Linear-optical quantum information processing: A few experiments

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1 Linear-optical quantum information processing: A few experiments Miloslav Dušek Lucie Čelechovská, Karel Lemr, Michal Mičuda, Antonín Černoch, Jaromír Fiurášek, Miroslav Ježek, ek, Jan Soubusta, Radim Filip, Konrad Kieling, Jens Eisert,, Helena Fikerová,, Martina Miková Department of Optics, Palacký University, Olomouc, Joint Laboratory of Optics of Palacký acký University and Institute of Physics of Academy of Sciences of the Czech Republic, Institute of Physics and Astronomy, University of Potsdam, Institute for Advanced Study, Berlin.

2 Quantum information processing Information has a physical character what we can do with it depends on the physical system which carries it. Quantum systems behave more strangely than classical ones (superpositions of states, entanglement, non-locality, intrinsic randomness and non-linearity of quantum measurements). Quantum effects offer the solution of some tasks which cannot be solved in classical information theory or whose classical solution is unknown (secure distribution of cryptographic key, factorization of large numbers in polynomial time and other hard computational problems). Classical bit: 0 or Quantum bit: α 0 + β Quantum register superposition of states of the whole register, e.g. ( 00 + ) 2 (entangled state)

3 Quantum information processing Quantum cryptography Secret communication: security is guarantied by the laws of physics (secure key distribution, eavesdropping can be detected) Can profitably use: quantum repeaters, distillation of entanglement, quantum memories, etc. Quantum computation Efficient (polynomial) algorithms for factorization (using quantum Fourier transform), discrete logarithm, database search, etc. Can exploit: quantum correction codes, preparation of complex entangled states, quantum teleportation, etc. To built quantum circuits we need quantum gates (analogous to classical logical gates) Qubits in quantum gates must interact with each other in a controlled way but must not interact with the environment (in order to keep superpositions). This is a difficult task. Many platforms are considered: trapped ions, atoms in a cavity, electrons in a quantum dot, light etc.

4 Linear-optical quantum information processing Photons are good carriers of quantum information but they do not interact with each other well. Quantum gates need controlled interaction between qubits. Interaction of qubits requires non-linearity. Quantum measurement is nonlinear (breaks superpositions) it can emulate nonlinearity in linear optical quantum gates (after the measurement on an auxiliary system the state of the whole system collapses). But quantum measurement is also probabilistic (gives random results) linear optical quantum gates are probabilistic (sometimes fail). Fortunately, for many small-scale quantum computing tasks this is not a key problem. Especially for quantum information processing immediately after the quantum transmission the probability of success lower than one is not essential because the losses on the quantum channel are usually a few orders of magnitude higher. Linear-optical gates are experimentally feasible. They works directly with photons without the necessity to transfer the quantum state of a photonic qubit into another quantum system like an ion etc. (photons are useful for communication purposes).

5 D Programmable unambiguous discriminator of weak coherent states L. Bartůšková,, A. Černoch, J. Soubusta,, M. Dušek, Phys. Rev.. A 77,, (2008) An unknown state can equal to either one of the two, in general non-orthogonal, program states. [M. Sedlák et al., Phys. Rev. A 76, (2007)] clicks: α = α, D clicks: α = α? 2 2? T 0 =, T = 2, T 2 = nm, 4 ns Active stabilization

6 Programmable unambiguous discriminator of weak coherent states p η exp α α, η is detector efficiency 3 η 2 suc = α α 2 η The number of program states can be increased quantum database search [M. Sedlák,, et al., Phys. Rev. A 76, (2007)]

7 Encoding two qubits into a single qutrit L. Bartůšková,, A. Černoch, R. Filip, J. Fiurášek,, J. Soubusta,, M. Dušek, Phys. Rev. A 74,, (2006) Alice encodes an arbitrary pure product state of two qubits into a state of one qutrit.. Bob can then restore either of the two encoded qubit states but not both of them simultaneously. Both the encoding and decoding are probabilistic but error free. β } β 2 2 Ψ = α 0 + Ψ = α Ψ Ψ Φ = N αα + α β + β β 2 ( ) [A.Grudka and A.Wojcik, Phys.Lett. A 34, 350 (2003)]

8 Encoding two qubits into a single qutrit 3 T α α ( R T ) α β η Rβ β2 00, R =, T =, η = f f f f f f4 f f 2 f Photon pairs at 826 nm: SPDC in LiIO 3, cw pump If D3 clicks there is at most one photon in f, f2, f4. These three modes constitute a qutrit.

