis of occupation number states

>

Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity)

(2.6.1)

>	$Dagger(subs({alpha = beta, n = k}, Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha) = Sum(exp(-(1/2)abs(alpha)^2)alpha^n*Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity)))$

Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta) = Sum(exp(-(1/2)*abs(beta)^2)*conjugate(beta)^k*Physics:-Bra(A, k)/factorial(k)^(1/2), k = 0 .. infinity)

(2.6.2)

Taking the `*` product of these two expressions

>

Physics:-`*`(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = Physics:-`*`(Sum(exp(-(1/2)*abs(beta)^2)*conjugate(beta)^k*Physics:-Bra(A, k)/factorial(k)^(1/2), k = 0 .. infinity), Sum(exp(-(1/2)*abs(alpha)^2)*alpha^n*Physics:-Ket(A, n)/factorial(n)^(1/2), n = 0 .. infinity))

(2.6.3)

Perform the attachment of Bras and Kets on the right-hand side by replacing `*` by `.`, evaluating the sum and simplifying the result

>

Physics:-`*`(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2+alpha*conjugate(beta))

(2.6.4)

•	Overview of the real and imaginary part of

In most cases, and are complex valued numbers. Below, the plots assume that and are both real. To take into account the general case, the possibility to tune a phase difference between and is explicitly added, so that (2.6.4) becomes

>

%Bracket(Physics:-Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Physics:-Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2+alpha*conjugate(beta)*exp(I*theta))

(2.6.5)

>

Explore(plot3d(Re(rhs(%Bracket(Bra(`#msup(mi("a"),mo("&uminus0;"))`, beta), Ket(`#msup(mi("a"),mo("&uminus0;"))`, alpha)) = exp(-(1/2)*abs(beta)^2-(1/2)*abs(alpha)^2+alpha*conjugate(beta)*exp(I*theta)))), alpha = -10 .. 10, beta = -10 .. 10, view = -1 .. 1, orientation = [-12, 74, 3], axes = boxed), parameters = [theta = 0 .. 2*Pi], initialvalues = [theta = (1/10)*Pi])

Download Coherent_States_in_Quantum_Mechanics.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft
Editor, Computer Physics Communications

Tensor products in Quantum Mechanics usi...

Posted: ecterrab 14837 Product: Maple

December 30 2018

6 0

Tensor product of Quantum States using Dirac's Bra-Ket Notation - 2018

There has been increasing interest in the details of the Maple implementation of tensor products using Dirac's notation, developed during 2018. Tensor products of Hilbert spaces and related quantum states are relevant in a myriad of situations in quantum mechanics, and in particular regarding quantum information. Below is a presentation up-to-date of the design and implementation, with input/output and examples, organized in four sections:

•	The basic ideas and design implemented

•	Tensor product notation and the hideketlabel option

•	Entangled States and the Bell basis

•	Entangled States, Operators and Projectors

Part of this development is present in Maple 2018.2. To reproduce what you see below, however, you need a more recent version, as the one distributed within the Maplesoft Physics Updates (version 272 or higher).

The basic ideas and design implemented

Suppose and are quantum operators and are, respectively, their eigenkets. The following works since the introduction of the Physics package in Maple

>

(1.1)

>

(1.2)

>

(1.3)

In previous Maple releases, all quantum operators are supposed to act on the same Hillbert space. New: suppose that and act on different, disjointed, Hilbert spaces.

1) To represent that situation, a new keyword in Setup , , is introduced. With it you can indicate the quantum operators that act on a Hilbert space, say as in with the meaning that the operator acts on one Hilbert space while acts on another one.

The Hilbert space thus has no particular name (as in 1, 2, 3 ...) and is instead identified by the operators that act on it. There can be one or more, and operators acting on one space can act on other spaces too. The disjointedspaces keyword is a synonym for hilbertspaces and hereafter all Hilbert spaces are assumed to be disjointed.

