Gilles Brassard. Université de Montréal

Transcription

1 Gilles Brassard Université de Montréal

2 Gilles Brassard Université de Montréal

3 VOLUME 76, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 29 JANUARY 1996 Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels Charles H. Bennett, 1, * Gilles Brassard, 2, Sandu Popescu, 3, Benjamin Schumacher, 4, John A. Smolin, 5,k and William K. Wootters 6, 1 IBM Research Division, Yorktown Heights, New York Département IRO, Université de Montréal, C.P. 6128, Succursale centre-ville, Montréal, Québec, Canada H3C 3J7 3 Physics Department, Tel Aviv University, Tel Aviv, Israel 4 Physics Department, Kenyon College, Gambier, Ohio Physics Department, University of California at Los Angeles, Los Angeles, California Physics Department, Williams College, Williamstown, Massachusetts (Received 24 April 1995) Two separated observers, by applying local operations to a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity (e.g., near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield D M of pure singlets jc 2 distillable from mixed states M, showing D M. 0 if C 2 jmjc PACS numbers: Bz, Dv, c The techniques of quantum teleportation [1] and quantum data compression [2,3] exemplify a new goal of quantum information theory, namely, to understand the kind and quantity of channel resources needed for the transmission of intact quantum states, rather than classical information, from a sender to a receiver. In this approach, the quantum source S is viewed as an ensemble of pure states c i, typically not all orthogonal, emitted with known probabilities p i. Transmission of quantum information through a channel is considered successful if the channel outputs closely approximate the inputs as quantum states. Because nonorthogonal states, in principle, cannot be observed without disturbing them, their faithful transmission requires that the entire transmission processes be carried out by a physical apparatus that functions obliviously, that is, without knowing or learning which c i are passing through. Just as classical data compression techniques allow data from a classical source to be faithfully transmitted using a number of bits per signal asymptotically approaching the source s Shannon entropy, 2 P i p i log 2 p i, quantum data compression [2,3] allows quantum data to be transmitted, with asymptotically perfect fidelity, using a number of 2- state quantum systems or qubits (e.g., spin- 1 2 particles) asymptotically approaching the source s von Neumann entropy S r 2Trr log 2 r, where r X p i jc i c i j. (1) i Quantum teleportation achieves the goal of faithful transmission in a different way, by substituting classical communication and prior entanglement for a direct quantum channel. Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally entangled qubits (e.g., two spin- 1 2 particles in a pure singlet state) previously shared between sender and receiver, and a 2-bit classical message from the sender to the receiver. Both quantum data compression and teleportation require a noiseless quantum channel in the former case for the direct quantum transmission and in the latter for sharing the entangled particles yet available channels are typically noisy. Since quantum information cannot be cloned [4], it would perhaps appear impossible to use redundancy in the usual way to correct errors. Nevertheless, quantum error-correcting codes have recently been discovered [5] which operate in a subtler way, essentially by embedding the quantum information to be protected in a subspace so oriented in a larger Hilbert space as to leak little or no information to the environment, within a given noise model. We describe another approach in which the noisy channel is not used to transmit the source states directly, but rather to share entangled pairs (e.g., singlets) for use in teleportation. But before they can be used to teleport reliably, the entangled pairs must be purified converted to almost perfectly entangled states from the mixed entangled states that result from transmission through the noisy channel. We show below how the two observers can accomplish this purification, by performing local unitary operations and measurements on the shared entangled pairs, coordinating their actions through classical messages, and sacrificing some of the entangled pairs to increase the purity of the remaining ones. Once this is done, the resulting almost perfectly pure, almost perfectly entangled pairs can be used, in conjunction with classical messages, to teleport the unknown quantum states c i from sender to receiver with high fidelity. The overall result is to simulate a noiseless quantum channel by a noisy one, supplemented by local actions and classical communication. Let M be a general mixed state of two spin- 1 2 particles, from which we wish to distill some pure entanglement. The state M could result, for example, (5) 722(4)$ The American Physical Society

