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Unitary matrices are the complex analogues of orthogonal matrices, and both are very common in the theory of Lie groups and Lie algebras. Orthogonal matrices are the matrix representations of real linear maps that preserve distance. A unitary matrix is a square matrix of complex numbers, whose inverse is equal to its both conjugate and transpose. This is basically the complex version of real orthonormal matrices. All rows of unitary matrices are ortho normal and all columns are again ortho normal to each other. The magnitude of unitary matrix in most of the times unity and mainly used to indicate the phase in the physical system. They are mathematically related with unitary transformation. In general the unitary operator can be expressed by U. Then by definition U U † = Ӏ or U † =U -1. Unitary matrices are the complex versions, and they are the matrix representations of linear maps on complex vector spaces that preserve "complex distances”.

If U is the unitary operator acts on function H then unitary transformation can be expressed as UH=H. [1-3] It was found that Unitary matrix can be the frame to describe the different complex particles and wave functions .[4] We observe the increase in the fidelity of the output quantum state both in a quantum emulation experiment where all protecting unitaries are perfect and in a real experiment with a cloud-accessible quantum processor where protecting unitaries themselves are affected by the noise. [5] To qualify as the fundamental quantum variables of a physical system, a set of operators must suffice to construct all possible quantities of that system. Such operators will therefore be identified as the generators of a complete operator basis. Unitary operator bases are the principal subject . Here exists two state vector space coordinate systems and a rule of correspondence define a unitary operator.[6] several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space are discussed. Then using the uncertainty relation for the unitary operators we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anti-commuting observables in higher dimensions. Our discussion mainly aims to reveals the following properties of the magical functions on Time evaluation, Elementary particle representation, Utility in signal processing using Fourier Matrix and application for Mathematical reduction in complexity.

The theoretical discussion gives  the real need of unitary matrices and some scientists  call the  unitary operator is a magical operator.

Theoretical  Description :

Origin of  Unit Matrix

According to quantum mechanics in state space there are collections of  vectors. These vectors can represent the waves and  in state space there is no time. Time is not considered as a vector in any vector space or any function in function space. Time is coming just to indicate the change of vectors in vector space and it is completely auxiliary in vector space.The state of a particular phenomena at instant of time t₀  can be expressed as |ψ(r,⃗ t_o)⟩  and after time t-t₀  the state of the phenomena  becomes  |ψ(r,⃗ t)⟩   for t > t_o.  The state moves at every instant of time.  If we normalize the state then  the  tip of the state vector will trace  on the surface of the unit sphere. This movement of state vector keeping its magnitude and  fixed can be expressed by an special operator  which basically changing angle. This  can be properly expressed by the    unitary operator.

This can be expressed by |ψ ,t_o⟩  =U(t,t₀)|ψ ,t⟩ . This operator can be expressed by  matrix also, same like rotation operator just rotating some vectors. R(θ)V₁=V₂  . Here  we can conclude some properties of unitary time evaluation operator in similarity with general unitary operator.

Here  U acts as time evaluation operator.[7] The time evaluation operator has certain properties.

Property: 1

 U(t,t₀))† U(t,t₀) = 1  , that means U(t,t₀))† =[U(t,t₀)] ^-1

Property: 2

|Ψ, t₂) = U(t₂, t₁)|Ψ, t₁) = U(t₂, t₁)U(t₁, t₀)|Ψ, t₀).

|Ψ, t₂) = U(t₂, t₀)|Ψ, t₀)  ,

That means  U(t₂, t₀)=U(t₂, t₁) U(t₁, t₀).

This chain length multiplication is possible only since the unitary operator has unity length in magnitude. Its significance is in its phase and it has phase only. Next we find what actually unitary operator is. We use this operator on Schrodinger equation.

-iћ[∂Ψ /∂t]=HΨ 

Where H is total energy and we call it as Hamiltonian. Hamiltonian in general express total energy and it is an operator also. The  Hamiltonian can be expressed as Hermitian matrix. Hermitian matrix is always relevant in quantum mechanics due to its real eigen values. Now we replace the wave function Ψ  by the Unitary operator U .

  -iћ[∂U/∂t]=HU

Solution gives U=exp[-iHt/ћ] or U(t,t₀)=exp[-iH(t-t₀)/ћ]. This result is very much expected because we know the unitary operator which  has phase only. We can also express in another form.

Let we  have two vectors in complex plane of same magnitude. This may be  P=x+iy  and Q=x-iy .

