INTERNATIONAL WORKSHOP ON QUANTUM INFORMATION QUANTUM TELEPORTATION Invited Talk by HARI PRAKASH INDIAN INSTITUTE OF INFORMATION TECHNOLOGY, ALLAHABAD, INDIA and PHYSICS DEPARTMENT, UNIVERSITY OF ALLAHABAD, ALLAHABAD, INDIA WORKERS IN THIS FIELD IN THE RESEARCH GROUP: Prof. Naresh Chandra, Prof. Ranjana Prakash, Ajay K. Maurya, Ajay K. Yadav, Vikram Verma & Manoj K. Mishra WORKERS IN OTHER FIELDS IN THE RESEARCH GROUP: Dr. Devendra K. Singh. Dr. Rakesh Kumar Dr. Pankaj Kumar, Dr. Devendra K. Mishra & Namrata Shukla INTERNATIONAL WORKSHOP ON QUANTUM INFORMATION HRI, Feb. 26, 2012

Experimental Quantum Teleportation (Institut fu¨r Experimentalphysik, Universita¨t Innsbruck) The experimental set-up: A pulse of ultra- violet radiation passing through a nonlinear crystal creates the ancillary pair of photons 2 and 3. After retroflection during its second passage through the crystal the ultraviolet pulse creates another pair of photons, one of which will be prepared in the initial state of photon 1 to be teleported, the other one serving as a trigger indicating that a photon to be teleported is under way. Alice then looks for coincidences after a beam splitter BS where the initial photon and one of the ancillaries are superposed. Bob, after receiving classical Information that Alice obtained coincidence count in detectors f1 and f2 identifying Bell state, knows that his photon 3 is in the initial state of photon 1 which he then can check using polarization analysis with polarizing beam splitter PBS and detectors d1 and d2. The detector p provides Information that photon 1is under way. [ Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter & Anton Zeilinger, Nature 390,575 (1997)]

Experimental Quantum Teleportation with a Complete Bell State Measurement using Nonlinear Interaction Principle schematic of quantum teleportation with a complete BSM. Nonlinear interactions (SFG) are used to perform the BSM. and represent the respective horizontal and vertical orientations of the optic axes of the crystals. [Yoon-Ho Kim, Sergei P. Kulik, and Yanhua Shih, Phys. Rev. Lett. 86, 1370 (2001)]

optical sum frequency generation (SFG) (or “up-conversion”). BSM is based on nonlinear interactions: optical sum frequency generation (SFG) (or “up-conversion”). First type-I SFG crystal: Second type-I SFG crystal: First type-II SFG crystal: Second type-II SFG crystal:

QUANTUM TELEPORTATION: SIMPLE THEORY INFORMATION: ENTANGLED STATE: BELL STATES COMPOSITE STATE: BOB’S UNITARY TRANSFORMATION: QUALITY OF TELEPORTATION: FIDELITY = , where is the Teleported state, gives the quality of Teleportation.

MAVFI ≥ MASFI ≥ C (equality for C = 1, only) and MAVFI × MASFI ≥ C. STANDARD QUANTUM TELEPORTATION For Information state of particle 1 and Entangled state of particles 2 and 3, particle 1 & 2 go to Alice for BSM and particle 3 goes to Bob where it takes up later, after a unitary transformation dependent on result of BSM, the information state. CONTROLLED QUANTUM TELEPORTATION For Information state of particle 1 and Entangled state of particles 2, 3 and 4, particle 1 & 2 go to Alice for BSM, particle 3 goes to Charlie for determination of its state (ground or excited) and particle 4 goes to Bob, where it takes up later, after a unitary transformation dependent on results of BSM of Alice and state determination by Charlie, the information state. USE OF NON-MAXIMALLY ENTANGLED STATES The Fidelity for each outcome of the BSM is seen to depend on the initial state and the minimum value of Fidelity, the MASFI is seen to be 2C/(1+C) in terms of concurrence C. The average value of average Fidelity, MAVFI, is seen to satisfy MAVFI ≥ MASFI ≥ C (equality for C = 1, only) and MAVFI × MASFI ≥ C. Since MAVFI ≈ 0.5 and MASFI ≈ 2C, for C<<1, MASFI obviously gives the best idea about the measure of quality of Teleportation when F begins to depend on the initial state. H.Prakash & V. Verma, Q. Inf. Processing, Online Dec. 2011

H. Prakash & Vikram Verma arXiv:1110.1220v1 [quant-ph] SQT of N-Qubits It has been showed that SQT of N Qubits requires Entangled state of at least 2N qubits, Prakash H, Chandra N, Prakash R, Dixit A., MOD. PHYS. LETT. B . 21(2007)  2019   H. Prakash & Vikram Verma arXiv:1110.1220v1 [quant-ph]

H. Prakash & Ajay K. Maurya, Opt. Comm.284(2011)5024. MAGIC BASIS Bennett et al defined a magic Basis with the following states Name was given by Hill & Wootters If entangled state is called CONCURRENCE. C = 1 leads to SQT with F=1 and C=0 gives SQT where MASFI = 0. We tried to find if the Magic Bases exist for states with larger number of entangled qubits and found that For even qubit states with N > 2, no magic bases exist but Magic Partial Bases exist. The number of basis states for Magic Partial Bases is n < , but if holds, gives perfect SQT. H. Prakash & Vikram Verma arXiv:1110.1220v1 [quant-ph] For 3 entangled qubits, magic basis is seen to exist for both SQT and CQT if the destinations of qubits are fixed. But, if destinations of qubits are not fixed, a Magic Partial basis is seen to exist H. Prakash & Ajay K. Maurya, Opt. Comm.284(2011)5024.

