QUANTUM ENTANGLEMENT AND IMPLICATIONS IN INFORMATION PROCESSING: Quantum TELEPORTATION K. Mangala Sunder Department of Chemistry IIT Madras

Contents 1. Introduction 2. Bits / Qubits/ Quantum Gates 3. Entanglement 4. Teleportation / Teleportation through gates / Experimental Realization 5. Application of teleportation

Introduction What are quantum computation and quantum information processing? They are concerned with computations and the study of the information processing tasks that can be accomplished using quantum mechanical systems V. Vedral et al, Prog. Quantum Electron 22 (1998), 1-39 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

Toc H Institute of Science and Technology, Kerala4 Introduction One must understand the difference between classical computation and quantum computation. V. Vedral et al, Prog. Quantum Electron 22 (1998), 1-39 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

Toc H Institute of Science and Technology, Kerala5 Introduction However, quantum computation offers an enormous advantage over classical computation in terms of the available data that a computer can handle. conventional computer can do anything a quantum computer is capable of. V. Vedral et al, Prog. Quantum Electron 22 (1998), 1-39 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

Toc H Institute of Science and Technology, Kerala6 Introduction Also by Moore’s Law quantum effects will show up in the functioning of electronic devices as they are made smaller and smaller V. Vedral et al, Prog. Quantum Electron 22 (1998), 1-39 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

Toc H Institute of Science and Technology, Kerala7 Bits / Qubits bit is a fundamental unit of classical computation and classical information only possible values for a classical bit are and quantum analogue of classical bit : quantum bit or qubit

Bits / Qubits qubits are correctly described by two quantum states – can be a linear combination of and. The two states here are the only possible outcomes when you measure the state of the qubit.

Toc H Institute of Science and Technology, Kerala9 Bits / Qubits where and Remember that there are other quantum states where the number of outcomes can be one of many rather than one of two. Those are not considered here. a matrix representation for the states.

where and are complex numbers such that state of a qubit is a vector of unit length in a two dimensional complex vector space

Toc H Institute of Science and Technology, Kerala11 state of a qubit cannot be determined : i.e. a and b cannot be determined from a single measurement. using the normalization condition there is nothing one can experimentally do to them to reveal their states The result above can be derived from simple quantum mechanics of spins

single qubit operation can be described within the Bloch sphere picture where and represent a point on the unit three dimension sphere known as Bloch sphere M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

Multiple Qubits two classical bits can take four different possible values two qubit system has four computational basis states thus the state can be represented as such that

Bell States : play an important role in information processing schemes and Bits qubits 0,1 00,01,10,11 000,001,010,100, 011,101,110,111 This means that quantum computer can, in only one computational step, perform the same mathematical operation on different input numbers encoded in coherent superposition of n qubits

what distinguishes classical and quantum computing is how the information is encoded and manipulated conventional gates are irreversible in operation. Quantum gates are reversible. (will elaborate later) existing real world computers dissipate energy as they run quantum parallelism F =

Quantum Gates quantum computer : quantum circuit containing wires and elementary quantum gates classical computer : electrical circuit containing wires and logic gates Irreversible (except NOT Gate) reversible quantum analogue of NOT Gate is X-Gate it acts on the state of the qubit to interchange the role of computational basis state X X

in matrix form Z Gate : it leaves unchanged but flips the sign of Z in matrix form Z

its operation is just a rotation of about the y-axis followed by a reflection through x-y plane Hadamard Gate H turns and halfway between and and in matrix form

can all 2 dimensional matrices be appropriate for quantum gates for single qubits? Think about it. matrix representing the single qubit gate must be unitary Controlled Not gate two qubit gate : consists of two input qubits known as controlled qubit and target qubit control wire target wire ABA’B’

in matrix form operation will be reversed by merely repeating the gate B’ = A XOR B

Entanglement entanglement is a quantum mechanical phenomena in which the quantum states of two or more particles have to be described collectively without being able to identify individual states entanglement is at the heart of the quantum computation and information processing it introduces the correlation between the particles such that measurement on one particle will affect the state of other algebraically if a composite state is not separable it is called as an entangled state separable state Non-separable for specific values of coeffs. A. Peres, Phys. Rev. Lett. 77 (1996), M. Horodecki et al, Phys. Lett. A 223 (1996), 1-8

