Download Quantum Mechanics is Real Black Magic Calculus

Using Principles of Quantum Mechanics
for Separating Mixed Signals
Issues to discuss
■ What is quantum mechanics (QM)?
■ Why QM framework is useful for signal processing (SP)?
■ How do we use QM for solving SP problems?
Famous Quotes
■
Albert Einstein:
Quantum Mechanics is Real Black Magic Calculus
■
Richard Feynman:
I think it is safe to say that no one understands Quantum Mechanics
One does not, by knowing all the physical laws as we know them today,
immediately obtain an understanding of anything much.
The more you see how strangely Nature behaves, the harder it is to make
a model that explains how even the simplest phenomena actually work.
B. S. Atal 1
What is Quantum Mechanics
■ Quantum mechanics is the branch of physics that describes
how systems behave at their most fundamental level.
■ According to quantum mechanics, the ‘‘state’’ of a system is
completely specified by a ‘‘state function.’’ The dynamics of
the system is described by the time dependence of this state
function.
■ The ‘‘state function’’ is in general a complex function of
x and t. The ‘‘nature’’ of the system is imbedded in the
physical interpretation of the state function.
■ Rules of quantum mechanics are simple but counter-intuitive.
B. S. Atal 2
Qubits in Quantum Mechanics
■ Quantum computation and quantum information are
built upon the concept of quantum bit or qubit.
■ The state of a qubit is a vector in a two-dimensional
complex vector space.
■ A bit represents one of two points, but a qubit represents any
point on the unit circle in the complex plane.
■ N bits can represent 2**N integer values. N qubits can represent
any complex vector of unit length in 2**N dimensions.
B. S. Atal 3
The Superposition Principle in Quantum Mechanics
B. S. Atal 4
Quantum Mechanical Framework
for Signal Processing
Unitary transform for creating huge constructive interference
Amplify the correct answer while suppressing all other possibilities
B. S. Atal 5
Quantum Noiseless Coding Theorem
By an ingenious argument, the quantum noiseless coding theorem runs parallel
to Shannon’s noiseless coding theorem, using much the same mathematical
ideas. If we consider a long sequence of N systems drawn from the quantum
source, their joint state will be
ρ⊗N = ρ1 ⊗ ρ2 ⊗ . . . ⊗ ρN,
where ρi is the density operator for the ith system. For large enough N, the joint
state ρ⊗N will have support on two orthogonal subspaces, one of which, the
typical subspace, will carry the vast majority of the weight of ρ⊗N, whilst the
other subspace will have vanishingly small weight as N → ∞. Because of this,
the state ρ⊗N may be transmitted with arbitrarily small error by being encoded
onto a channel system of only NS(ρ) qubits. These channel systems may then
be sent to the receiver and the original state recovered with near perfect fidelity.
Reference:
■
Quantum information theory and the foundations of quantum mechanics
Ph. D. thesis – C. G. Timpson, The Queen’s College Oxford University, Trinity 2004
■
Quantum information theory: Results and open problems by Peter Shor
Visions in Mathematics - Towards 2000 conference in Tel Aviv, September 1999
B. S. Atal 6
References
■ Quantum Computation and Quantum Information
Michael A. Nielsen, Isaac L. Chuang
Cambridge University Press 2000
■ Classical and Quantum Computation
A. Yu. Kitaev, A. H. Shen, M. N. Yvalyi
American Mathematical Society 2002
■
Physical justification for using the tensor product to describe
two quantum systems as one joint system
Diederik Aerts and Ingrid Daubechies
Helvetica Physica Acta, 51, 661-675, 1978.
B. S. Atal 7
Leave The Beaten Track
Leave the beaten track occasionally
and dive into the woods.
You will be certain to find something
that you have never seen before.
Alexander Graham Bell
B. S. Atal 8
When One Door Closes, Another Opens
When one door closes, another opens.
But we so often look so long and so regretfully upon the
closed door that we do not see the one which has
opened for us.
Alexander Graham Bell
B. S. Atal 9