Case Study 2: From Spin to Qubits — The Birth of Quantum Information

Overview

The spin-1/2 system — the simplest nontrivial quantum system — is not merely a pedagogical toy. It is the physical foundation of quantum information science. The qubit, the fundamental unit of quantum computation, is mathematically identical to a spin-1/2 particle. The Bloch sphere that we use to visualize spin states is the same Bloch sphere that quantum computer engineers use to visualize qubit states. The Pauli matrices that describe spin operators are the same matrices that define quantum gates.

This case study traces the conceptual path from the physics of spin to the science of quantum information, showing how the formalism of Chapter 13 becomes the language of quantum computing.

Part 1: The Qubit as a Spin-1/2 System

The Definition

A qubit (quantum bit) is any two-level quantum system. The two basis states are conventionally written:

$$|0\rangle \equiv |+\rangle, \qquad |1\rangle \equiv |-\rangle$$

The general qubit state is:

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1$$

This is identical to the spin-1/2 state $|\chi\rangle = \alpha|+\rangle + \beta|-\rangle$. The mathematics is the same; only the notation and physical implementation differ.

Physical Implementations

While spin-1/2 is the conceptual prototype for a qubit, actual quantum computers use a variety of two-level systems:

Despite the different physical substrates, every qubit is described by the same $2 \times 2$ matrix algebra we developed for spin-1/2. This universality is one of the most powerful ideas in quantum information.

Classical Bits vs. Qubits

A classical bit is either 0 or 1. A qubit can be in any superposition $\alpha|0\rangle + \beta|1\rangle$. The key differences:

1. Superposition: A qubit can be "both 0 and 1 simultaneously" (though this language is dangerously misleading — it is better to say the qubit is in a state with definite amplitudes for both outcomes).

2. Phase: The relative phase $\phi$ in $|0\rangle + e^{i\phi}|1\rangle$ has no classical analogue. It is physically measurable and computationally exploitable.

3. Measurement collapse: Measuring a qubit yields $|0\rangle$ or $|1\rangle$ with probabilities $|\alpha|^2$ and $|\beta|^2$, and the qubit collapses to the measured state. This is exactly the Stern-Gerlach measurement from Section 13.5.

4. No cloning: An unknown qubit state cannot be perfectly copied (the no-cloning theorem). This is a direct consequence of the linearity of quantum mechanics and has no classical analogue — you can always copy a classical bit.

Part 2: Quantum Gates as Spin Rotations

Single-Qubit Gates

Every single-qubit gate is a $2 \times 2$ unitary matrix — exactly the kind of operator we studied for spin-1/2 rotations. The most important gates are built from the Pauli matrices:

Pauli Gates:

$$X = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}: \quad |0\rangle \to |1\rangle, \quad |1\rangle \to |0\rangle \quad \text{(NOT gate / bit flip)}$$

$$Y = \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}: \quad |0\rangle \to i|1\rangle, \quad |1\rangle \to -i|0\rangle$$

$$Z = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}: \quad |0\rangle \to |0\rangle, \quad |1\rangle \to -|1\rangle \quad \text{(phase flip)}$$

The $X$ gate is the quantum analogue of the classical NOT gate, but with the crucial difference that it also works on superpositions: $X(\alpha|0\rangle + \beta|1\rangle) = \beta|0\rangle + \alpha|1\rangle$.

Hadamard Gate:

$$H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$

This maps the computational basis to the superposition basis:

$$H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}} = |+\rangle_x, \qquad H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} = |-\rangle_x$$

In spin language, the Hadamard gate is a rotation by $\pi$ about the axis $(\hat{x} + \hat{z})/\sqrt{2}$. On the Bloch sphere, it swaps the $z$-axis and $x$-axis.

Phase Gate and T Gate:

$$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} = e^{i\pi/4}R_z(\pi/2), \qquad T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} = e^{i\pi/8}R_z(\pi/4)$$

These are rotations about the $z$-axis of the Bloch sphere by $\pi/2$ and $\pi/4$ respectively.

General Rotation Gates

The rotation operators from Section 13.4 translate directly into parameterized quantum gates:

$$R_x(\phi) = e^{-i\phi\sigma_x/2} = \cos\frac{\phi}{2}\,I - i\sin\frac{\phi}{2}\,\sigma_x = \begin{pmatrix} \cos\frac{\phi}{2} & -i\sin\frac{\phi}{2} \\ -i\sin\frac{\phi}{2} & \cos\frac{\phi}{2} \end{pmatrix}$$

$$R_y(\phi) = e^{-i\phi\sigma_y/2} = \begin{pmatrix} \cos\frac{\phi}{2} & -\sin\frac{\phi}{2} \\ \sin\frac{\phi}{2} & \cos\frac{\phi}{2} \end{pmatrix}$$

$$R_z(\phi) = e^{-i\phi\sigma_z/2} = \begin{pmatrix} e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2} \end{pmatrix}$$

The Euler angle decomposition guarantees that any single-qubit unitary $U$ can be written as:

$$U = e^{i\alpha} R_z(\beta) R_y(\gamma) R_z(\delta)$$

for some angles $\alpha, \beta, \gamma, \delta$. This is the quantum computing analogue of the Euler angle decomposition for rotations in three dimensions, and it means that $R_y$ and $R_z$ (or equivalently, rotations about any two non-parallel axes) form a universal set for single-qubit operations.

