BA1(t)

BA2(t)

BB2(t)

BB1(t)

A1

C

A2

B2

B1

JB(t)

JA(t)

…

…

C

A1

A2

B2

B1

“phantom” spins

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.

- The Problem:- How can a fixed network of interacting spins be used as a conduit for high-fidelity quantum communication, when only limited external control is available?
- We consider two types of “limited external control”:
- 1) The sender encodes the message onto several spins,
- or,
- 2) The sender and receiver modulate the interactions on a just two spins each.

- We assume:
- -- The Hamiltonian conserves total Z-spin
- -- The system can be initialised in all-|0i

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:

A

B

(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)

“extended Bob”

“extended Alice”

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:

Alice

Bob

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

JA(t)

JB(t)

Derived control functions:

(All fixed bonds have strength 1)

--- JA(t)

--- JB(t)

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