1.1.1 What is a Qubit#

Prompts

What distinguishes quantum physics from classical physics in the way it describes physical systems? How does quantum mechanics represent states and predict outcomes?
What is a qubit, and why is it the simplest quantum system? How does it illustrate the core ideas of quantum mechanics?
How do classical bits and qubits differ in terms of what information they can hold and how we learn about that information through measurement?
Why can the same quantum system (like an electron’s spin) be realized in many different physical platforms, yet all obey identical mathematical rules? What does this universality tell us about quantum mechanics itself?

Lecture Notes#

Overview#

Quantum mechanics is famously abstract. Unlike gravity or fluid flow, you cannot build intuition from everyday experience—no one has ever “seen” a superposition or “felt” an entangled state. The quantum world operates on scales and principles so far from daily life that we cannot simply model it by analogy. Instead, we must rely on mathematical structures and learn what nature does from observation data.

This is not as alien as it sounds. Modern machine learning has shown us a powerful paradigm: everything is a vector. Images, words, proteins, music—all are embedded as vectors in high-dimensional spaces, and all transformations (classification, generation, translation) are operators acting on those vectors. Quantum mechanics arrived at the same insight a century earlier: every physical state is a vector, and every physical process—measurement, time evolution, symmetry—is an operator.

What is Physics? Description and Prediction#

Two Pillars of Physics

Physics rests on two fundamental objectives:

Description: How do we represent the state of a physical system?
Prediction: Given a state, how do we predict the outcome of an experiment?

For a classical system (a ball rolling down a ramp), we describe the state by specifying position and velocity. We predict the future using Newton’s laws.

For a quantum system, the description and prediction mechanisms are radically different.

Quantum mechanics answers:

Aspect	Classical	Quantum
Description	Definite values: \(x\), \(v\), \(E\)	Abstract vector in Hilbert space: \(\vert \psi\rangle\)
Prediction	Equations of motion (Newton, Lagrange)	Operators acting on vectors; time evolution (Schrödinger)
Role of measurement	Non-invasive: observe without disturbing	Invasive: measurement collapses state and changes outcome
Information	Deterministic: same initial state → same outcome	Probabilistic: same state → distribution of outcomes

The Qubit: Mathematical Definition#

Qubit Definition

A qubit is a quantum system that lives in a two-dimensional Hilbert space over the complex numbers. Any pure state is a unit vector (a “ket”):

(1)#\[ \vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle, \quad |\alpha|^2 + |\beta|^2 = 1 \]

where \(\vert 0\rangle\) and \(\vert 1\rangle\) are orthonormal basis states, and \(\alpha, \beta \in \mathbb{C}\).

Key properties:

Basis states \(\vert 0\rangle\) and \(\vert 1\rangle\) are orthonormal: \(\langle 0\vert 1\rangle = 0\), \(\langle 0\vert 0\rangle = 1\).
Superposition: Unlike a classical bit (which is 0 or 1), a qubit can be in a superposition of both. The coefficients \(\alpha\) and \(\beta\) describe the quantum state.
Normalization: The condition \(|\alpha|^2 + |\beta|^2 = 1\) ensures the state is normalized. We interpret \(\vert\alpha\vert^2\) as the probability of measuring the qubit in state \(\vert 0\rangle\).
Global phase: States that differ by a global phase (e.g., \(\vert \psi\rangle\) and \(\mathrm{e}^{\mathrm{i}\phi}\vert \psi\rangle\)) are physically identical. Thus a qubit is truly a two-parameter system (after factoring out global phase and normalization).

Physical Realizations of Qubits#

A qubit is not merely a mathematical abstraction. Many physical systems naturally obey the two-state model:

Platform	\(\vert 0\rangle\)	\(\vert 1\rangle\)	\(\vert 0\rangle\)–\(\vert 1\rangle\) splitting	Gates via
Electron spin	\(\vert \downarrow\rangle\)	\(\vert \uparrow\rangle\)	\(\mu_B B \sim 0.1\) meV (at \(\sim\)1 T)	RF/microwave pulses
Photon polarization	horizontal \(\vert H\rangle\)	vertical \(\vert V\rangle\)	0 (degenerate)	Waveplates (passive optics)
Trapped ion	hyperfine ground \(\vert g\rangle\)	excited \(\vert e\rangle\)	\(\sim\mu\)eV (GHz hyperfine)	Laser pulses
Superconducting circuit	transmon ground	first excited	\(\sim\mu\)eV (\(\sim\)5 GHz)	Microwave pulses
Nuclear spin	\(m_I = -1/2\)	\(m_I = +1/2\)	\(\mu_N B \sim\) neV (at \(\sim\)1 T)	RF pulses (NMR)