In a physical spin-1/2 system, $R_z(\phi)$ is implemented by Larmor precession in a $z$-directed field for time $t = \phi/\omega_0$. $R_x(\phi)$ is implemented by a transverse magnetic field pulse. Together, they can produce any single-qubit gate.

Part 3: The Bloch Sphere as the Qubit State Space

Geometry of Quantum Gates

On the Bloch sphere, every single-qubit gate is a rotation. The Bloch sphere makes the action of quantum gates geometrically intuitive:

Measurement as Projection

Measuring a qubit in the computational basis ($|0\rangle$/$|1\rangle$) corresponds to projecting the Bloch vector onto the $z$-axis: - If the Bloch vector has a positive $z$-component, the measurement is more likely to return $|0\rangle$. - Measuring in the $x$-basis ($|+\rangle_x$/$|-\rangle_x$) projects onto the $x$-axis. - Measuring in any basis $\hat{n}$ projects onto that axis.

The probability formula $P(0) = \cos^2(\theta/2) = (1 + n_z)/2$ is Malus's law for spin — now reinterpreted as the Born rule for qubit measurement.

State Tomography

To fully characterize an unknown qubit state, we need to determine the Bloch vector $\hat{n} = (n_x, n_y, n_z)$. This requires measurements along three independent axes:

$$n_x = 2\langle\hat{S}_x\rangle/\hbar = \langle\sigma_x\rangle$$ $$n_y = 2\langle\hat{S}_y\rangle/\hbar = \langle\sigma_y\rangle$$ $$n_z = 2\langle\hat{S}_z\rangle/\hbar = \langle\sigma_z\rangle$$

Each expectation value requires many repeated measurements on identically prepared copies. With $N$ measurements per axis and three axes, the Bloch vector can be estimated with uncertainty $\sim 1/\sqrt{N}$ per component.

This is quantum state tomography for a single qubit — the procedure for reconstructing the full quantum state from measurement data. For multi-qubit systems, the procedure scales exponentially (requiring $3^n$ measurement bases for $n$ qubits), which is one of the fundamental challenges of quantum information.

Part 4: From One Qubit to Many — Entanglement Preview

The Tensor Product Structure

Two qubits live in a four-dimensional Hilbert space $\mathcal{H}_2 = \mathcal{H}_1 \otimes \mathcal{H}_1$, spanned by:

$$|00\rangle, \quad |01\rangle, \quad |10\rangle, \quad |11\rangle$$

In spin language, these are the two-spin states $|++\rangle$, $|+-\rangle$, $|-+\rangle$, $|--\rangle$ that we first encountered in Chapter 11 (tensor products).

Entangled States

The Bell states — maximally entangled two-qubit states — are:

$$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} = \frac{|++\rangle + |--\rangle}{\sqrt{2}}$$

$$|\Phi^-\rangle = \frac{|00\rangle - |11\rangle}{\sqrt{2}} = \frac{|++\rangle - |--\rangle}{\sqrt{2}}$$

$$|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}} = \frac{|+-\rangle + |-+\rangle}{\sqrt{2}}$$

$$|\Psi^-\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}} = \frac{|+-\rangle - |-+\rangle}{\sqrt{2}}$$

In spin language, $|\Psi^-\rangle$ is the singlet state and $\{|\Phi^+\rangle, |\Psi^+\rangle, |\Phi^-\rangle\}$ are the three triplet states — exactly the states constructed in Chapter 11 by adding two spin-1/2 angular momenta.

The connection is precise: the Clebsch-Gordan decomposition $\frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1$ corresponds to the decomposition of two-qubit Hilbert space into the singlet and triplet subspaces. The formalism of angular momentum addition (Chapter 14) and the formalism of two-qubit quantum computing are the same mathematics in different clothing.

The CNOT Gate

The controlled-NOT gate is the fundamental two-qubit gate:

$$\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$

It flips the second qubit if and only if the first qubit is $|1\rangle$. Combined with single-qubit gates (which we now know are just spin rotations), CNOT forms a universal gate set — any quantum computation can be built from single-qubit rotations and CNOT.

Creating the Bell state $|\Phi^+\rangle$ from $|00\rangle$:

$$|00\rangle \xrightarrow{H \otimes I} \frac{|00\rangle + |10\rangle}{\sqrt{2}} \xrightarrow{\text{CNOT}} \frac{|00\rangle + |11\rangle}{\sqrt{2}} = |\Phi^+\rangle$$

The Hadamard gate creates a superposition on the first qubit. The CNOT entangles the two qubits. This two-step procedure — a single-qubit rotation followed by an entangling gate — is the fundamental motif of quantum circuit design.

Part 5: Historical Arc — From Stern-Gerlach to Shor's Algorithm

1922: Stern-Gerlach

The discovery of spatial quantization. Stern and Gerlach did not know they had observed spin — the concept would not exist for three more years.

