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Entanglement in Quantum Information Processing

Entanglement in Quantum Information Processing. 25 April, 2004 Les Houches. Samuel L. Braunstein University of York. Superposition between two rays in Hilbert space. Entanglement between (distant) objects. Classical/Quantum State Representation. Bit has two values only: 0, 1

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Entanglement in Quantum Information Processing

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  1. Entanglement in Quantum Information Processing 25 April, 2004 Les Houches Samuel L. Braunstein University of York

  2. Superposition between two rays in Hilbert space Entanglement between (distant) objects Classical/Quantum State Representation Bit has two values only:0, 1 Information is physical BITS  QUBITS Many qubits leads to ...

  3. Fast Quantum Computation (Shor) (Grover) (slide with permission D.DiVincenzo)

  4. PSPACE NP BQP BPP P factoring* primality testing Computational Complexity Computational complexity: how the `time’ to complete an algorithm scales with the size of the input. Quantum computers add a new complexity class: BQP† For machines that can simulate each other in polynomial time. *Shor,35th Proc. FOCS, ed.Goldwasser (1994) p.124 †Bernstein & Vazirani, SIAM J.Comput.25, 1411 (1997).

  5. e.g., Bell state Picturing Entanglement Pure states are entangled if (picture from Physics World cover)

  6. Evolves via Computation as Unitary Evolution Any unitary operatorUmay be simulated by a set of 1-qubit and 2-qubit gates.* e.g., for a 1-qubit gate: *Barenco, P. Roy. Soc. Lond. A449, 679 (1995).

  7. State unentangled if “Can a quantum system be probabilistically simulated by a classical universal computer? … the answer is certainly, No!” Richard Feynman (1982) “Hilbert space is a big place.” Carlton Caves 1990s “Size matters.” Anonymous Entanglement as a Resource Theorem: Pure-state quantum algorithms may be efficiently simulated classically, provided there is a bounded amount of global entanglement. Jozsa & Linden, P. Roy. Soc. Lond. A459, 2011 (2003). Vidal, Phys. Rev. Lett.91, 147902 (2003).

  8. Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement must grow with the size of the input. • Caveats: • Converse isn’t true, e.g., Gottesman-Knill theorem • Doesn’t apply to mixed-state computation, e.g., NMR • Doesn’t apply to query complexity, e.g., Grover • Not meaningful for communication, e.g., teleportation

  9. The Pauli groupPnis generated by the n-fold tensor product • of , , , and factors±1and±i.  Pn stabilizes • Gates: , , , , , map subgroups ofPnto subgroups ofPn.  any computation restricted to these gates may be simulated efficiently within the stabilizer formalism. Gottesman-Knill theorem* • Subgroups ofPnhave compact descriptions. stabilizes . *Gottesman, PhD thesis, Caltech (1997).

  10. Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement must grow with the size of the input. • Caveats: • Converse isn’t true, e.g., Gottesman-Knill theorem* • Doesn’t apply to mixed-state computation, e.g., NMR • Doesn’t apply to query complexity, e.g., Grover • Not meaningful for communication, e.g., teleportation

  11. Since write mixture so For on unentangled if: , otherwise entangled. Mixed-State Entanglement

  12. Consider a positive map that is not a CPM s.t. entangled For = partial transpose, this is necessary & sufficient on 2x2 and 2x3 dimensional Hilbert spaces. But positive maps do not fully classify entanglement ... Test for Mixed-State Entanglement  negative eigenvalues in entangled. Peres, Phys.Rev.Lett.77, 1413 (1996). Horodecki3, Phys.Lett.A223, 1 (1996).

  13. Utilizes so-called pseudo-pure states which occur in NMR experiments with small For any unitary transformation is pseudo-pure with replaced by The algorithm unfolds as usual on pure state perturbation for traceless observables , Each molecule is a little quantum computer. Liquid-State NMR Quantum Computation (figure from Nature 2002)

  14. NMR Quantum Computation (1997 - ) Selected publications: Nature (1997), Gershenfeld et al., NMR scheme Nature (1998), Jones et al., Grover’s algorithm Nature (1998), Chuang et al., Deutsch-Jozsa alg. Science (1998), Knill et al., Decoherence Nature (1998), Nielsen et al., Teleportation Nature (2000), Knill et al., Algorithm benchmarking Nature (2001), Lieven et al., Shor’s algorithm But mixed-state entanglement and hence computation is elusive. Physics Today (Jan. 2000), first community-wide debates ...

