Universal Optical Operations in Quantum Information Processing Wei-Min Zhang ( Physics Dept, NCKU )

Quantum Information Processing (QIP) Definition of QIP: Uses quantum mechanics for information storage, computation and communication. Implementation of QIP: Is accomplished by applying quantum gates and measurements to prepared qubits. QIP network (to process information): 1. Initializing qubits (quantum state preparation) 2. Quantum gates (quantum state evolutions) 3. Readout (quantum state measurements)

State Preparation for Qubits Qubits: Any two-level physical system |   =  |0  +  |1 , |  | 2 +|  | 2 =1 Typical qubit systems: 1. Photon’s polarization; Photon number 2. Electron spin; Nuclei’s spin 3. The ground and excited states of atoms Requirements: 1. Nearly isolated from environment (avoiding decoherence of qubits) 2. Easy to manipulate (controlling qubits)

Reliable Physical Systems Optical systems One photon and two photon devices, (Linear and nonlinear quantum optics) Donor-atom nuclear spins, electron spin, quantum dots,… Tunable exchange interactions, local magnetic field (Heisenberg systems),… Cavity QED, trapped ions, quantum dots, Josephson-junction,… Laser beam pulses (electron-photon coupling),…

Universal Quantum gates Quantum gates: time evolutions of the designed Hamiltonians acting on qubits. Universal quantum computing: all unitary evolutions can be approximated up to a global phase. Logic gates: universal gate sets acting on one or two qubits.

Measurement (Read-out) Methods: Typically, we have 1. Photons: Homodyne detector…… 2. Spins: Stern-Gerlach apparatus…… But the most useful measurement method for QIP should be the so called Non-demolition measurement using entangled state between the measured system with the “meter” system.

Advantages of QIP: Implementing quantum information with quantum states  Large improvement in computational efficiency and communication security via Superposition principle and non-classical correlations (=entanglement) of quantum mechanics.

Optical computing (As an example of QIP): Basic qubit evolutions in quantum optics: 1. One photon devices: (e.g. Lasers…) H=  a  a –ig * ae i  t + iga  e –i  t 2. Two-photon devices: (e.g. Parametric amplifiers …) H=  1 a 1  a 1 +  2 a 2  a 2 –i  * a 1 a 2 e i  t + i  a 1  a 2  e –i  t

Universal optical operations: Consider two-mode photonic system: States: |n 1 n 2   Fock states of photons Operators: { A i } = { a 1  a 1, a 2  a 2, a 1 , a 1, a 2 , a 2, a 1  a 1 , a 2  a 2 , a 1 a 1, a 2 a 2, a 1  a 2 , a 1 a 2, a 1  a 2, a 2  a 1 }  one and two photon operators General unitary transformations: U = exp {  i b i A i }, b i is a parameter

Disentangling Theorem: ( also called Baker-Campbell-Hausdorff formula ) U = exp {  i b i A i } = P 2 (  )P 1 (  )B 12 ( ,  ) S 12 (  ) S 2 (  )S 1 (  )D 2 (z)D 1 (z) where D i (z)=exp(z i a i  – z i * a i ) S i (  )= exp(  i a i  a i  –  i * a i a i ) S 12 (  )= exp(  a 1  a 2  –  * a 1 a 2 ) B 12 ( ,  )= exp(  e i  a 1  a 2 –  e -i  a 2  a 1 ) P i (  ) = exp(i  i a i  a i ) Zhang, Feng, Gilmore Rev. Mod. Phys. 62, 867 (1990) A subspace of SP(6)

Implementation of optical computing (I): Laser beam: D(  )=exp(  a  –  * a) |0   |  = D(  )|0  = e - |  |/2  n |n  Squeezer: S(  )= exp(  a  a  –  * aa) |0   |   = S(  )|0 ,  = |  |e i  |   | ,  = S(  )|  Qubit preparation : |01   | 0 , |10   | 1   |   = c 0 | 0  +  c 1 | 1 

Implementation of optical computing (II): Parametric amplifiers: S 12 (  )= exp (  a 1  a 2  –  * a 1 a 2 ) Beam splitters: B 12 ( ,  )= exp (  e i  a 1  a 2 –  e - i  a 2  a 1 ) t = cos , r = sin  Phase shifters: P(  ) = exp (i  a  a) Note: The degrees of polarizations of photons can also be easily implemented in these operations.

Implementation of optical computing (III): Homodyne detector: “ the local oscillator has the same frequency as that of the detected signal.” Quantum non-demolition measurement: “to monitor a single observable that can be measured many times with the same result, identical to the first precise result if no external perturbation is applied.” Beam splitters

Applications: Entangled state generation: Using parametric down conversion to generate entangled photon states is currently the most reliable implementation of QIP. For single photon source : Exp. on quantum teleportation: D. Bouwmeester et al. Nature, 390, 575, (1997)

For a laser beam : The relevant Hamiltonian H=  a 1  a 2  –  * a 1 a 2

Teleportation using squeezed states: A. Furusawa, et al. Science, 282, 706 (1998) |   = exp (  a 1  a 2  –  * a 1 a 2 )|0 

Entanglement concentration and purification: Entanglement is the most important resource in QIP. Entanglement concentration, purification and distillation are the central issue in the implementation of entangled states.

Quantum key distribution : Quantum secure communication is based on quantum key distribution: Generating m entangled pairs between distant sides Alice and Bob [ Ekert’s scheme, PRL, 67, 661 (1991 )]. Alice and Bob make their local photon number measurements (using homodyne detectors) of the two-mode polarized squeezed states they shared, and then compared their result via classical communication, and keep the photon states with the same n. Then Alice and Bob share multiphoton maximal entangled states. These multiphoton maximal entangled states can be used not only for quantum key distribution, but also for quantum error correction, etc. AliceBob

Optical Manipulation of Quantum computation: 1. One qubit gates: evolutions of single qubit ( |0  =|01 , |1  =|10  ) which can be manipulated simply by the use of phase shifter: P(  ) = exp [i  (a 1  a 1 - a 2  a 2 ) ]  exp [i   z /2] beam splitter: B 12 (  )= exp [  (a 1  a 2 –a 2  a 1 )]  exp [i   y ]  Any simple qubit rotation R R( , ,  ) = P(  ) B 12 (  ) P(  ) e.g. Hadamard gate:

Superposition of qubits:

2. Two qubit controlled gates: Photonic two-qubit controlled gates are much more difficulties to be implemented due to the fact that photons do not interact each other. Historically, the first scheme of implementing optical CNOT gates is the use of nonlinear effect of Kerr medium [Chuang & Yamamoto, PRA52, 3489 (1995)] A scheme of universal quantum computation using single-qubit operation and teleportation is proposed [Gottesman & Chuang, Nature, 402, 390 (1999)] : |out  =CNOT|   |   A non-deterministic scheme of quantum computation with linear optics is proposed [Knill, Laflamme & Milburn, Nature, 409, 46 (2001)]

Challenge of optical controlled gates :

Summary and Future Works: Only a few reliable operations (available in Lab.) are enough for implementing various QIP protocols: the universal set of one and two qubit logical gates, entangled state generations, concentration, quantum key distribution, quantum teleportation quantum error corrections …… Two important problems are waited to work out: Efficient two qubit optical controlled gates Entanglement distillation of mixed photon states The above procedure is also useful to analyze the universal operations of electron-photon coupling systems and spin systems.