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? B. S. Atal 1 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.

What is Quantum Mechanics ■ 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. ■ Rules of quantum mechanics are simple but counter-intuitive. ■ 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. ■ Quantum mechanics is the branch of physics that describes how systems behave at their most fundamental level. 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 B. S. Atal 5 Unitary transform for creating huge constructive interference Amplify the correct answer while suppressing all other possibilities

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 Quantum Noiseless Coding Theorem B. S. Atal 6

■ 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, , References 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