## 4.12 *Remarks and exercises*

### 4.12.1 Projector?

Consider two unit vectors

### 4.12.2 Knowing the unknown

Suppose you are given a *single* qubit in some entirely unknown quantum state

Can you determine

|\psi\rangle , using as many measurements as you want?Say you measure the qubit in the standard basis, and register outcome

|0\rangle . What does this tell you about the pre-measurement state|\psi\rangle ?How many real parameters do you need to determine

|\psi\rangle ? Would you be able to reconstruct|\psi\rangle from\langle\psi|X|\psi\rangle ,\langle\psi|Y|\psi\rangle , and\langle\psi|Z|\psi\rangle ?*Hint: it may help you to visualise*|\psi\rangle as a Bloch vector.You are given zillions of qubits, all prepared in the same quantum state

|\psi\rangle . How would you determine|\psi\rangle ?

### 4.12.3 Measurement and idempotents

The

Consider the measurement associated to any Hermitian operator

### 4.12.4 Unitary transformations of measurements

In our quantum circuits, unless specified otherwise, all measurements are assumed to be performed in the standard basis. This is because any measurement can be reduced to the standard measurement by performing some prior unitary transformation.

Show that any two orthonormal bases

\{|e_1\rangle,\ldots,|e_n\rangle\} and\{|d_1\rangle,\ldots,|d_n\rangle\} are always related by some unitaryU .*Hint: it suffices to show that*\sum_k |d_k\rangle\langle e_k| is unitary — why?Suppose that the projectors

P_k define the standard measurement. Show that, for any unitaryU , the projectorsUP_kU^\dagger also define a measurement.

### 4.12.5 Optimal measurement

The optimal measurement to distinguish between the two equally likely non-orthogonal signal states, *not* equally likely: say

### 4.12.6 Alice knows what Bob did

Alice prepares a qubit in any state of her choosing and gives it to Bob, who secretly measures either

*This is a simplified version of a beautiful quantum puzzle proposed in 1987 by Lev Vaidman, Yakir Aharonov, and David Z. Albert in a paper with the somewhat provocative title “How to ascertain the values of \sigma_x, \sigma_y, and \sigma_z of a spin-\frac{1}{2} particle”. For the original, see *

*Phys. Rev. Lett.*

**58**(1987), p. 1385.