TeachersBruno BenedettiLorenzo Orecchia

StudentStefano FrancoBari, 26/07/2013

“God does not throw dice”(Einstein, 4 December 1926)

Summary● Introduction● Basic Quantum Mechanics● Qubit● EPR Paradox● Bell's inequality● Example● References● Conclusions

Introduction

A quantum computer is a computation device that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.

Why Quantum Computation?● No limitations on computation imposed by the extended Church-Tuing thesis ● Random number ● No cloning● Quantum teleportation● Non-locality (entanglement)● Cryptography

What is Quantum Computation?

Basic Quantum Mechanics (postulates)

1. Superposition Principle

2. Measurement Principle

3. Unitary evolution

Qubits (or quantum bit)The basic entity of quantum information (analogue of the bit for classical computation)

Sfera di Bloch

Curiosity: how many information can be stored by a qubit?

Exactly 2, like a classical bit(Holevo, 1973)

Two Qubits

0 0

1 1

Can we say what the state of each of the individual qubits is? NO: entanglement!

Bell's states (or EPR pairs) maximally entangled states of two qubits

EPR Paradox (1935)

Can quantum mechanics be complete?

Einstein Podolsky RosenAssumption

1. Physics reality

2. Locality

3. Completeness

Bell's state

There exist local hidden variables!

Bell's Inequality (1964)(experimentally Aspect and co-workers, 1981)

“There does not exist any local hidden variable theory consistent with outcomes of quantum physics”

Consequences● Entanglement is not paradossal● Quantum correlations in an EPR pair are

“stronger” than classical correlations

Example: more efficient information processing by use of shared entanglement

Classical Computation

a b

YOU WIN 75% OF THE TIMES

Quantum Computation

Protocol:

EPR pair

CLAIM:

Recall (superpositional principle and rotation matrix)In general, rotation of a state

by an angle in the two-dimensional state space gives the rotaded state

where

Hence the probability of measuring a 0 for the rotated state is given by

By calculating:

Let's start:

Conclusion:

With Quantum Computer you can win more often!

YOU WIN 85% OF THE TIMES

References● Wikipedia - Paradosso EPR- Teoria delle variabili nascoste- Teorema di Bell- Qubit- Entanglement quantistico- Notazione bra-ket- Informatica quantistica- Ampiezza di probabilità

● Introduction, Axioms, Bell Inequalities (Lecture 1, Spring 2007, CS 294-2)

● Qubit gates and EPR (Lecture 5, Fall 2007, C/CS/Phys C191)

● Entanglement can facilitate information processing (Lecture 5, Fall 2005, C/CS/Phys C191)

ConclusionsAbout the courseVery interesting course, in many respects. These activities improve people and institutions. I hope it will be the first of many others.

About Quantum ComputationI think that the current paradoxes about quantum mechanics are comparable to Zenone's paradoxes. One day, perhaps, all things will be clearer.

“God does not throw dice”But we really love doing it!

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