MAGIC STATES FOR TWO ENTANGLED QUBITS Introduced by Bennett et al (Phys. Rev. A54(1996)3824) Obviously, these define matrices and have the property that if C=1 gives SQT with F=1 and C=0 gives F dependent on and MASFI ≡ This can be derived by demanding for where Bell states are defined by matrices and It gives real, imaginary,

H Prakash and A K Maurya, Opt. Commun. 284 (2011) 5024.

· Van Enk & Hirota Scheme of QT Entangled coherent states are stronger than standard bi-photonic states against possible photon transfer to reservoir modes

Van Enk & Hirota noted that One of the counts is always 0. If the other counts are odd, teleportation is successful. If the other counts are even, teleportation fails. Success of teleportation is ½.

Our Scheme differs from the above We divide counts in to: Zero, nonzero even and odd Zero counts gives failure Odd counts gives perfect QT Nonzero even counts gives almost perfect QT For |α|² = 1, 2 or 5 minimum average fidelity = 0.73, 0.9987, 0.9999 respectively

For photon counting modes 4 & 6 and Bob’s mode 2 One of the modes 4 and 6 is and Hence one of the count is always zero For count in the other mode Van Enk & Hirota wrote We write where

In the callouts:

These lead to ~ denotes unnormalized state Fidelity defined by or where and

Information ( i.e., and ) - arbitrary Consider MASFI (Minimum Assured Fidelity), the minimum of F for variation of If Pi is probability for occurrence of case I, we will define average fidelity as This has minimum value, which we call minimum average fidelity (MAVFI), For

Teleportation with non-maximally entangled coherent state Recently H. Prakash and Manoj K. Mishra noted a very interesting thing that MAVFI increases on decreasing entanglement. Use of non-maximally entangled coherent state, in place of maximally entangled coherent state, gives, MAVFI =0 for both zero count, F=1 for one nonzero even count and F≈1 for one odd count. Minimum average fidelity in this case is greater then that when maximally entangled state used by an amount, It is seen that this is important as there is a marked increase in MAVFI at low |α|, and it is difficult to generate superposed coherent states with large |α|.

Wang’s Teleportation of Bipartite State Phys. Rev. A 64(2001)022302

Our Scheme for QT of Bipartite State

TELEPORTATION OF 2-QUBIT INFORMATION ENCRYPTED IN SINGLE MODE SUPERPOSED COHERENT STATE are superposed coherent states with 4n, 4n+1, 4n+2, 4n+3 photons respectively. Entangled state: Photon counting is done in four modes. Almost perfect fidelity is obtained if three counts are nonzero and one count zero Minimum average fidelity is ≥ 0.99 for |α| ≥ 3.2

SCHEME FOR TELEPORTATION OF 2-QUBIT INFORMATION

Taking Decoherence into Account

Case with Nonzero Even Counts Figure 4: Variation of MAF with for different values of

Critical value of = 0.738 Case with Odd Counts Figure 6: Variation of MAF with for different values of Critical value of = 0.738

Figure7: Variation of average fidelity with for different values of at = 0.9.

Entanglement Diversion

LONG DISTANCE QT WITH PERFECT FIDELITY AND AS GOOD SUCCESS AS DESIREED USING REPEATED ATTEMPTS For QT using ECS Photon-counting in two modes – one count always zero Fidelity F = 0 when the other count is zero F= 1 when other counts are odd MASFI ≈ 1 when other counts is nonzero-even and |α|2 is appreciable In photon-counting information is destroyed and measurements cannot be repeated to increase success Repeated attempts can be possible only if the information is not destroyed in measurements We present another scheme of QT of state of a qubit with Alice on to a qubit with Bob using (i) a light pulse in even coherent state, (ii) two light pulses in coherent state (iii) Beam splitter assemblies and light detectors. This suits long distance Quantum Teleportation serving as a link between two quantum processors at long distances the photons fly and the quantum state of qubit is teleported.

Idea is based on results of Wang & Duan (2005, Phys. Rev Idea is based on results of Wang & Duan (2005, Phys. Rev. A) for reflection of light in coherent state from a single atom cavity Atomic states: ground state , very low lying excited state and an excited state ; are resonantly coupled to the cavity mode driven by a resonant optical pulse in coherent state ; states are largely detuned. Cavity reflects state to (i) state if the atom is in the state and to (ii) state if the atom is in the state . Thus, , and therefore and the atom and light are entangled. We need 3 BEAM SPLITTER ASSEMBLIES (BSA) and 4light detectors. BSA consists of symmetric Beam splitter and phase Shifters so as to give right output phases.

Thank You