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Complex Vector State
Notes from Quantum Computing for Computer Scientists book by Noson S. Yanofsky and Micro A. Mannucci
Set of Vectors
Primary example of a complex vector space is set of vectors of a fixed length with complex enteries. These vectors describe the states of a quantum systems and quantum computers. Example: set of vectors of length 4.
a typical element of it will look like:
To simply put means a matrix of ( 1D array )having complex numbers as in it.
Operations
All operations that we can perform on Real vector space can be performed on complex vector space. taking example of
Addition
Consider:
That is
Properties followed by addition operator:
commutative Associative Additive Inverse, ie. here
C^4 = C × C × C × C
6 − 4 i 7 + 3 i 4.2 − 8.1 i −3 i
C n^ n × 1
6 − 4 i 7 + 3 i 4.2 − 8.1 i −3 i
16 + 2.3 i −7 i 6 −4 i
22 − 1.7 i 7 − 4 i 10.2 − 8.1 i −7 i
V + W ∈ C^4
V + W = W + V
( V + W ) + X = V + ( W + X )
V + Z = 0 Z = − V
Scalar multplication
some other Scaler multplication properties:
, the set of all m-by-n matrices, with complex enteries in it.
Basis and Dimension
A set of vectors is called a basis of a (complex) vector space if both
every, can be written as a liner combination of vectors from and is linearly independent,each of the vectors in the set cannot be written as a combination of the others in the set.
Example : we can say is a liner combination of
as
The dimension of a (complex) vector space is the number of elements in a basis of the vector space. For example, if is a complex vector space with a basis , then the dimension of is.
has dimension n as a real vector space. has dimension n as a complex vector space
(3 + 2 i ). =
6 + 3 i 0 5 + 1 i 4
12 + 12 i 0 13 + 13 i 12 + 8 i
1. V = V
c 1. ( c 2. V ) = ( c 1 × c 2 ). V c. ( V + W ) = c. V + c. W ( c 1 + c 2 ). V = c 1. V + c 2. V
C m^ ×^ n
B = { V 0, V 1,... , V n − 1} ⊆ V V
V ∈ V B B { V 0 , V 1 , … , Vn −1}
[45.3, −2.9, 31.1] T
, , , and
V B = { v 1 , v 2 , … , vn } V n
Rn C n
if then and are orthogonal , ie. perpendicular to each other.
A basis is called an Ortho-Gonal basis, if vectors are pairwise orthogonal to each other, ie. , where.
if norm of every vector pairwise is equal to 1 , then it's an Ortho-Normal basis.
In , we can write any as
A Hilbert space is a complex inner product space that is complete.
Eigen-Values and Eign-Vectors
For a matrix in , if there is a number c in and a vector V \ne 0 with such that
then,
c is eign-value of A , V is eign-vector of A.
Hermitian and Unitary Matrices
A n-by-n matrix is called hermitian if , also known as self-adjoint
If is a hermitian n-by-n matrix, then for all we have
Try proof, solution at page 81.
For a given hermitian matrix , distinct eigenvectors that have distinct eigenvalues are orthogonal. Page 82
An n-by-n matrix is Unitary if
if is a unitary matrix and and are in , then
ie, preservers the distance(An operator that preserves distances is called an isometry ).
d ( V 1 , V 2 ) = | V 1 − V 2 | = √⟨ V 1 − V 2 , V 1 − V 2 ⟩
⟨ V 1 , V 2 ⟩ = 0 V 1 V 2
β = { V 0 , V 1 , … , Vn −1} ⟨ Vj , Vk ⟩ = 0 j ≠ k
C n^ V V = ⟨ E 0 , V ⟩ E 0 + ⟨ E 1 , V ⟩ E 1 + ⋯ + ⟨ En −1, V ⟩ En −
A C n × n^ C C n
AV = c. V
A A †^ = A
A V , V ∈ C n ⟨ AV , V ′⟩ = ⟨ V , AV ′⟩
U ⋆ U †^ = U †^ ⋆ U = In
U V 1 V 2 C n d ( UV 1 , UV 2 ) = d ( V 1 , V 2 ) U
If is unitary and , then we can easily form and multiply both sides of the equation by to get. In other words, because is unitary, there is a related matrix that can und the action that performs. takes the result of ’s action and gets back the original vector. In the quantum world, all actions (that are not measurements) are “undoable” or “reversible” in such a manner.
U UV = V U †
U †^ U † UV = U † V or V = U † V U U U † U