1925: Uhlenbeck and Goudsmit

The proposal of electron spin, explaining the anomalous Zeeman effect and the two-spot Stern-Gerlach result.

1927: Pauli

Formalization of the spin-1/2 formalism. The Pauli matrices become the standard language for spin.

1928: Dirac

The Dirac equation derives spin from the demand for Lorentz covariance. Spin is not ad hoc — it is required by relativity.

1935: Einstein, Podolsky, and Rosen (EPR)

The EPR paper uses entangled spin states to argue that quantum mechanics is incomplete. The paper introduces the idea that quantum correlations might require "hidden variables."

1964: Bell

John Bell shows that no local hidden-variable theory can reproduce all quantum mechanical predictions for entangled spin-1/2 pairs. The Bell inequalities can be tested experimentally using exactly the Stern-Gerlach-type measurements we studied in this chapter. (Chapter 24 is devoted to this.)

1982: Feynman

Richard Feynman proposes that quantum systems might be efficiently simulated by quantum computers, rather than classical ones. The two-level system (spin-1/2/qubit) is the fundamental building block.

1994: Shor

Peter Shor discovers a quantum algorithm for factoring large integers exponentially faster than any known classical algorithm. The algorithm uses qubits — spin-1/2 systems — manipulated by quantum gates — spin rotations.

1996: Quantum Error Correction

Shor and Steane independently discover quantum error-correcting codes, showing that quantum information can be protected from decoherence. The codes work on spin-1/2 (qubit) systems.

2019: Google "Quantum Supremacy"

Google's Sycamore processor, using 53 superconducting qubits (two-level systems, each mathematically identical to a spin-1/2 particle), performs a computation in 200 seconds that would take a classical supercomputer approximately 10,000 years.

2023-2025: The NISQ Era

Noisy Intermediate-Scale Quantum devices with 50-1000+ qubits are used for quantum chemistry simulations, optimization problems, and machine learning experiments. Every qubit is a spin-1/2 system. Every gate is a spin rotation.

The line from Stern and Gerlach's silver atoms in 1922 to modern quantum processors is direct and unbroken. The mathematics is the same: Pauli matrices, spinors, the Bloch sphere, unitary rotations. What has changed is our understanding of what can be done with these simple quantum systems when they are combined, entangled, and controlled.

Part 6: Quantum Teleportation — Spin States Across Space

To illustrate the power of spin-1/2 entanglement, consider the quantum teleportation protocol (Bennett et al., 1993):

Goal: Alice wants to send an unknown qubit state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ to Bob, using only classical communication and a shared entangled pair.

Resources: Alice and Bob share the Bell state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. Alice has the qubit to be teleported.

Protocol:

1. The total state of three qubits (Alice's unknown qubit $A$, Alice's half of the Bell pair $B_1$, Bob's half $B_2$):

$$|\Psi\rangle_{AB_1B_2} = |\psi\rangle_A \otimes |\Phi^+\rangle_{B_1B_2} = (\alpha|0\rangle + \beta|1\rangle) \otimes \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$

2. Alice performs a Bell-basis measurement on her two qubits ($A$ and $B_1$). She obtains one of four outcomes with equal probability $1/4$.

3. Alice sends the measurement result (2 classical bits) to Bob.

4. Bob applies the appropriate Pauli correction to his qubit:

In every case, Bob ends up with $\alpha|0\rangle + \beta|1\rangle$ — Alice's original state. The state has been "teleported" from Alice to Bob without any physical qubit traveling between them.

The correction operators $\{I, X, Y, Z\}$ are precisely the Pauli matrices (plus identity). The protocol works because the Bell states form a complete basis for two qubits, and because the Pauli matrices form a complete basis for single-qubit operations. Both facts follow from the spin-1/2 algebra.

Discussion Questions

1. A classical bit can be in state 0 or 1. A qubit can be in a superposition $\alpha|0\rangle + \beta|1\rangle$. Does this mean a qubit stores "more information" than a classical bit? (Hint: Consider what happens when you measure the qubit.)

2. The no-cloning theorem says you cannot perfectly copy an unknown quantum state. This is a direct consequence of the linearity of quantum mechanics. Explain why classical bits can be copied and why the analogous argument fails for qubits.

3. Quantum teleportation transfers a state without transferring the physical system. Does this violate the no-communication theorem (the principle that entanglement alone cannot transmit information faster than light)? Why or why not?

4. The Bloch sphere representation of a single qubit is three-dimensional (a sphere in $\mathbb{R}^3$). How many real parameters are needed to describe the state of $n$ qubits? Why does the Bloch sphere picture fail for multi-qubit systems?

5. Spin-1/2 was discovered by accident (Stern-Gerlach), formalized out of necessity (Pauli), derived from fundamental principles (Dirac), and now forms the basis of quantum computing. What does this historical trajectory suggest about the relationship between foundational physics and practical technology?

6. In quantum error correction, logical qubits are encoded in many physical qubits. Each physical qubit is a spin-1/2 system. If we think of the logical qubit as also having a "Bloch sphere," how does this relate to the Bloch spheres of the constituent physical qubits?