NOTE: noncommutative quantum operators acting on disjointed spaces commute between themselves, so after setting - for instance - , automatically, become quantum operators satisfying (see comment (ii) on page 156 of ref.[1])

2) Product of Kets and Bras that belong to different Hilbert spaces, are understood as tensor products satisfying (see footnote on page 154 of ref. [1]):

while

3) All the operators of one Hilbert space act transparently over operators, Bras and Kets of other Hilbert spaces. For example

and the same for the Dagger of this equation, that is

Hence, when we write the left-hand sides of the two equations above and press enter, they are automatically rewritten (returned) as the right-hand sides.

4) Every other quantum operator, set as such using Setup , and not indicated as acting on any particular Hilbert space, is assumed to act on all spaces.

5) Notation:

•

Tensor products formed with operators, or Bras and Kets belonging to different Hilbert spaces (set as such using Setup and the keyword hilbertspaces), are now displayed with the symbol 5 in between, as in instead of , and instead of . The product of an operator of one space and a Ket of another space however, is displayed , without 5.

•	A new Setup option hideketlabel , makes all the labels in Kets and Bras to be hidden at the time of displaying Kets, Bras and Bracket, so when you set it entering ,

is displayed as

This is the notation frequently used when working with angular momentum or in quantum information, where tensor products of Hilbert spaces are used.

	Design details

Tensor product notation and the hideketlabel option

According to the design section, set now two disjointed Hilbert spaces with operators acting on one of them and on the other one (you can think of )

>

(2.1)

Consider a tensor product of Kets, each of which belongs to one of these different spaces, note the new notation using

>

(2.2)

•	As explained in the Details of the design section, the ordering of the Hilbert spaces in tensor products is now preserved: Bras (Kets) of the first space always appear before Bras (Kets) of the second space. For example, construct a projector into the state (2.2)

>

(2.3)

You see that in the product of Bras, and also in the product of Kets, A comes first, then B.

Remark: some textbooks prefer a diadic style for sorting the operands in products of Bras and Kets that belong to different spaces, for example, instead of the projector sorting style of (2.3). Both reorderings of Kets and Bras are mathematically equal.

•	Because that ordering is preserved, one can now hide the label of Bras and Kets without ambiguity, as it is usual in textbooks (e.g. in Quantum Information). For that purpose use the new keyword option hideketlabel

>

(2.4)

The display for (2.3) is now

>

(2.5)

Important: this new option only hides the label while displaying the Bra or Ket. The label, however, is still there, both in the input and in the output. One can "see" what is behind this new display using show, that works the same way as it does in the context of CompactDisplay . The actual contents being displayed in (2.5) is thus (2.3)

>

(2.6)

Operators of each of these spaces act on their eigenkets as usual. Here we distribute over both sides of an equation, using `*` on the left-hand side, to see the product uncomputed, and `.` on the right-hand side to see it computed:

>

(2.7)

>

(2.8)

•	The tensor product of operators belonging to different Hilbert spaces is also displayed using 5

>

(2.9)

•

As mentioned in the preceding design section, using the commutativity between operators, Bras and Kets that belong to different Hilbert spaces, within a product, operators are placed contiguous to the Kets and Bras belonging to the space where the operator acts. For example, consider the delayed product represented using the start `*` operator

>

(2.10)

Release the product

>

(2.11)

The same operation but now using the dot product `.` operator. Start by delaying the operation

>

(2.12)

Recalling that this product is mathematically the same as (2.11), and that

>

(2.13)

by releasing the delayed product (2.12) we have

>

(2.14)

Reset hideketlabel

>

(2.15)

	Implementation details

Entangled States and the Bell basis

With the introduction of disjointed Hilbert spaces in Maple it is possible to represent entangled quantum states in a simple way, basically as done with paper and pencil.