  15. most negative eigenvalue 4n-1(-2) = -22n-1  whereas for ,  is unentangled Does NMR Computation involve Entanglement?

  16. For NMR states so if  unentangled   unentangled In current liquid-state NMR experiments ~ 10-5, n < 10 qubits  no entangled states accessed to-date …or is there? Braunstein et al, Phys.Rev.Lett.83, 1054 (1999).

  17. Can there be Speed-Up in NMR QC? For Shor’s factoring algorithm, Linden and Popescu* showed that in the absence of entanglement, no speed-up is possible with pseudo-pure states. Caveat: Result is asymptotic in the number of qubits (current NMR experiments involve < 10 qubits). For a non-asymptotic result, we must move away from computational complexity, say to query complexity. *Linden & Popescu, Phys.Rev.Lett.87, 047901 (2001).

  18. Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement must grow with the size of the input. • Caveats: • Converse isn’t true, e.g., Gottesman-Knill theorem* • Doesn’t apply to mixed-state computation, e.g., NMR • Doesn’t apply to query complexity, e.g., Grover • Not meaningful for communication, e.g., teleportation

  19. Classically, finding x0 takes O(N) queries of . Grover’s searching algorithm*on a quantum computer only requiresO(N) queries. 0 1 2 20 Grover’s Search Algorithm* Suppose we seek a marked number from satisfying: *Grover, Phys.Rev.Lett.79, 4709 (1997).

  20. At stepk In Schmidt basis Project onto . is entangled when . Since projection cannot create entanglement, if unentangled  . Can there be Speed-up without Entanglement?

  21. At step k, the probability of success must be amplified through repetition or parallelism (many molecules). Each repetition involves k+1function evaluations. `Unentangled’ query complexity (using ) We find that entanglement is necessary for obtaining speed-up for Grover’s algorithm in liquid-state NMR. Braunstein & Pati, Quant.Inf.Commun.2, 399 (2002).

  22. Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement must grow with the size of the input. • Caveats: • Converse isn’t true, e.g., Gottesman-Knill theorem* • Doesn’t apply to mixed-state computation, e.g., NMR • Doesn’t apply to query complexity, e.g., Grover • Not meaningful for communication, e.g., teleportation

  23. Alice Bob rin rout In the absence of entanglement, the fidelity of the output state F = is bounded. e.g., for teleporting qubits, F  2/3 whereas for the teleportation of coherent states in an infinite-dimensional Hilbert space F  1/2.* Fidelities above these bounds were achieved in teleportation experiments (DiMartini et al, 1998 for qubits; Kimble et al 1998 for coherent states). Entanglement matters! Absence of entanglement precludes better-than-classical fidelity (NMR). NB Teleportation only uses operations covered by G-K (or generalization to infinite-dimensional Hilbert space†). Simulation is not everything ... Entanglement Entanglement in Communication: Teleportation *Braunstein et al, J.Mod.Opt.47, 267 (2000) †Braunstein et al, Phys.Rev.Lett.88, 097904 (2002)

  24. Summary The role of entanglement in quantum information processing is not yet well understood. For pure states unbounded amounts of entanglement are a rough measure of the complexity of the underlying quantum state. However, there are exceptions … For mixed states, even the unentangled state description is already complex. Nonetheless, entanglement seems to play the same role (for speed-up) in all examples examined to-date, an intuition which extends to few-qubit systems. In communication entanglement is much better understood, but there are still important open questions.

  25. Entanglement in communication • The role of entanglement is much better understood, • but there are still important open questions … • Theorem:* • additivity of the Holevo capacity of a quantum channel. • additivity of the entanglement of formation. • strong super-additivity of the entanglement of formation. If true, then we would say that wholesale is unnecessary! We can buy entanglement or Holevo capacity retail. *Shor, quant-ph/0305035 some key steps by: Hayden, Horodecki & Terhal, J. Phys. A34, 6891 (2001). Matsumoto, Shimono & Winter, quant-ph/0206148. Audenaert & Braunstein, quant-ph/030345

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