Heisenberg-Type Uncertainty Relation for Qubits

angle between a and b

ϑ

0.209

additional settings

show symmetrized approximators

show triangle of best approximations

show locations of best approximation

Consider a pair of sharp qubit observables

A,B

. These observables do not commute and cannot be measured jointly and precisely; they are incompatible. However, allowing for imprecision in the measurement makes it possible to measure approximations

C,D

to the noncommuting pair

A,B

. The approximators

C,D

would ideally be chosen so that they give the best possible approximation to

A,B

without being incompatible themselves. In this Demonstration, you can explore this choice of

C,D

: representing

A,B,C,D

by vectors

a,b,c,d

on a vertical slice of the Bloch sphere, the described physical structure translates into geometric constraints. The vectors

a

and

b

(both represented by black arrows) are to be approximated by the vectors

c

(blue arrow) and

d

(red arrow), while

c

must be contained in the blue ellipse and

d

must be contained in the red ellipse. You can drag the vectors

c,d

. The scale on the right-hand side indicates how well the current choice of

c,d

is approximating

a,b

. No choice of approximators

c,d

can do better than the minimum of this scale.