2. with probability ½ and post measurement state responsible for the exponential nature of quantum parallelism singlet state neither of the subsystem in singlet state can be attributed by a pure state any measurement on subsystem one leads to two possibilities for the first qubit any subsequent measurement on second sub system will yield in former case and in latter case respectively E. Rieffel et al, ACM Computing Surveys, 32 (2000), with probability ½ and post measurement state

Quantum network to prepare two and three particle entangled states H Bell states H GHZ states G. Brassard et al, Physics D, 120 (1998), Quantum Teleportation and multiphoton Entanglement, Thesis by J. W. Pan, Univ. of Sc. And Tech., China

using single qubit operations and controlled not gates a suitable quantum network can be constructed to produce maximally entangled states e.g. Bell states for two particle system H gate C-NOT 12 inputoutput in a similar way all the four Bell states can be visualized 1 Play a flash movie here for one of them)

H Gate on 1st C-NOT on 1-2

Entangled state analysis photon 1 is in input mode e and photon 2 is in input mode f state of photon 1 state of photon 2 each photon has the same probability to get either reflected or transmitted by beam splitter Photon 1 Photon 2 e f g h

four different possibilities are:

Teleportation branch of quantum information processing transmission of information and reconstruction of the quantum state of the system over arbitrary distances process in which an object can be transported from one location to another remote location without transferring the medium containing the unknown information and without measuring the information content on either side of transport techniques for moving things around, even in the absence of communication channels linking the sender of quantum state to the recipient process in which an object disintegrates at one place and its perfect replica occurs at some other location – no cloning C. H. Bennett et al, Phys. Rev. Lett. 70 (1993),

Bob (receiver)Alice (sender) Alice wants to communicate enough information about to Bob how to do this? quantum systems cannot be fully determined by measurements unknown state

particle 3 particle 2 to couple her particle 1 with EPR pair Alice performs Bell state measurement on her particles complete state of three particles before Alice’s measurement is 23 Quantum carrier to be used

measurement basis expressing each direct product in Bell operator basis all the four outcomes are equally likely, occurring with equal probability 1/4 having Alice tell Bob her measurement outcome, he can recover the unknown state If the outcome is Bob has to do nothing

in all other cases Bob has to use appropriate unitary transformation

Teleportation through gates Quantum circuit for teleporting a qubit here is the unknown state to be teleported and M is probabilistic classical bit G. Brassard et al, Physics D, 120 (1998), 43-47

the state input into the circuit is where first two qubits belongs to Alice and third qubit belongs to Bob Alice sends her qubits through a C-NOT gate, obtaining she then sends the first qubit through the Hadamard gate, obtaining i.e.

if Alice performs a measurement and obtain a result then Bob’s qubit will be in state which is identical to input state with Alice thus knowing the measurement outcome Bob can fix up his state by the application of appropriate gate operation as depending on Alice’s measurements Bob’s qubit will end up in one of these four equally likely possible states with probabilities 1/4

C-NOT on qubit 12 with 3 remaining unchanged: The unitary transformation is Use this in the next two transparencies!!

the state input into the circuit is C-NOT 1-2 H gate on 1

Experimental Quantum Teleportation Pictorial representation of original scheme Click D. Bouwmeester et al, Nature 390 (1997), 575

Pictorial representation of actual set-up D. Bouwmeester et al, Nature 390 (1997), 575 P. G. Kwait et al, Phys. Rev. Lett. 75 (1995),

Application quantum communication is centered on the ability to send data over large distances quickly teleportation has practical application in the field of quantum information processing that hold promises for making computing both much faster and secure it exploits the concept of quantum entanglement which is at the heart of quantum computing during the process the quantum state of an object will be destroyed and not the original object

The most obvious practical application of teleportation is in cryptography. It can provide a completely secure communication between two distant components. Sending photons entangled in a quantum state makes it impossible for an eavesdropper to intercept the message because even if intercepted the message would be unintelligible unless it was intended for a specific recipient.

Toc H Institute of Science and Technology, Kerala44 I wish to thank Mr. Atul Kumar, my Ph. D. Scholar for his enthusiasm to learn this and do further research in this area. Thank you all.