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

2. Quantum Tensor Product Structures H =quantum state space, d=dim(H) d =prime numberNo TPS Very many possible TPSs Unitary mapping Question : How a particular TPS is singled out? “local observable algebras” Answer : it’s all about operational resources!

3. The Bell basis Example 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)

4. 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 Identical Particles

5. II-quantized Modes Vacuum (no particles) Creation-Annihilation ops 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. Virtual Subsystems & Sub-Algebras (ii) (iii) (i) Eachis independently implementable Dynamical independence Completeness (i) (ii) (iii) P.Z., D. Lidar and S. Lloyd, PRL (2004) Collection of *-subalgebras of = algebra of finear operators over the quantum state-space H

7. Noiseless Quantum Subsystems & QIP Fighting Decoherence & control Errors in Quantum Information Processing Error Correction Error Avoiding Error Suppression Noiseless Quantum Subsystem E. Knill et al, PRL 84, 2525 (2000); PZ, 63, PRA 12301 (2001) Unifying concept beneath Example of observable-induced TPS

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

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

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

11. 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 Processing = Creation of excitations, braiding, fusing non-trivial Op on the GS manifold C 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 !

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

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

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