Why the same mathematics everywhere: All two-level quantum systems obey the same Schrödinger equation with a \(2 \times 2\) Hamiltonian. The specific physics differs (magnetic fields, photonic transitions, circuit dynamics), but the abstract qubit rules are universal.

Discussion: why two-level systems matter

Why do physicists care about two-level systems when nature has many more states?

A realistic electron has both orbital angular momentum and spin, giving many possible energy levels. A real atom has infinitely many states (different orbitals). Why restrict attention to qubits?

In many experimental conditions (low temperature, weak fields, far from resonance with other transitions), only the two lowest energy levels are significantly populated. The two-level approximation is valid when \(k_B T \ll \Delta E\) and the driving field frequency is resonant with only one transition.
The qubit is the minimal quantum system that exhibits superposition, entanglement, measurement, and unitary evolution. All quantum computation can (in principle) be built from qubits.
Show that if you restrict a three-level system to only two of its three states, you lose nothing: any unitary transformation on the two-level subspace can be applied without mixing in the third state (under what conditions is this true?).

Classical Bit vs. Quantum Qubit#

Property	Classical Bit	Quantum Qubit
Values	0 or 1	\(\alpha\vert 0\rangle + \beta\vert 1\rangle\) (superposition)
State space	2 possible values	Infinitely many (Bloch sphere: \(\mathbb{CP}^1\))
Measurement	Non-invasive; outcome pre-exists	Invasive; outcome is random, state collapses
Repeatability	Measuring the same bit twice gives the same result	Measuring the same qubit twice gives the same result, but the first measurement destroys the superposition
Copying	Trivial: copy the bit	Impossible: no-cloning theorem forbids copying an unknown quantum state
Information capacity	1 bit of classical information	\(\leq 1\) bit of classical information per measurement, but encodes continuous parameters

Poll: Superposition vs. uncertainty

A qubit state \(\vert\psi\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\) is in a superposition. What does this tell you about measurement?

(A) The outcome is uncertain; we cannot predict it before measuring.

(B) The outcome is determined before measurement, but we don’t know it.

(D) Measurement will give 0 or 1 with equal probability, encoding probabilities via superposition.

Summary#

A qubit is the simplest quantum system: A two-dimensional vector in complex Hilbert space that can exist in superposition of basis states \(|0\rangle\) and \(|1\rangle\).
Qubits are physically universal: Electron spins, photon polarizations, trapped atoms, and superconducting circuits all behave as qubits—same mathematics, different physics.
Quantum mechanics is descriptive and predictive: States are vectors, observables are operators, outcomes are eigenvalues, and probabilities are given by the Born rule.
Measurement is invasive: Unlike classical measurement, measuring a qubit collapses its superposition and irreversibly changes the state for future measurements.

Homework#

1. Born rule and probabilities. A qubit is in state \(\vert\psi\rangle = \tfrac{3}{5}\vert 0\rangle + \tfrac{4}{5}\vert 1\rangle\).

(a) Use Born’s rule to find \(P(0)\) and \(P(1)\).

(b) Verify that \(P(0) + P(1) = 1\).

(c) Now consider \(\vert\phi\rangle = \tfrac{3}{5}\vert 0\rangle - \tfrac{4}{5}\vert 1\rangle\). Are \(\vert\psi\rangle\) and \(\vert\phi\rangle\) the same physical state? Compute \(P(+)\) and \(P(-)\) in the \(X\) basis \(\{\vert\pm\rangle = \tfrac{1}{\sqrt{2}}(\vert 0\rangle \pm \vert 1\rangle)\}\) for both states, and use the result to justify your answer.

2. State reconstruction from probabilities. Construct a qubit state \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) whose measurement statistics satisfy:

\(Z\)-basis: \(P(0) = 0.3\), \(P(1) = 0.7\).
\(X\)-basis: \(P(+) = 0.5\), where \(\vert\pm\rangle = \tfrac{1}{\sqrt{2}}(\vert 0\rangle \pm \vert 1\rangle)\).

