This instructional presentation utilizes a "Quantum Detective" persona to demonstrate the process of quantum state tomography for a two-level spin-1/2 system. The objective is the empirical determination of four unknown quantum states—$\psi_1$ through $\psi_4$—using a Stern-Gerlach experimental framework. The methodology relies on sequential measurements across the X, Y, and Z axes to resolve the complex coefficients of the state vectors.
The analysis begins with the determination of amplitudes through X-direction measurements, which, for all four states, yielded a 50/50 probability distribution, indicating equal weighting of the basis states. The session then details the mapping of these states onto a Bloch sphere, where latitude represents amplitude (theta) and longitude represents the relative phase (phi). By applying the Born Rule (the fourth postulate of quantum mechanics), the presenter derives the relative phases for each state from the probability differences observed in the Y and Z orientations. The process concludes with the formal calculation of the complex state vectors, successfully identifying $\psi_1$ (zero phase), $\psi_2$ (phase of $3\pi/2$), and specific phases for $\psi_3$ and $\psi_4$.
Quantum State Tomography: Determining Unknown Spin-1/2 States
0:26 Stern-Gerlach Apparatus: The experimental setup utilizes a particle source producing spin-1/2 particles, which are passed through a Stern-Gerlach magnet to be sorted into "Spin Up" and "Spin Down" states at the detectors.
1:07 Initial X-Axis Measurements: Measurements in the X-direction for all four unknown states show an identical 5,000/5,000 split over 10,000 particles, establishing that the probabilities for these basis states are equal ($P = 0.5$).
2:42 Two-Level System Foundations: The states are defined within a two-level Hilbert space, where any state $|\psi\rangle$ is a linear combination of the basis vectors $| \text{up} \rangle$ and $| \text{down} \rangle$, represented by complex coefficients $c_{\text{up}}$ and $c_{\text{down}}$.
3:59 Constraints and Normalization: To fully define the state, two conditions must be met: the normalization condition (the sum of the squares of the amplitudes must equal 1) and the realization that only the relative phase between coefficients is physically significant.
5:41 Bloch Sphere Visualization: Every quantum state of a two-level system corresponds to a unique point on the surface of a Bloch sphere. The "poles" represent the eigenbasis, while any other point represents a superposition defined by the angles $\theta$ (latitude/amplitude) and $\phi$ (longitude/phase).
8:41 Application of the Born Rule: Utilizing the fourth postulate of quantum mechanics, the probability of a measurement outcome is calculated as the square of the scalar product between the basis vector and the state vector ($P = |\langle \text{basis} | \psi \rangle|^2$).
10:22 Amplitude Resolution: Based on the X-axis measurements of $0.5$ probability, the amplitudes $A_{\text{up}}$ and $A_{\text{down}}$ for all four states are determined to be $1/\sqrt{2}$.
10:51 Phase Determination Strategy: Relative phases ($\Delta \phi$) are extracted by analyzing the difference in probabilities between "up" and "down" counts in the Y and Z axes, where $\Delta P_y$ is proportional to $\cos(\Delta \phi)$ and $\Delta P_z$ is proportional to $\sin(\Delta \phi)$.
$\psi_1$: Phase $\Delta \phi = 0$; coefficients are both $1/\sqrt{2}$.
$\psi_2$: Phase $\Delta \phi = 3\pi/2$; coefficients are $1/\sqrt{2}$ and $-i/\sqrt{2}$.
$\psi_3$: Phase $\Delta \phi = 0.34$.
$\psi_4$: Phase $\Delta \phi = \pi/6$.
16:08 Final State Synthesis: The unknown states are successfully reconstructed as complex vectors by combining the calculated amplitudes and phase terms, effectively "solving" the quantum mystery.