13.1 Qubits and their properties 285where
ψ 1 ψ 2 =[a| 0 〉+b| 1 〉] 1 [c| 0 〉+d| 1 〉] 2. (13.6)
Herecanddare additional arbitrary constants, and the futility of at-
tempting to determine these constants quickly becomes obvious if you
try it. Generally, we do not bother with the subscript used to denote the
particle, so| 0 〉 1 | 1 〉 2 ≡| 01 〉and| 1 〉 1 | 1 〉 2 ≡| 11 〉, etc. Multiple-particle sys-
tems that have wavefunctions such as eqn 13.4 that cannot be written as
a product of single-particle wavefunctions are said to beentangled.This
entanglement in systems with two, or more, particles leads to quantum
properties of a completely different nature to those of a system of clas-
sical objects—this difference is a crucial factor in quantum computing.
Quantum computation uses qubits that aredistinguishable,e.g.ionsat
well-localised positions along the axis of a linear Paul trap. We can label
the two ions as Qubit 1 and Qubit 2 and know which one is which at any
time. Even if they are identical, the ions remain distinguishable because
they stay localised at certain positions in the trap. For a system of dis-
tinguishable quantum particles, anycombination of the single-particle
states is allowed in the wavefunction of the whole system:
Ψ=A| 00 〉+B| 01 〉+C| 10 〉+D| 11 〉. (13.7)The complex amplitudesA,B,CandDhave arbitrary values. It is
convenient to write down wavefunctions without normalisation, e.g.
Ψ=| 00 〉+| 01 〉+| 10 〉+| 11 〉, (13.8)
Ψ=| 00 〉+2| 01 〉+3| 11 〉, (13.9)
Ψ=| 01 〉+5| 10 〉. (13.10)Two of these three wavefunctions possess entanglement (see Exercise
13.1). We encounter examples with three qubits later (eqn 13.12).
In the discussion so far, entanglement appears as a mathematical prop-
erty of multiple-particle wavefunctions, but what does it mean physi-
cally? It is always dangerous to ask such questions in quantum me-
chanics, but the following discussion shows how entanglement relates
to correlations between the particles (qubits), thus emphasising that
entanglement is a property of the systemas a wholeand not the indi-
vidual particles. As a specific example consider two trapped ions. To
measure their state, laser light excites a transition from state| 1 〉(the
upper hyperfine level) to a higher electronic level to give a strong flu-
orescence signal, so| 1 〉is a ‘bright state’, while an ion in| 0 〉remains
dark.^3 Wavefunctions such as those in eqns 13.4 and 13.10 that contain^3 This is similar to the detection of
quantum jumps in Section 12.6, but
typically quantum computing experi-
ments use a separate laser beam for
each ion to detect them independently.
only the terms| 10 〉and| 01 〉always give one bright ion and the other
dark, i.e. an anticorrelation where a measurement always finds the ions
in different states. To be more precise, this corresponds to the following
procedure. First, prepare two ions so that the system has a certain ini-
tial wavefunction Ψin, then make a measurement of the state of the ions
by observing their fluorescence. Then reset the system to Ψinbefore
another measurement. The record of the state of the ions for a sequence