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October 23, 2002 Theoretical Aspects of Experimental Entanglement and Teleportation Horace P. Yuen, Max Raginsky, Mauro D’Ariano, Masanao Ozawa Entanglement and Teleportation • • • • Qubit Entanglement Qumode Entanglement Entanglement Concentration Quantum Repeaters – Device & System Imperfections – Possible Applications vs. Classical 2 Entanglement • Qubit: Ψ − = 1 2 (0 1 1 2 − 1 0 1 2 ) maximally entangled — measure of degree of entanglement E Bell states Ψ µ (basis) Φµ ∆ • Qumode: 1 2 (0 1 0 2 2-mode squeezed state Ψ AB [( = exp r a a − a A a B λ = tanh r E(Ψ ) † A † B )] 0 = 1− λ 2 µ 1 3 1 ∞ 2 λ ∑ n n =0 can be increased by QND measurement of NA+NB 1 A 2 n ) B Effect of Imperfection • Error model of – 1-qubit operations p1 – 2-qubit operations p2 – 1-qubit measurement η 1− p 1 2 1−η error propabilities • Imperfect Bell-measurement described by – 1 imperfect 2-qubit operation – 2imperfect 1-qubit measurements ⇒ Fmax maximum attainable F, Fmax < 1 Fmin minimum F necessary for improvement F~1-ε ε linear is imperfection 4 General Problem • If state is totally unknown, can(not) use teleportation – Qubit Case: ↑1→ 2− →1↑ 2 0 Need to know relative polarization to make the appropriate Bell measurement – Qumode case: • q0 q1 q2 p1 p2 Need to know relative phase to make the appropriate homodyne measurement 5 Coherence and Teleportation • • Optical Coherence Requirement for Continuous Variable Quantum Teleportation – Rudolph and Sanders PRL 87, 2001 Optical Coherence: a convenient fiction – Molmer PRA 55, 1997 ? quantum coherence * uniform random phase in (laser) coherent state amplitude ⇒ diagonal in number-state representation ⇒ no coherent 2-mode squeezed state * usual coherence a result of same source 6 Coherence cont. • The Quantum State of propagation laser field – van Enk and Fuchs q-ph Nov 2001 (Journal of quantum Information) Field state inside cavity vs. outside • The Partitum Ensemble Fallacy Fallacy – Nemoto and Braunstein q-ph Jul 2002 Phase unobservable in-principle *privileged role of coherent state 7 Space-time coherence • Any single-mode excited field is fully space-time coherent – So ρ diagonal in n O.K. – “Coherent states” special in that a multimode CS field is equivalent to a single-mode one – Only fixed relative phases between modes essential for homodyne detection • Field expansion (quantum Karhunen-Loeve) E( x, t ) = ∑ akΨ k (x, t ) k Joint ρ for all k-modes, α k classically 8 Difference between LED & LASER • Multimode random variables {αk } vs single appropriate mode (random phase or not) • Phase Φ(t) fluctuates slowly for lasers − a fixed unknown parameter Φ0 over observation time • Φ0 can be measured – Split strong beam and measure the phase – Laser field does have a definite phase 9 Squeezed States Teleportation c = µa + νb d = µb + νa † † [a,a ] = I † [a, b] = 0 etc. ν → νe jφ φ pump laser phase • Only need to look at appropriately phase-locked • Phase of state to be teleported defined also by 10 { a,b} modes { a,b} Fallacies: • Confuse unknown parameter with a random variable • Confuse ensemble average with time average • Confuse relative phase with absolute phase – cancellation of fluctuation 11 Entanglement at distance: Qubits vs. Qumodes • Problem: – Obtain maximally entangled states suitable for teleportation in the presence of loss. – Compare efficiency of the two following schemes for establishing N shared ebits (entangled qubit pairs) in the presence of loss: • (a) N singlets: N single ebits with qubits encoded on polarization; • (b) 1 twin beam: 1 pair of entangled qumodes + LOCC to obtain a maximally entangled state of dim 2N, with qubits encoded on photon number. • Result: – Scheme (a) is exponentially more efficient than scheme (b). Moreover: – Scheme (a) allows knowingly successful teleportation with a fixed protocol/machine. G. M. D’Ariano and M. F. Sacchi, quant-ph/0205073 (to appear on Forscr. der Phys.) 12 Anonymous Key Identification (AKI) • If quantum memory is available, one can have 1. A identifies to B without on-line third party 2. B cannot pretend to be A 3. Can solve the N2 problem in key management as follows: challenge A | θA + θB 〉, | θ’B 〉 B | θA 〉 stored, θA known only to A, picks random θB, θ’B response A B | θB 〉 | θA + θ’B 〉 to replenish 13 Checks | θ B 〉, stores | θA 〉 Two Distinct Issues • Whether there is, for US QBC: – An impossibility theorem – A protocol provably unconditionally secure 14 A priori No Impossibility Theorem without QBC Definition • Similar to: – No Church-Turing theorem (only Thesis) without definition of “algorithm” • Different Forms of algorithms: – Turing, Post, Markov, etc. • Different Types of QBC protocols: – 1,2,3,4,… from cheating detection, quantum games, evidence questioning etc. 15 Fundamental Stability Issues for Large-Scale Quantum Computers • As we have shown, any physical channel can be modeled arbitrarily closely by a Strictly Contractive Quantum Channel (SCC) – SCC causes any initial state to evolve exponentially to a unique final state – Maximally-mixed-state (MMS) preserving SCC strictly increases the entropy of any other state • • Goal: Determine stability of large scale quantum computers under the influence of MMS-preserving SCCs Results: – Derived lower bounds on increase of entropy due to such SCCs – Derived limits on run-time and size of circuit-based quantum computers – Cellular automata (interacting particle systems) provide a more robust medium for quantum information processors M. Raginsky, "Entropy production rates of bistochastic SCC on a matrix algebra,” mathph/0207041; to appear in J. Phys A. “Entropy-energy balance in noisy quantum computers,” QCMC’02 Proceedings, to appear. “Almost any quantum spin system with short-range interactions can support toric codes,” Phys. Lett. A 294, 153 (2002). 16 Quantum Limits on Quantum Computing Induced by Conservation Laws • Assume: – (i)Computational basis is represented by a component of spin – (ii)Physical realizations obey the angular momentum conservation laws • Then, controlled NOT gates cannot be realized with error probability less than 1/4S2, where S is the size of the ancilla. – Fermionic ancilla: S = # of qubits included in the ancilla particles – Bosonic ancilla: S2 = 4 x (the average photon # in the ancilla field) • To prevent this limitation altogether, we need to code the computational basis to commute with the angular momentum. M. Ozawa, “Conservation laws, uncertainty relations, and quantum limits of measurements,” Phys. Rev. Lett. 88, 050402 (2002); “Conservative Quantum Computing,” Phys. Rev. Lett. 89, 0507902 (2002). 17 Future Plan I. Study the possibility of robust teleportation of a qubit vs qumode II. Develop appropriate direct criterion for imperfect teleportation of quantum states with corresponding efficient entanglement concentration schemes III. Quantify AKI key management protocol performance 18