When you measure a qubit cos(φ)|0> + sin(φ)|1>, the result has variance (1-cos(4φ))/4. (I’m skipping over the trig calculations here and below.) If you have a choice between either measuring the qubit, or rotating it by an angle ψ and then measuring it, then the sum of variances of the two operations is at least (1-|cos(2ψ)|)/2. In particular, if ψ=π/4, the sum of variances is at least 1⁄2.
So no matter what state a qubit is in, and how precisely you know its state, there are two possible measurements such that the sum of variances of their results is at least 1⁄2. The reason is that the bases corresponding to these measurements are at an angle to each other, not aligned. Apparently in the real world, an object’s “position” and “momentum” are two such measurements, so there’s a limit on their joint precision.
You can also carry out one measurement and then the other, but that doesn’t help—after the first measurement you have variance in the second. Moreover, after the second you have fresh variance in the first. This lets you get an infinite stream of fair coinflips from a single qubit: start with |0>, rotate by π/4, measure, rotate by π/4, measure...
(12/?)
The uncertainty principle, or something like it.
When you measure a qubit cos(φ)|0> + sin(φ)|1>, the result has variance (1-cos(4φ))/4. (I’m skipping over the trig calculations here and below.) If you have a choice between either measuring the qubit, or rotating it by an angle ψ and then measuring it, then the sum of variances of the two operations is at least (1-|cos(2ψ)|)/2. In particular, if ψ=π/4, the sum of variances is at least 1⁄2.
So no matter what state a qubit is in, and how precisely you know its state, there are two possible measurements such that the sum of variances of their results is at least 1⁄2. The reason is that the bases corresponding to these measurements are at an angle to each other, not aligned. Apparently in the real world, an object’s “position” and “momentum” are two such measurements, so there’s a limit on their joint precision.
You can also carry out one measurement and then the other, but that doesn’t help—after the first measurement you have variance in the second. Moreover, after the second you have fresh variance in the first. This lets you get an infinite stream of fair coinflips from a single qubit: start with |0>, rotate by π/4, measure, rotate by π/4, measure...