Recalling the Hilbert spaces set at this point are,

>

(3.1)

where acts on the tensor product of the spaces where and act. A state of can then always be written as

>

(3.2)

where is a matrix of complex coefficients. Bra states of are formed as usual taking the Dagger

>

(3.3)

•

By definition, all states that can be written exactly as , that is, the product of a arbitrary state of the subspace A and another of the subspace B, are product states, and all the other ones are entangled states. Entangelment is a property that is independent of the basis used in (3.2).

The physical interpretation is the standard one: when the state of a system constituted by two subsystems A and B is represented by a product state, the properties of the subsystem A are well defined and all given by while those for the subsystem B by . When the system is in an entangled state one typically cannot assign definite properties to the individual subsystems A or B, each subsystem has no independent reality.

To determine whether a state is or not entangled it then suffices to check the rank R of the matrix (see LinearAlgebra:-Rank ): when the state is a product state, otherwise it is an entangled state. When the state being analized belongs to the tensor product of two subspaces, .is equivalent to having the determinant of equal to 0. The condition , however, is more general, and suffices to determine whether a state is a product state also on a Hilbert space that is the tensor product of three or more subspaces: , in which case the matrix M will have more rows and columns and a determinant equal to 0 would only warrant the possibility of factorizing one Ket.

Example: the Bell basis for a system of two qubits

Consider a 2-dimensional space of states acted upon by the operator , and let act upon another, disjointed, Hilbert space that is a replica of the Hilbert space on which acts. Set the dimensions of , and respectively equal to 2, 2 and 2x2 (see Setup)

>

(3.4)

The system C with the two subsystems A and B represents the a two qubits system. The standard basis for C can be constructed in a natural way from the basis of Kets of A and B, , by taking their tensor products:

>

(3.5)

Set a more mathematical display for the imaginary unit

>

The four entangled Bell states also form a basis of C and are given by

>

(3.6)

>

Physics:-Ket(`&Bscr;`, 0) = (Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))/sqrt(2)

(3.7)

>

Physics:-Ket(`&Bscr;`, 1) = (Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 1))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 0)))/sqrt(2)

(3.8)

>

Physics:-Ket(`&Bscr;`, 2) = I*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 1))-Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 0)))/sqrt(2)

(3.9)

>

Physics:-Ket(`&Bscr;`, 3) = (Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))-Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))/sqrt(2)

(3.10)

There is no standard notation for denoting a Bell state (the linar combinations of the right-hand sides above). The convention used here relates to the definition of the Bell states related to the Pauli matrices shown below. Regardless fo the convention used, this basis is orthonormal. That can be verified by taking dot products, for example:

>

(3.11)

In steps, perform the same operation but using the star (`*`) operator, so that the contraction is represented but not performed

>

(3.12)

Evaluate now the result at `*` = `.`, that is transforming the star product into a dot product

>

(3.13)

>

(3.14)

>

(3.15)

The Bell basis and its relation with the Pauli matrices

The Bell basis can be constructed departing from using the Pauli matrices . For that purpose, using a Vector representation for ,

(3.16)

>

Physics:-Ket(B, 0) = Vector[column](%id = 18446744078301209294), Physics:-Ket(B, 1) = Vector[column](%id = 18446744078301209414)

(3.17)

Multiplying by each of the Pauli Matrices and performing the matrix operations we have

>

[Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 0)) = Physics:-Psigma[1].Vector[column](%id = 18446744078301209294), Physics:-`*`(Physics:-Psigma[2], Physics:-Ket(B, 0)) = Physics:-Psigma[2].Vector[column](%id = 18446744078301209294), Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(B, 0)) = Physics:-Psigma[3].Vector[column](%id = 18446744078301209294)]

(3.18)

>

[Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 0)) = Vector[column](%id = 18446744078376366918), Physics:-`*`(Physics:-Psigma[2], Physics:-Ket(B, 0)) = Vector[column](%id = 18446744078376368838), Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(B, 0)) = Vector[column](%id = 18446744078376358606)]

(3.19)

In this result we see that and flip the state, transforming into , also multiplies the state by the imaginary unit , while leaves the state unchanged.

