Quantum information processing

1. Quantum information processing Myungshik Kim Queen’s University, Belfast

2. What is quantum information processing? • A research in quantum information processing is to understand how quantum mechanics can improve acquisition, transmission and processing of information. • Review article: Bennett & DiVincenzo, Nature 404, 247 (2000) Who may be involved? • Computer scientists – What are difficult problems ? • Mathematicians – Coding and information theory • Electrical engineers – electronic quant info processor • Chemists – Chemical quantum devices • Physicists – Developer of quantum theory

3. Quantum coherence PBS • Example 1: • By focusing a /2 pulse on a two level atom • Example 2: • Schroedinger cat states •  denotes the amplitude of a coherent state • By nonlinear Kerr interaction ; Kerr medium

4. Why quantum coherence? • Consider the density operator • Recall Young’s double slit interference • Allows massive parallelism in quantum computation Classical Quantum

5. What is entanglement ? • Two important ingredients • Random • Deterministic Type II nonlinear crystal Energy & momentum conservation One photon is polarised & the other polarised  polarised  or   or   polarised Hong+Mandel, PRL 59, 2044 (1987); Kwiat et al., PRL 75, 4337 (1995)

6. There are four basis states • They are so called the Bell states • Braunstein et al. PRL 68, 3259 (1992) • States maximally violating Bell’s inequality Bell

7. How about ? • Is it entangled? Yes, as far as a0 & b1. • How do we define a degree of entanglement? • For a pure state, the von Neumann entropy is a useful tool. • Find  for two particles • Find the reduced density operator for one particle: • Substitute 1 into the definition of the von Neumann entropy to find out the degree of entanglement: S = -Tr 1 ln 1.

8. How to manufacture an arbitrary entangled state? White & Kwiat (PRL 83, 3103 (1999)) a) Twin photons are emitted at two identical down-conversion crystals. The crystals are oriented so that the optic axis of the first (second) lies in the vertical (horizontal) plane b) Adjustable quarter- & half-wave plates and polarising beam splitters allow polarisation analysis in any basis.

9. Experimentally reconstructed density matrices of states that are a) b) c)

10. pure states mixed states Is a mixed state always classical? • Eh? What is a mixed state? • Why mixed states? As soon as a quantum state is embedded in an environment, the pure state becomes mixed.

11. A mixed state is not always classical. • Consider • For a=b=1/2 the density operator describes a pure state and for b=0 a classical mixture. • There should be some critical point when the system loses quantum nature (entanglement). • We need to define a measure of entanglement for a mixed state.

12. D2 D1 /2 T D3 BS BBO PBS (transmits  reflects ) Multi-particle entanglementBouwmeester+Zeilinger, PRL 82, 1345 (1999) • Three-particle entanglement • so called GHZ(Greenberger-Horne-Zeilinger) state • Generation: • Use type II interaction • Two pairs are generated • GHZ for four simultaneous clicks • How do we define a degree of entanglement? Any use? Sackett & Wineland, Experimental entanglement of four paritcles, Nature 404, 256 (2000)

13. Higher-dimensional entanglement • Quantum information processing for qubits. • Quantum effects for a d-dimensional system:qudits • Most quantum systems require the use of an infinite dimensional vector space. • Possibility to test experimentally. • Laser • Provide a test ground for paradoxes in quantum mechanics. • Photonic state: Defined in phase space-continuous variables • Braunstein “error correction for continuous quantum variables”, PRL 80, 4084 (1998) • Braunstein, “quantum error correction for communication with linear optics”, Nature 394, 47 (1998) • Braunstein&Lloyd, “quantum computation over continuous variables”, PRL 82, 1784 (1999) • Boto&Braunstein, “quantum interferometric optical lithography”, PRL 85, 2733 (2000)