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Entangling Quantum Virtual Subsytems

February 23 2005 Universita’di Milano. Entangling Quantum Virtual Subsytems. Paolo Zanardi ISI Foundation. Unitary mapping. “local observable algebras”. d = prime number. No TPS. Very many possible TPSs. Quantum Tensor Product Structures. H = quantum state space , d=dim(H).

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Entangling Quantum Virtual Subsytems

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  1. February 23 2005 Universita’di Milano Entangling Quantum Virtual Subsytems Paolo Zanardi ISI Foundation

  2. Unitary mapping “local observable algebras” d =prime number No TPS Verymany possible TPSs Quantum Tensor Product Structures H =quantum state space, d=dim(H) Question : How a particular TPS is singled out? Answer : it’s all about operational resources!

  3. Let’s write them as New TPS on with local subalgebras The “swap” operator Un-entangling in the old TPS is now maximally entangling I.e., (C-phase shift) The Bell basis Example

  4. Identical Particles Single particle state-space (L levels) Two particle state-space In general += bosons, -= Fermions (Anti)symmetrization postulate No Natural N-partite TPS over the state-space of N identical particles Fock Space Bosons (L Harmonic oscillators) Fermions: L qubits

  5. II-quantized Modes • Creation-Annihilation ops • Vacuum (no particles) • Occupation number basis Bogoliubov Transformation • Different TPSs on the Fock Space : • Quantum entanglement is relative to the given choice of modes PZ, Phys. Rev. A 65, 042101 (2002)

  6. Collection of *-subalgebras of = algebra of finear operators over the quantum state-space H (i) Each is independently implementable (ii) (iii) (i) (ii) (iii) Virtual Subsystems & Sub-Algebras Dynamical independence Completeness P.Z., D. Lidar and S. Lloyd, PRL (2004)

  7. Noiseless Quantum Subsystems & QIP Fighting Decoherence & control Errors in Quantum Information Processing Error Correction Unifying concept beneath Error Avoiding Noiseless Quantum Subsystem Error Suppression Example of observable-induced TPS E. Knill et al, PRL 84, 2525 (2000); PZ,63, PRA12301(2001)

  8. C H D N Decoherence-free Decoherence-full Errors restricted to C = trivial TENSOR non-trivial Symmetry Duality! Control Operations =non-trivial TENSOR trivial N = Decoherence-Free Subspace For D =1dim e.g., Global SU(2)-singlets for Collective decoherencePZ & Rasetti 1997 Noiseless Quantum Subsystem = factor N of a subspace Cof the state-space H unaffected by unwanted interactions I.e., errors (KLV 1999)

  9. J=irrep label in each Canonical Algebra Pairs: Errors & Control The Errors Symmetry Duality The control = Noisecommutant State-space splits according irreps of theError Algebra A Noise/Control algebras define a bunch of QVSs

  10. Control Algebra = linear combinations of permutations I.e., algebra generated by the permutation group The Prototype: Collective Decoherence (N qubits) Error Algebra= Totally Symmetric Operators (permutation Symmetry) I.e., algebra generated by Collective SU(2) J = total angular momentum N=4one Noiseless qubit sub-space PZ & Rasetti PRL 79, 3306 (1997) N=3 one Noiseless qubit sub-system KLV PRL 84, 2525 (1999) Experimental verification: Ion Traps, Q-Optics, NMR,…

  11. Processing= Creation of excitations, braiding, fusing non-trivial Op on the GS manifold C C= representation space for the Braid-group I.e, b=braid X(b):C C (modular functor) {X(b)}b=Topologically Robust Operations ! Another Example: TOPOLOGICAL QIP (Kitaev 1997, Freedman 2000) Encoding= degenerate & Gapped Ground-state C Degeneration Topologically robust against local perturbations i.e., tunneling amongs GSs exponentially suppressed Leakage gap-suppressed

  12. Error Algebra= local perturbations O trivial topological content NB Thermodynamical limit, f does not depend on i,j Control Algebra = braiding operations= Holonomies over a statistical connectionNon-trivial topological content NBnon-trivial topology of the ambient space e.g., torus (Kitaev 1997) TOP-QIP is based on topologically generatedNS over which Robust computations are performed by means of holonomies P.Z. , S. Lloyd, Phys. Rev. Lett. 90, 067902 (2003)

  13. Nested Subalgebras: Iteration of the irrep decomposition Satisfy (ii), (iii) Virtual multipartiteness: Sub-algebras Chain Example N=6 Qubits Su(2)-triplet 6-Perm irrep We got the tripartite term 3x3-Perm irrep

  14. Conclusions & Summary • Quantum subsystems I.e., TPS, are observable-induced • Quantum entanglement is relative • Noise/Control are canonical (useful) examples • Topological QIP is based on induced TPS Freely drawn from P.Z., PRL. 87, 077901(2001) P.Z. , S. Lloyd, PRL,90, 067902 (2003) P. Z, D. Lidar, S. Lloyd, PRL. 92, 060402 (2004) Thanks for the attention!

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