We can rewrite all that by removeing from (3.19) the Vector representations of (3.17). For that purpose, create a list of substitution equations, replacing the Vectors by the Kets

>

[Vector[column](%id = 18446744078301209294) = Physics:-Ket(B, 0), Vector[column](%id = 18446744078301209414) = Physics:-Ket(B, 1), Vector[column](%id = 18446744078376351494) = I*Physics:-Ket(B, 0), Vector[column](%id = 18446744078376351734) = I*Physics:-Ket(B, 1)]

(3.20)

So the action of in is given by

>

(3.21)

For , the same operations result in

>

[Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 1)) = Physics:-Psigma[1].Vector[column](%id = 18446744078301209414), Physics:-`*`(Physics:-Psigma[2], Physics:-Ket(B, 1)) = Physics:-Psigma[2].Vector[column](%id = 18446744078301209414), Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(B, 1)) = Physics:-Psigma[3].Vector[column](%id = 18446744078301209414)]

(3.22)

>

[Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 1)) = Vector[column](%id = 18446744078464860518), Physics:-`*`(Physics:-Psigma[2], Physics:-Ket(B, 1)) = Vector[column](%id = 18446744078464862438), Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(B, 1)) = Vector[column](%id = 18446744078464856182)]

(3.23)

>

(3.24)

To obtain the other three Bell states using the results (3.21) and (3.24), indicate to the system that the Pauli matrices operate in the subspace where operates

>

(3.25)

Multiplying given in (3.7) by each of the three we get the other three Bell states

>

Physics:-Ket(`&Bscr;`, 0) = (1/2)*2^(1/2)*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.26)

>

Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*Physics:-`*`(Physics:-Psigma[1], Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.27)

Substitute in this result the first equations of (3.21) and (3.24)

>

(3.28)

>

(3.29)

>

Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*Physics:-`*`(Physics:-Psigma[1], Physics:-`*`(Physics:-Ket(A, 0), Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 1)))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(B, 0))))

(3.30)

>

Physics:-`*`(Physics:-Psigma[1], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 1))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 0)))

(3.31)

This is defined in (3.8)

>

Physics:-Ket(`&Bscr;`, 1) = (1/2)*2^(1/2)*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 1))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 0)))

(3.32)

>

(3.33)

Multiplying now by and substituting using the equations of (3.21) and (3.24) we get

>

Physics:-`*`(Physics:-Psigma[2], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*Physics:-`*`(Physics:-Psigma[2], Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.34)

>

(3.35)

>

(3.36)

>

(3.37)

The above is defined in (3.9)

>

(3.38)

>

(3.39)

Finally, multiplying by

>

Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*Physics:-`*`(Physics:-Psigma[3], Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))+Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.40)

Substituting

>

(3.41)

>

(3.42)

We get

>

Physics:-`*`(Physics:-Psigma[3], Physics:-Ket(`&Bscr;`, 0)) = (1/2)*2^(1/2)*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))-Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.43)

which is

>

Physics:-Ket(`&Bscr;`, 3) = (1/2)*2^(1/2)*(Physics:-`*`(Physics:-Ket(A, 0), Physics:-Ket(B, 0))-Physics:-`*`(Physics:-Ket(A, 1), Physics:-Ket(B, 1)))

(3.44)

>

(3.45)

Entangled States, Operators and Projectors

Consider a fourth operator, , that is Hermitian and acts on the same space of , and then it has the same dimension,

>

[disjointedspaces = {{A, C, H}, {B, C, H}, {B, C, Physics:-Psigma}}, hermitianoperators = {H}, quantumbasisdimension = {A = 2, B = 2, C[1] = 2, C[2] = 2, H[1] = 2, H[2] = 2}]

(4.1)