Definitely we can  express     U=[P/Q] since magnitude is unity or ‖IU‖=1

=[(x+iy)/(x-iy)]   Let x=Acosθ  and y=Asinθ

=[ (cosθ+isinθ)/(cosθ-isinθ)]

     =exp(iθ)/exp(-iθ)

=exp(-2iθ)

     =e−iφ      where φ = 2θ

Relations  Between Unitary operator with Hamiltonian;

Let |Ψ, t) = U(t,t₀)|Ψ, t₀)

∂/∂t  [|Ψ, t)]=  ∂/∂t[U(t,t₀)|Ψ,t₀)]

=  ∂/∂t[U(t,t₀)U(t₀,t)|Ψ,t)]

=  ∂/∂t[U(t,t₀)U(t,t₀)^†|Ψ,t)]

 =   Λ(t,t₀)|Ψ,t)     where Λ(t,t₀) = ∂/∂t[U(t,t₀)]U(t,t₀)^†

Now Λ(t,t₀) = ∂/∂t[U(t,t₀)U(t,t₀)^†]

                    =∂/∂t[U(t,t₀)] U(t₀,t₁) U(t₀,t₁) ^† U(t,t₀)^†]

                    =∂/∂t[U(t,t₁)] U(t₁,t₀) U(t₀,t)]

                    =∂/∂t[U(t,t₁)] U(t₁,t)

                    = Λ(t,t₁)

This shows Λ is independent of t and t₁ or any time interval.  Most important thing is that  the unit of ћ Λ is energy. Using all the similarities we can say  i ћ Λ= Hamiltonian. We can write Schrodinger equation by different ways as follows using condition  ∂/∂t |Ψ, t) = Λ(t)|Ψ, t)

1.             iћ ∂/∂t |Ψ, t) =iћ Λ(t)|Ψ, t)

2.             iћ ∂/∂t |Ψ, t)=H|Ψ, t)

So  if our unitary operator is known then definitely our Hamiltonian will be known. This helps us to determine the  future state with the help of  initial state  from time evaluations of the system.

Unitary Operator  Can Act as a Basis in   More Extended Complex Plane

In  our ordinary 3D space , we require three basis vectors to indicate any 3D vectors. Here three  basis vectors are  given and these are mostly taken and  these three can form a ortho-normal basis.

[8]Now to express the  complex particles like electron we need  more extended  and  complex vector space. In this space we in general use  2x2  complex matrices which must be unitary. The unitary nature is mandatory because the unit magnitude normalize the space.  Here the new space is completely orthonormal. Here the well known Pauli Matrices  can act as basis  and most beautiful thing is that all the Pauli matrices are orthogonal and they can be made orthonormal. These  matrices  are complex 2x2 matrices  . The 2x2 identity  matrix with these Pauli matrices  can form the basis.

Unitary Operator  Can act as a Representation of  Elementary Particles

Elementary particlescan be broadly classified into two groups. Particles with  integer spin is called bosons (ex photons ) and  particles with half of odd integral multiple of spin called fermions (ex electron ). In general the  bosonic state can be expressed by rotational matrix and the fermonic state can be expressed by unitary matrix or by unitary operator.    The particles undergoing  strong interactions behave in such a way that all the   particleswhich are subjected to strong interactions  must belong to a particular family. These particles may have multiplates but they follow the conservation principle  of iso-spin. These conservation principle can  follow a group property. These groups are unitary in nature. The  most of the of elementary particles can follow particular patterns. Here eight particles are forming the picture  and corresponding symmetry is called the eightfold way. The process of  forming different isospinmultiplates is carried out a special type of trans formation  known as a special unitary transformation. These multiplates  transformation  be expressed by 2x2 unitary matrices. These  generators originate from the  three Pauli matrices with identiy.

According to conventional idea  protons and neutrons  are completely different particles. But irrespective of  their great difference in   charge and  a little difference in mass they are projections of  some fundamental entity in two different plane. Here the conservation  of isospin is observed. Since the conservation is coming so automatically a group is formed and this group is basically the  unitary group.

 and it express the state of a system of a identical entity. When this identical entity is projected at an particular angle q then this particular state can express elementary particles which is one of the member of octave. This member can be expressed by an unitary group member which can be express by U (q) = exp [-s₁q/2]. This form basically SU (2) group.

 In the similar way for other ser of  elementary particles  3x3 unitary matrix can form the basis.  These basis  are also very useful   to express different multi-plate  and conservation principle of  mostly heavy elementary particles called baryons. These are the origin of SU(3) groups. Here  the generators and total group is shown.

The continuous group has infinite number of elements. In general the elements of the continuous groups are produced from a generator. Most  of the generators are unitary in nature though the formula of generators have no limitations to select the unitary matrix. 

