Entanglement in Quantum Information Processing 25 April 2004

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

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

Classical/Quantum State Representation Bit has two values only: 0, 1 Information is physical BITS

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. . .

Fast Quantum Computation (Shor) (Grover) (slide with permission D. Di. Vincenzo)

Fast Quantum Computation (Shor) (Grover) (slide with permission D. Di. Vincenzo)

Computational Complexity Computational complexity: how the `time’ to complete an algorithm scales with the

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

Picturing Entanglement Pure states are entangled if e. g. , Bell state (picture from

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

Computation as Unitary Evolution Evolves via U Any unitary operator may be simulated by

Computation as Unitary Evolution Evolves via U Any unitary operator 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).

Entanglement as a Resource State unentangled if “Can a quantum system be probabilistically simulated

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

Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement

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

Gottesman-Knill theorem* • The Pauli group Pn is generated by the n-fold tensor product

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

Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement

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

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

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

Test for Mixed-State Entanglement Consider a positive map s. t. that is not a

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

Liquid-State NMR Quantum Computation Utilizes so-called pseudo-pure states which occur in NMR experiments with

Liquid-State NMR Quantum Computation 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. (figure from Nature 2002)

NMR Quantum Computation (1997 - ) Selected publications: Nature (1997), Gershenfeld et al. ,

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

Does NMR Computation involve Entanglement? most negative eigenvalue 4 n-1(-2) = -22 n-1 whereas

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

For NMR states so if unentangled In current liquid-state NMR experiments ~ 10 -5,

For NMR states so if 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).

Can there be Speed-Up in NMR QC? For Shor’s factoring algorithm, Linden and Popescu*

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).

Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement

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

Grover’s Search Algorithm* Suppose we seek a marked number from satisfying: Classically, finding x

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

Can there be Speed-up without Entanglement? At step k In Schmidt basis Project onto

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

At step k, the probability of success must be amplified through repetition or parallelism

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 ) 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).

Entanglement as a Prerequisite for Speed-up Naively, to get an exponential speed-up, the entanglement

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

Entanglement in Communication: Teleportation Alice rin In the absence of entanglement, the fidelity of

Entanglement in Communication: Teleportation Alice rin In the absence of entanglement, the fidelity of the output state F= is bounded. Bob rout Entanglement 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 (Di. Martini 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)

Summary The role of entanglement in quantum information processing is not yet well understood.

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 todate, an intuition which extends to few-qubit systems. In communication entanglement is much better understood, but there are still important open questions.

Entanglement in communication The role of entanglement is much better understood, but there are

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. Û additivity of the entanglement of formation. Û 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