To operate in a practical way with these operators, Bras and Kets, however, bracket rules reflecting their relationship are necessary. From the definition of as acting on the tensor product of spaces where and act (see (3.2)) and taking into account the dimensions specified for , and we have

>

Physics:-Ket(C, a, b) = Sum(Sum(M[a, j, b, p]*Physics:-`*`(Physics:-Ket(A, j), Physics:-Ket(B, p)), j = 0 .. 1), p = 0 .. 1)

(4.2)

>

Physics:-Bracket(Physics:-Bra(A, k), Physics:-Ket(C, a, b)) = Sum(M[a, k, b, p]*Physics:-Ket(B, p), p = 0 .. 1)

(4.3)

>

Physics:-Bracket(Physics:-Bra(B, k), Physics:-Ket(C, a, b)) = Sum(M[a, j, b, k]*Physics:-Ket(A, j), j = 0 .. 1)

(4.4)

>

(4.5)

The bracket rules for , and are the first two of these; Set these rules, so that the system can take them into account

>

[bracketrules = {%Bracket(%Bra(A, k), %Ket(C, a, b)) = Sum(M[a, k, b, p]*Physics:-Ket(B, p), p = 0 .. 1), %Bracket(%Bra(B, k), %Ket(C, a, b)) = Sum(M[a, j, b, k]*Physics:-Ket(A, j), j = 0 .. 1)}]

(4.6)

If we now recompute (4.5), the left-hand side is also computed

>

(4.7)

>

(4.8)

Suppose now that you want to compute with the Hermitian operator , that operates on the same space as , both using C using the operators and , as in

where = when is a product (not entagled) state.

For it suffices to set a bracket rule

>

[bracketrules = {%Bracket(%Bra(A, k), %Ket(C, a, b)) = Sum(M[a, k, b, p]*Physics:-Ket(B, p), p = 0 .. 1), %Bracket(%Bra(B, k), %Ket(C, a, b)) = Sum(M[a, j, b, k]*Physics:-Ket(A, j), j = 0 .. 1), %Bracket(%Bra(C, a, b), H, %Ket(C, c, d)) = `&Hscr;`[a, b, c, d]}, realobjects = {`&Hscr;`, x, y, z}]

(4.9)

After that, the system operates taking the rule into account

>

(4.10)

Regarding , since belongs to the tensor product of spaces A and B, it can be an entangled operator, one that you cannot represent just as a product of one operator acting on A times another one acting on B. A computational representation for the operator (that is not just itself or as abstract) is not possible in the general case. For that you can use a different feature: define the action of the operator on Kets of and .

Basically, we want:

A program sketch for that would be:

if H is applied to a Ket of A or B then

    if H itself is indexed then
        return H accumulating its indices, followed by the index of the Ket
    else

return H indexed by the index of the Ket;
otherwise
return the dot product operation uncomputed, unevaluated

In Maple language, that program-sketch becomes

>

"H := K -> if K::Ket and op(1, K)::'identical(A,B)' then if procname::'indexed' then if nops(procname) <4 then H[op(procname), op(2, K)] #` accumulate indices` else 'H . K' fi else H[op(2, K)] fi else 'procname . K' fi:"

Let's see it in action. Start erasing the Physics performance remember tables, that remember results like computed before the definition of

>

(4.11)

Recalling that is Hermitian,

>

(4.12)

>

(4.13)

>

(4.14)

>

(4.15)

Note that the definition of as a procedure does not interfer with the setting of an bracket rule for it with , that is still working

>

(4.16)

but the definition takes precedence, so if in it you indicate what to do with a Ket, that will be taken into account before the bracket rule. Finally, In the typical case, the first four results, (4.11), (4.12), (4.13) and (4.14) are operators while (4.15) is a scalar; you can always represent the scalar aspect by substituing the noncommutative operator by a related scalar, say H.