9 Programmable gate for an arbitrary rotation of a single qubit along the z axis M. čuda, M. žek, M. šek, J. ášek ek,, Phys. Mi Je Du Fiur Phys Rev 78 M. Mič Mičuda, M. Jež Ježek, M. Duš Dušek, J. Fiuráš Fiurášek, Phys.. Rev. Rev.. A A 78, 78,, (2008) (2008) It applies a unitary phase shift to a data qubit qubit.. The value of the phase shift is determined by the state of a program qubit qubit.. [G. Lett [G. Vidal Vidal et et al., al., Phys. Phys. Rev. Rev. Lett. Lett.. 88, 88, (2002)] (2002)] Polarization encoding is used Photon pairs at 84 nm: SPDC in LiIO33, cw pump (CUBE)

10 Programmable gate for an arbitrary rotation of a single qubit along the z axis out 2 ( i φ ) ( H V ) Input: ψ = α H + β V, ψ = H + e V D If the program qubit is found in state Then the data qubit is in state ψ = α H ± β e V P D ± = ± P 2 i φ H e V Reconstructed process matrices (real part left, imaginary part right). Process fidelity exceeds 97%. An exact specification of the phase shift would require infinitely many classical bits. Here the information is encoded into a single qubit. Using a different set of program states the device can operate as a programmable partial polarization filter.

11 Partial-SWAP gates including entangling square-root root of SWAP A. Černoch, J. Soubusta,, L. Bartůšková,, M. Dušek, J. Fiurášek, Phys. Rev. Lett. 00, 8050 (2008) i φ U = Π + e Π, where Π = Ψ Ψ, Π = I Π φ + + (to implement U ϕ three CNOT gates are required in general) Our scheme can reliably implement a whole class of inequivalent operations by changing the length of one interferometer arm. φ = 0... identity φ = π... SWAP 2 φ = π... SWAP Photon pairs at 826 nm: SPDC in LiIO 3, cw pump The gate is successful only if we detect a single photon in each output port.

12 Partial-SWAP gates including entangling square-root root of SWAP Output 2-qubit 2 state tomography Input state: V H 2 ( V H i H V ) Quantum process tomography for ϕ =π/2 (matrix 6x6)

13 Partial symmetrization and anti- symmetrization of two-qubit states A. Černoch, J. Soubusta, L. Bartůšková, M. Dušek, J. Fiurášek, New J. Phys., (2009). The filter enables an arbitrary attenuation of either the symmetric or anti-symmetric part of the input two-qubit state. i V = T Π + e φ T Π, where Π = Ψ Ψ, Π = I Π S + A +

14 Partial symmetrization and anti- symmetrization of two-qubit states 36 input states, 9 measurement bases, ϕ = 0 Tomography of output states, Process tomography Partial anti-symmetrization, T A = Partial symmetrization, T S = Utilization for optimal universal asymmetric quantum cloning A. Černoch, J. Soubusta, L. Čelechovská, M. Dušek, J. Fiurášek, Phys. Rev. A 80, (2009),

15 Optimal controlled phase gate with an arbitrary phase shift K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, M. Dušek, submitted to Phys. Rev. Lett. The phase shift can be tuned to any given value. 0, 0 0, 0 0, 0, For each phase shift the gate operates at the, 0, 0 maximum success probability achievable within the i, e framework of linear optics. ϕ, [K. Kieling, J.L. O'Brien, J. Eisert, New J. Phys. 2, 3003 (200)] Bulk optical elements Polarization encoding of qubit states

16 Optimal controlled phase gate with an arbitrary phase shift Re Choi matrix for φ =π Reconstructed: Ideal: Choi matrix for φ =π /2 Reconstructed: Ideal: Re Im The optimum success probability is not monotonous in the phase.

17 Programmable phase gate increasing probability of success H. Fikerová, M. Miková, M. Dušek It applies a unitary phase shift to a data qubit. The value of the phase shift is determined by the state of a program qubit. Theoretical limit of the success probability: 50% Success probab.. of our orig. implementation: 25% If the program qubit is found in state Then the data qubit is in state 2 ( 0 ) i α 0 βe φ We ignored these results, but we can change +

18 Programmable phase gate increasing probability of success Electrronic feed-forward forward The output quantum state is changed according to the measurement result (useful also for other experiments) Success probability: 50% Fiber-optics implementation - spatial-mode encoding is used Photon pairs at 84 nm: SPDC in LiIO 3, cw pump U det U λ / 2 5 V,.5 V

19 Other experiments Optimal symmetric and asymmetric phase-covariant quantum cloning J. Soubusta,, L. Bartůšková,, A. Černoch, M. Dušek, J. Fiurášek, Phys. Rev. A 78, (2008) J. Soubusta,, L. Bartůšková,, A. Černoch, J. Fiurášek,, M. Dušek, Phys. Rev. A 76,, (2007) Bartůšková,, M. Dušek, A. Černoch, J. Soubusta,, J. Fiurášek, Phys. Rev. Lett. 99,, (2007) A. Černoch, L. Bartůšková,, J. Soubusta,, M. Ježek, ek, J. Fiurášek,, M. Dušek, Phys. Rev.. A 74,, (2006) Programmable quantum-state discriminator and Phase-covariant quantum multimeter J. Soubusta,, A. Černoch, J. Fiurášek,, M. Dušek, Phys. Rev.. A 69,, (2004)

20 Thank you for your attention