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

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

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

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

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

6. Computation as Unitary Evolution Any unitary operator U may be simulated by a set of 1-qubit and 2-qubit gates. * e.g., for a 1-qubit gate: * Barenco, P. Roy. Soc. Lond. A 449, 679 (1995). Evolves via

7. Entanglement as a Resource “Can a quantum system be probabilistically simulated by a classical universal computer? … the answer is certainly, No!” Richard Feynman (1982) “Size matters.” Anonymous “Hilbert space is a big place.” Carlton Caves 1990s 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. A 459, 2011 (2003). Vidal, Phys. Rev. Lett. 91, 147902 (2003). State unentangled if

8. 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 Entanglement as a Prerequisite for Speed-up

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

10. 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 Entanglement as a Prerequisite for Speed-up

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

12. Test for Mixed-State Entanglement s.t. Consider a positive map that is not a CPM  entangled  negative eigenvalues in  entangled. Peres, Phys.Rev.Lett. 77, 1413 (1996). Horodecki 3, Phys.Lett.A 223, 1 (1996). For = partial transpose, this is necessary & sufficient on 2x2 and 2x3 dimensional Hilbert spaces. But positive maps do not fully classify entanglement...

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

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

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

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. * Linden & Popescu, Phys.Rev.Lett. 87, 047901 (2001). 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.

18. 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 Entanglement as a Prerequisite for Speed-up

19. Grover’s Search Algorithm * Suppose we seek a marked number from satisfying: * Grover, Phys.Rev.Lett. 79, 4709 (1997). Classically, finding x 0 takes O ( N ) queries of. Grover’s searching algorithm * on a quantum computer only requires O (  N ) queries. 00 1 2020 2

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

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

22. 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 Entanglement as a Prerequisite for Speed-up

23. Entanglement in Communication: Teleportation Alice Bob Entanglement  out  in 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... * 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.

26. 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. A 34, 6891 (2001). Matsumoto, Shimono & Winter, quant-ph/0206148. Audenaert & Braunstein, quant-ph/030345