•	You can set the projectors for all these operators / spaces. For example,

>

`&Iopf;__A` := Projector(Ket(A, i)); `&Iopf;__B` := Projector(Ket(B, i)); `&Iopf;__C` := Projector(Ket(C, a, b))

Sum(Physics:-`*`(Physics:-Ket(A, n), Physics:-Bra(A, n)), n = 0 .. 1)

Sum(Physics:-`*`(Physics:-Ket(B, n), Physics:-Bra(B, n)), n = 0 .. 1)

Sum(Sum(Physics:-`*`(Physics:-Ket(C, a, b), Physics:-Bra(C, a, b)), a = 0 .. 1), b = 0 .. 1)

(4.17)

Since the algebra rules for computing with eigenkets of , and were already set in (4.6), from the projectors above you can construct any subspace projector, for example

>

Sum(Sum(Sum(M[a, m, b, p]*Physics:-`*`(Physics:-Ket(B, p), Physics:-Bra(C, a, b)), p = 0 .. 1), a = 0 .. 1), b = 0 .. 1)

(4.18)

>

Sum(Sum(Sum(conjugate(M[a, m, b, p])*Physics:-`*`(Physics:-Ket(C, a, b), Physics:-Bra(B, p)), p = 0 .. 1), a = 0 .. 1), b = 0 .. 1)

(4.19)

The conjugate of is due to the contraction or attachment from the right of (4.18), that is with

>

Physics:-Bra(C, a, b) = Sum(Sum(conjugate(M[a, j, b, p])*Physics:-`*`(Physics:-Bra(A, j), Physics:-Bra(B, p)), j = 0 .. 1), p = 0 .. 1)

(4.20)

The coefficients satisfy constraints due to the normalization of Kets of and . One can derive these contraints by inserting the unit operator constructing this identity

Sum(Sum(Physics:-`*`(Physics:-Ket(C, a, b), Physics:-Bra(C, a, b)), a = 0 .. 1), b = 0 .. 1)

(4.21)

>

Sum(Sum(conjugate(M[a, r, b, s])*M[a, m, b, n], a = 0 .. 1), b = 0 .. 1) = Physics:-KroneckerDelta[m, r]*Physics:-KroneckerDelta[n, s]

(4.22)

Transform this result into a function P to explore the identity further

>

proc (m, n, r, s) options operator, arrow; sum(sum(conjugate(M[a, r, b, s])*M[a, m, b, n], a = 0 .. 1), b = 0 .. 1) = Physics:-KroneckerDelta[m, r]*Physics:-KroneckerDelta[n, s] end proc

(4.23)

The first and third indices refer to the quantum numbers of , the second and fourth to , so the the right-hand sides in the following are respectively 1 and 0

>

(4.24)

>

(4.25)

To get the whole system of equations satisfied by the coefficients , use P to construct an Array with four indices running from 0..1

>

_rtable[18446744078376377150]

(4.26)