Fourier Matrix and Fast Fourier Transform

The Fourier matrix is like square ortho normal columns  just we vave to take the transpose of components. It is defined as  F= ( F_n)_ij =ω_ij    where  i,j = 0 ,1,2,3…..n-1. Here  ω is a special number  and  ω_n=1=Exp (2Πi/n) . ω is describing  the unit circle in the complex plane.

The powers on the unit circle and angles are doubled with powers. For n=6 , θ=Π/3 and

for  n^{2     ;}            θ= 2 Π/3.  Now Fourier  Matrix  F₄   or 4x4  can be expressed  as given below

Columns of the matrix are orthogonal and  any columns are orthogonal to the other column. The inner product has to be taken in conjugate form. The length square is 4. So to make it orthonormal we need to divide by 1/4  . With the properties of Fourier Transform we can establish  a direct connection in between  F₄  and F₈  and  between F₈ and F₁₆  and so on. Here squaring is equal to double the angle.

Signals are transformed by fourier matrix and latter can be transformed into inverse fourier matrix. The main point is that fourier and inverse fourier matrix are very simple and their product is also very simple. FFT can perform the 1024 = 2¹⁰steps of any matrix multiplication by 5 steps only. It basically replaces the steps n²multiplication  by 0.5nl terms where n=2^l

The infinite sum of Fourier series can be expressed as a summation of  number of components. It basically changes from continuous to discrete form. This may be expressed as

Where

Finally, from the properties of matrix multiplication we can represent the DFT as a simple matrix operation. (If this is not immediately obvious, think about a simple case where N=K=2).

Let our original  R₄  matrix  and it can be arranged in a special form.

This special arrangement has all the unique terms irrespective of the size of each matrix. All the matrices  have common block of elements. Here we use the redundancy to reduce the number of operations required to compute the DFT . Fast-Fourier Transform (FFT).The blocks are shown  below.

Fourier Matrices in a this special form is arranged as a product of 3 building blocks of matrices as given below.

 

We can develop more larger dimension of fouriermatrix  satisfying the same sequence.

Here we have the R₁₆ expressed in terms of R₈that follows R₄ and R₂.

Repeating 16 x16 Fourier Matrix in terms of 4 x4 Fourier Matrix . Writing all the terms we can say that  R₁₆

The Fast-Fourier Transform (FFT) is a powerful tool. As its name it gives a very rapid process of multiplication  and Fourier Transform which is very much required in signal processing  technique.  This  includes  signal  production,  and  parity checking in  spectral analysis.

Unitary matrix in form of  House holder matrices;

By House holder transformation  we basically  reflects the vector about some mirror plane and  this matrix is basically reflection  matrix. If  in reflection in proper direction  the length remains same then the house holder matrix must be unitary. Let we have a vector x. Then the vector is reflected  by the mirror  which can be expressed by linear transformation Px.

The matrix P is house holder matrix and it follows the properties.

This is purely a unitary matrix. So we can get replacement of physical reflector by House holder matrix.

1. “Trace-wise Orthogonal Matrices 4” (Mathematica notebook dated 17 February 2012) and also “Aspects of the theory of Clifford algebras”: notes for a seminar prresented 27 March 1968 to the Reed College Math Club. 2. J. Schwinger, “Unitary operator bases,” PNAS 46, 570 (1960) 3. http://en.wikipedia.org/wiki/Butson-type–Hadamard–matrices. The original reference is A. T. Butson, “Generalized Hadamard matrices,” Proc. Amer. Math. Soc. 13, 894-898 (1962) 4. Suppressing Decoherence in Quantum State Transfer with Unitary Operations Maxim A. Gavreev, Evgeniy O. Kiktenko , Alena S. Mastiukova and Aleksey K. Fedorov , 30 December 2022 ,Entropy 5. Unitary operator bases , J. Schwinger, Harvard University , February 2, 1960 6. Uncertainty Relations for General Unitary Operators Shrobona Bagchi1, and Arun Kumar Pati1, Phys. Rev. A 94, 042104 – Published 5 October 2016. 7. MIT Open course ware – Barton Zebrich, Quantum Mechanics – II 8. Mathematical Methods for the physicists ,6e – Afken & Weber , Academic Press , 84 Theobqald’s Road , London WCIX8RR,UK. 9. Linear Algebra and Its Application , Gilbert Strang , Cengage learning India Private Limited, 418,FIE,Prataprgang,Delhi 1100092. India. 10. Elements of Group theory for Physiscists ,New Age International (P) Limited, Publishers 7/30 A Daryagang , New Delhi-110002.