4 Noisy Channel Alice Bob

5 Alice Bob 0 0

6 Alice Bob 0 0 Perfect with probability Randomized with prob 1/2 1/2

7

8

9 Werner State 2 qubits jointly in state with probability F (F = 5/8)

10 Distinguishing Bell States { { 0 1 or or 0 1

11 Manipulating Bell States - +

12 Manipulating Bell States and are unaffected (up to phase)

13

14

15 before after target example probability source target source target = Φ? F = 5/8 Φ + Φ + F 2 Φ + Φ + YES 25/64 Φ + Φ F (1 F )/3 Φ Φ YES 5/64 Φ + Ψ + F (1 F )/3 Φ + Ψ + 5/64 Φ + Ψ F (1 F )/3 Φ Ψ 5/64 Φ Φ + F (1 F )/3 Φ Φ + YES 5/64 Φ Φ (1 F ) 2 /9 Φ + Φ YES 1/64 Φ Ψ + (1 F ) 2 /9 Φ Ψ + 1/64 Φ Ψ (1 F ) 2 /9 Φ + Ψ 1/64 Ψ + Φ + F (1 F )/3 Ψ + Ψ + 5/64 Ψ + Φ (1 F ) 2 /9 Ψ Ψ 1/64 Ψ + Ψ + (1 F ) 2 /9 Ψ + Φ + YES 1/64 Ψ + Ψ (1 F ) 2 /9 Ψ Φ YES 1/64 Ψ Φ + F (1 F )/3 Ψ Ψ + 5/64 Ψ Φ (1 F ) 2 /9 Ψ + Ψ 1/64 Ψ Ψ + (1 F ) 2 /9 Ψ Φ + 1/64 Ψ Ψ (1 F ) 2 /9 Ψ + Φ YES 1/64

16 before after target example probability source target source target = Φ? F = 5/8 Φ + Φ + F 2 Φ + Φ + YES 25/64 Φ + Φ F (1 F )/3 Φ Φ YES 5/64 Φ + Ψ + Φ + Ψ Φ Φ + F (1 F )/3 Φ Φ + YES 5/64 Φ Φ (1 F ) 2 /9 Φ + Φ YES 1/64 Φ Ψ + Φ Ψ Ψ + Φ + Ψ + Φ Ψ + Ψ + (1 F ) 2 /9 Ψ + Φ + YES 1/64 Ψ + Ψ (1 F ) 2 /9 Ψ Φ YES 1/64 Ψ Φ + Ψ Φ Ψ Ψ + (1 F ) 2 /9 Ψ Φ + YES 1/64 Ψ Ψ (1 F ) 2 /9 Ψ + Φ YES 1/64 Prob keep P 40/64 Prob keep and Φ + Q 26/64 Prob Φ + if kept 13/20 P = F 2 + 2F (1 F ) 3 + 5(1 F )2 9 = 8F 2 4F Q = F 2 + (1 F )2 9 = 8F 2 4F F = 10F 2 2F + 1 8F 2 4F + 5 > F provided F > 1/2

17 Repeat Process? Round Fidelity

18 What's Wrong? After First Round: with probability 13/20 with probability 1/4 with probability 1/20 with probability 1/20 Not a Werner State!

19 Solution (1) Wernerize between rounds or or each with probability 1/3 After Wernerizing First Round: with probability 13/20 each with probability 7/60

20 Result of Wernerization Round Fidelity Rate % 65.0% 67.9% 71.2% 74.8% % 6

21 Manipulating Bell States and are unaffected (up to phase)

22 Solution (2) Macchiavellize between rounds Round Fidelity Rate % 65.0% 73.3% 82.6% 89.3% 96.7% 99.3% % 31.25% 9.06% 2.96% 1.13% 0.47% 0.22%!1/455

23 Wernerisation Soit ρ un état sur deux qubits distribués entre Alice et Bob tel que Ψ ρ Ψ = F. Si Alice et Bob choisissent au hasard l une de 12 transformations suivantes, s ils appliquent localement et indépendamment cette transformation sur leur qubit respectif et s ils oublient la transformation qu ils ont choisie, alors l état ρ résultant sera un état de Werner W Ψ F de même fidélité: et Ψ ρ Ψ = F Ψ + ρ Ψ + = Φ ρ Φ = Φ + ρ Φ + = (1 F )/3. Les 12 rotations bilatérales ( ) ( 0 i i 0 ) ( ) ( i 0 0 i ) ( i 1 + i i 1 i ) ( 1 1 i 1 i 2 1 i 1 + i ) 1 2 ( 1 i 1 + i 1 + i 1 + i ) ( 1 1 i 1 + i i 1 + i ) ( i 1 + i i 1 i ) ( 1 1 i 1 i 2 1 i 1 + i ) 1 2 ( 1 i 1 + i 1 + i 1 + i ) ( 1 1 i 1 + i i 1 + i )