Convert the whole Array into a set of equations

>

${abs(M[0, 0, 0, 0])^2+abs(M[1, 0, 0, 0])^2+abs(M[0, 0, 1, 0])^2+abs(M[1, 0, 1, 0])^2 = 1, abs(M[0, 0, 0, 1])^2+abs(M[1, 0, 0, 1])^2+abs(M[0, 0, 1, 1])^2+abs(M[1, 0, 1, 1])^2 = 1, abs(M[0, 1, 0, 0])^2+abs(M[1, 1, 0, 0])^2+abs(M[0, 1, 1, 0])^2+abs(M[1, 1, 1, 0])^2 = 1, abs(M[0, 1, 0, 1])^2+abs(M[1, 1, 0, 1])^2+abs(M[0, 1, 1, 1])^2+abs(M[1, 1, 1, 1])^2 = 1, conjugate(M[0, 0, 0, 0])*M[0, 0, 0, 1]+conjugate(M[1, 0, 0, 0])*M[1, 0, 0, 1]+conjugate(M[0, 0, 1, 0])*M[0, 0, 1, 1]+conjugate(M[1, 0, 1, 0])*M[1, 0, 1, 1] = 0, conjugate(M[0, 0, 0, 0])*M[0, 1, 0, 0]+conjugate(M[1, 0, 0, 0])*M[1, 1, 0, 0]+conjugate(M[0, 0, 1, 0])*M[0, 1, 1, 0]+conjugate(M[1, 0, 1, 0])*M[1, 1, 1, 0] = 0, conjugate(M[0, 0, 0, 0])*M[0, 1, 0, 1]+conjugate(M[1, 0, 0, 0])*M[1, 1, 0, 1]+conjugate(M[0, 0, 1, 0])*M[0, 1, 1, 1]+conjugate(M[1, 0, 1, 0])*M[1, 1, 1, 1] = 0, conjugate(M[0, 0, 0, 1])*M[0, 0, 0, 0]+conjugate(M[1, 0, 0, 1])*M[1, 0, 0, 0]+conjugate(M[0, 0, 1, 1])*M[0, 0, 1, 0]+conjugate(M[1, 0, 1, 1])*M[1, 0, 1, 0] = 0, conjugate(M[0, 0, 0, 1])*M[0, 1, 0, 0]+conjugate(M[1, 0, 0, 1])*M[1, 1, 0, 0]+conjugate(M[0, 0, 1, 1])*M[0, 1, 1, 0]+conjugate(M[1, 0, 1, 1])*M[1, 1, 1, 0] = 0, conjugate(M[0, 0, 0, 1])*M[0, 1, 0, 1]+conjugate(M[1, 0, 0, 1])*M[1, 1, 0, 1]+conjugate(M[0, 0, 1, 1])*M[0, 1, 1, 1]+conjugate(M[1, 0, 1, 1])*M[1, 1, 1, 1] = 0, conjugate(M[0, 1, 0, 0])*M[0, 0, 0, 0]+conjugate(M[1, 1, 0, 0])*M[1, 0, 0, 0]+conjugate(M[0, 1, 1, 0])*M[0, 0, 1, 0]+conjugate(M[1, 1, 1, 0])*M[1, 0, 1, 0] = 0, conjugate(M[0, 1, 0, 0])*M[0, 0, 0, 1]+conjugate(M[1, 1, 0, 0])*M[1, 0, 0, 1]+conjugate(M[0, 1, 1, 0])*M[0, 0, 1, 1]+conjugate(M[1, 1, 1, 0])*M[1, 0, 1, 1] = 0, conjugate(M[0, 1, 0, 0])*M[0, 1, 0, 1]+conjugate(M[1, 1, 0, 0])*M[1, 1, 0, 1]+conjugate(M[0, 1, 1, 0])*M[0, 1, 1, 1]+conjugate(M[1, 1, 1, 0])*M[1, 1, 1, 1] = 0, conjugate(M[0, 1, 0, 1])*M[0, 0, 0, 0]+conjugate(M[1, 1, 0, 1])*M[1, 0, 0, 0]+conjugate(M[0, 1, 1, 1])*M[0, 0, 1, 0]+conjugate(M[1, 1, 1, 1])*M[1, 0, 1, 0] = 0, conjugate(M[0, 1, 0, 1])*M[0, 0, 0, 1]+conjugate(M[1, 1, 0, 1])*M[1, 0, 0, 1]+conjugate(M[0, 1, 1, 1])*M[0, 0, 1, 1]+conjugate(M[1, 1, 1, 1])*M[1, 0, 1, 1] = 0, conjugate(M[0, 1, 0, 1])*M[0, 1, 0, 0]+conjugate(M[1, 1, 0, 1])*M[1, 1, 0, 0]+conjugate(M[0, 1, 1, 1])*M[0, 1, 1, 0]+conjugate(M[1, 1, 1, 1])*M[1, 1, 1, 0] = 0}$

(4.27)

	Reference

>

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Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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