Distinguishability of Qubit States

Classically, the outcomes of decision processes are always distinguishable:

it is taken for granted that a tossed coin will land either on heads or on tails

and upon looking at it, we can distinguish the different outcomes with cer-

tainty. In applications to quantum information processing too, we will usually

measure the output state after a process. If this state is to give us answers to

the problem we are trying to solve, it is important to be able to distinguish

alternate outcomes. In quantum, basis states can get transformed to superpo-

sitions. Alternate outcomes may be possible that must be distinguishable. It

is easy to see that this is possible if the states are orthogonal.

Suppose the possible final states are |ψ

1

i and |ψ

2

i that are not orthogonal,

hψ

2

|ψ

1

i 6= 0. This means that one can write the second state in terms of the

first and its orthogonal complement |ψ

1

i

⊥

:

|ψ

2

i = a|ψ

1

i + b|ψ

1

i

⊥

.

Thus on measuring the output, there is a probability |a|

2

that we get |ψ

1

i

even if the output state being measured was |ψ

2

i. There is a probability |a|

2

of getting the wrong outcome when measuring ψ

2

. The two output states as-

sumed here cannot therefore be distinguished reliably. This fact can be proved

rigorously by showing that one cannot invent any measurement operator that

gives distinct outcomes with certainty on measuring a set of states that are

not mutually orthogonal. This property is exploited in secure quantum key

distribution to make the communication safe.

Other means of distinguishing non-orthogonal states have been invented in

which the space of states is extended, and the notion of measurement is gen-

eralized. These so-called unambiguous state discrimination techniques allow

for the possibility of getting inconclusive results after measurement. However

if positive results are obtained then they do tell the two states apart.