B A1 (t). B A2 (t). B B2 (t). B B1 (t). A 1. C. A 2. B 2. B 1. J B (t). J A (t ). …. …. C. A 1. A 2. B 2. B 1. “phantom” spins. Optimal State Encoding for Quantum Walks and Quantum Communication over Spin Systems.
Optimal State Encoding for Quantum Walks and Quantum Communication over Spin Systems
Henry Haselgrove, School of Physical Sciences, University of Queensland, Australia
Quant-ph/0404152 – submitted to PRA
2 — DYNAMICAL CONTROL
In this scheme Alice and Bob control the interactions on just two spins each.
The interaction strengths on the control spins are modulated throughout the communication procedure.
1 — MESSAGE ENCODING
Alice wishes to send®|0i + ¯|1ito Bob.
They each control some subset, A and B, of the spins in a network. Example:
(Bulk of the spin network— fixed known interactions, arbitrary graph)
(graph edges represent fixed interactions between spins)
The communication procedure:
1. The state of the system is pre-prepared:
2. Alice encodes the message onto her spins:
The message state is placed onto this spin by Alice
The message is received by Bob here
(encoding of the|1imessage – to be optimised)
Method for deriving good control functions:
1) Imagine that Alice and Bob control many more spins (see figure, left)
2) Find the optimal encoding for |1ion these spins (see LHS of poster)
3) Simulate the evolution. Use the results to find control functions for actual system above.
NB: Steps 2 and 3 can be done efficiently, in a restricted subspace.
(encoding of the|0imessage – fixed at|0…0i, for convenience)
3. The system evolves for time T. If |1ENCiwas chosen well, the state is:
4. Bob decodes:
EXAMPLE: IrregularXY chain
Result: The best choice for |1ENCiis given by the first right-singular vector of wherePBandPAare the projectors onto:
Note: the B(·)(t) functions may be set to zero in this simple example
Derived control functions:
(All fixed bonds have strength 1)
A ´space of states of form
B ´space of states of form
EXAMPLE: The evolution of best encoding |1ENCi, when Alice and Bob are joined by a Heisenberg chain
Fidelity versus time, with control:
Fidelity versus time, no control (JA and JB fixed at 1):
Total # spins = 300
# control spins = 20 + 20
Avg. fidelity = 0.99999
NB:jis the coefficient of