(a) Use the \(Z\)-basis probabilities to determine \(\vert\alpha\vert\) and \(\vert\beta\vert\).

(b) Show that \(P(+) = 0.5\) pins down the relative phase between \(\alpha\) and \(\beta\). Find the allowed relative phase(s).

(c) Write down the most general such state up to a global phase. How many distinct physical states satisfy these constraints? In what sense are the \(Z\) and \(X\) measurements not enough to uniquely determine the state, and what additional measurement would suffice?

3. Global phase invariance. Show that two states differing by a global phase, \(\vert\psi'\rangle = \mathrm{e}^{\mathrm{i}\gamma}\vert\psi\rangle\) (with \(\gamma \in \mathbb{R}\)), give identical measurement probabilities \(\vert\langle m\vert\psi'\rangle\vert^2 = \vert\langle m\vert\psi\rangle\vert^2\) for every outcome \(\vert m\rangle\) and every choice of measurement basis. In one sentence, contrast with Problem 1(c) and explain why the same argument fails for a relative phase between components.

4. Bloch sphere parametrization. A generic qubit state \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) naively requires 4 real parameters \((\mathrm{Re}\,\alpha,\ \mathrm{Im}\,\alpha,\ \mathrm{Re}\,\beta,\ \mathrm{Im}\,\beta)\). Show that after imposing

(a) normalization \(\vert\alpha\vert^2 + \vert\beta\vert^2 = 1\) and

(b) global phase freedom, only 2 real parameters remain. How does this explain why the space of qubit states maps to a 2-dimensional sphere?

5. Quantum information. A qubit state \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) is specified by two continuous real parameters — seemingly more information than a single classical bit (0 or 1). Does a qubit therefore store more information than a classical bit? Address the following:

(a) How much classical information is extracted by a single measurement of the qubit?

(b) Can repeated measurements on identical copies extract more? Name the reconstruction procedure and state the upper bound on the classical information accessible per measurement, as quoted in the lecture’s comparison table.

(c) Identify the two quantum-mechanical resources beyond classical encoding — superposition (with interference) and entanglement — and name one example phenomenon or protocol that uses each.

6. Distinguishing non-orthogonal states. A qubit is prepared in one of two states with equal prior probability \(1/2\): either \(\vert\psi_1\rangle = \vert 0\rangle\) or \(\vert\psi_2\rangle = \tfrac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\). You perform a single projective measurement in a basis of your choice, then guess which state was prepared. The goal is to maximise the probability of guessing correctly.

(a) Compute the overlap \(\vert\langle\psi_1\vert\psi_2\rangle\vert\). Are the two states orthogonal?

(b) Suppose you measure in the \(\{\vert 0\rangle, \vert 1\rangle\}\) (\(Z\)) basis. What outcome — if any — uniquely identifies the state? What outcome is ambiguous, and which guess is more likely correct given that outcome?

(d) Explain in one sentence why no measurement strategy can achieve guessing probability \(1\), in contrast to the case of two orthogonal states.

7. Photon polarization vs. transmon qubit. A photon qubit uses polarization (\(\vert 0\rangle = \) horizontal, \(\vert 1\rangle = \) vertical). At room temperature (\(T = 300\,\mathrm{K}\)), thermal energy is \(k_BT \approx 26\,\mathrm{meV}\). A visible photon at \(\lambda = 800\,\mathrm{nm}\) has energy \(E = hc/\lambda\) (use \(hc = 1240\,\mathrm{eV\cdot nm}\)).

(a) Compute \(E\) in \(\mathrm{eV}\).

(b) Using a Boltzmann argument, argue whether the photon polarization qubit is susceptible to thermal noise at room temperature. State the relevant comparison of scales explicitly.

(c) A superconducting transmon qubit at \(5\,\mathrm{GHz}\) has energy gap \(\hbar\omega \approx 0.021\,\mathrm{meV}\). At what temperature \(T^*\) does \(k_BT^* = \hbar\omega\)? Estimate the operating temperature needed to keep the excited-state thermal population below \(1\,\%\), and compare to the typical dilution-refrigerator scale (\(\sim 10\text{--}20\,\mathrm{mK}\)).