This note is part of [[quantum/Practical Quantum Information System]]. > [!info] Course code > Use the companion repository for this lecture's runnable lab, helper functions, and regression checks: > - [notebooks/by_concept/basic_rules_of_quantum_mechanics.ipynb](https://github.com/montekkundan/quantum-code/blob/main/notebooks/by_concept/basic_rules_of_quantum_mechanics.ipynb) > - [qcourse/states.py](https://github.com/montekkundan/quantum-code/blob/main/qcourse/states.py) > - [tests/test_states_density_entanglement.py](https://github.com/montekkundan/quantum-code/blob/main/tests/test_states_density_entanglement.py) [TODO: add video - Basic Rules of Quantum Mechanics] ## What This Concept Is Once you accept that quantum theory works with amplitudes instead of raw probabilities, you need a mathematical language that can represent those amplitudes cleanly. That language is linear algebra. States become vectors, allowed closed-system evolutions become [[quantum/Glossary#Unitary|unitary]] transformations, and measurement uses the [[quantum/Glossary#Born rule|Born rule]] to convert amplitudes into probabilities. This lecture is where the notation stops being optional. If you are not comfortable with state vectors, basis states, and normalization here, the later circuit and entanglement lectures will feel like symbol pushing instead of reasoning. ## Foundation Terms You Need First A [[quantum/Glossary#Ket|ket]] such as $|\psi\rangle$ is notation for a state vector. A [[quantum/Glossary#Unitary|unitary]] is a transformation that preserves normalization and represents ideal closed-system evolution. [[quantum/Glossary#Global phase|Global phase]] is an overall phase factor on a state that does not change measurement statistics. Keep one distinction in mind from the beginning: a relative phase between basis components can matter physically, while a global phase multiplying the whole state does not. ## How The Idea Actually Works A one-qubit state is written as $ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle $ with $|\alpha|^2 + |\beta|^2 = 1$. That normalization condition is the statement that the outcome probabilities must sum to one after measurement. Unitary evolution acts linearly on the whole state. If you know what a gate does to $|0\rangle$ and $|1\rangle$, then linearity tells you what it does to every superposition of those states. This is one of the reasons quantum theory is elegant and restrictive at the same time. The same linear rule that gives you interference also forbids certain operations, such as universal cloning later in the course. Measurement is different from unitary evolution. A unitary changes amplitudes while preserving the total norm. A measurement does not simply rotate the state. It extracts classical information and, in the standard textbook picture, changes the post-measurement state according to the outcome observed. One of the most useful beginner checks is this: if two states differ only by a global phase, all measurement probabilities are the same. If they differ by a relative phase, later interference can expose that difference. That is why phase is fundamental even though some phases are unobservable. ## Why It Matters - It gives the mathematical rules that every later protocol and circuit obeys. - It clarifies why quantum evolution is linear but measurement has a different role. - It prepares you to reason with statevectors instead of only reading diagrams. ## Related Questions > [!question] Why must closed-system quantum gates be unitary? >> [!answer] A closed-system gate must preserve total probability for every input state. In vector language, that means preserving inner products and norms, which is exactly the role of unitary transformations. > [!question] Why is global phase invisible but relative phase observable? >> [!answer] Multiplying the whole state by the same phase leaves every measurement probability unchanged. Changing the phase between components can change how amplitudes interfere after later gates, so relative phase can become visible in output statistics. > [!question] Why does linearity matter beyond computation convenience? >> [!answer] Linearity tells you how a gate acts on every superposition once you know its action on a basis. It also rules out nonlinear operations such as a universal perfect cloning map for arbitrary unknown states. ## Additional Study Notes **Check yourself: inner products under gates** Any unitary $U$ preserves inner products: $\langle v|w\rangle = \langle Uv|Uw\rangle$. This is why unitary gates preserve distinguishability geometry rather than stretching state space arbitrarily. **Check yourself: global phase** The states $|0\rangle$ and $-|0\rangle$ cannot be perfectly distinguished by measurement. They differ only by global phase, so all measurement statistics are identical. **Key fact: Hermitian and unitary can overlap** A matrix can be both Hermitian and unitary. Pauli matrices such as $X$ and $Z$ are standard examples because they equal their own adjoint and square to the identity. **Key fact: unitary density matrices** No valid $2\times2$ density matrix is unitary. A density matrix is positive semidefinite with trace $1$, while a unitary matrix has eigenvalues of magnitude $1$; in two dimensions those requirements cannot both hold. **Question: Why can the North and South poles of the Bloch sphere be distinguished perfectly?** They correspond to orthogonal basis states, usually $|0\rangle$ and $|1\rangle$. Measuring in the computational basis separates them with certainty. ## Study Checks Use these after the explanation, not before it. ### Quick Checks - MCQ: What physical effect does multiplying an entire state by a global phase have? A. It changes all measurement probabilities B. It changes only X-basis probabilities C. It has no observable effect D. It creates entanglement **Answer:** C. It has no observable effect. - What is the difference between global phase and relative phase? **Answer:** Global phase multiplies the whole state and leaves all probabilities unchanged. Relative phase changes phase between components and can affect later interference. - What does the tensor product represent in quantum mechanics? **Answer:** It combines subsystem Hilbert spaces into the joint state space of a composite system. - MCQ: How many complex amplitudes describe a general pure state of n qubits before normalization and global phase conventions? A. n B. n^2 C. 2n D. 2^n **Answer:** D. 2^n. - What is the practical loop for each concept note? **Answer:** Read the note, run the per-concept notebook, inspect the qcourse helper, and check the corresponding test. ### Oral Exam Anchors > [!question] Oral exam anchor > Derive why global phase is unobservable but relative phase can matter. A good answer should say: Let |psi> be a normalized state and let e^{i theta}|psi> be the same state multiplied by a global phase. For any measurement effect or projector M, the probability is <psi| e^{-i theta} M e^{i theta} |psi> = <psi|M|psi>. The global phase cancels from every measurable probability, so it is not physical. Relative phase is different. For example, (|0>+|1>)/sqrt(2) and (|0>-|1>)/sqrt(2) have the same computational-basis probabilities, but applying a Hadamard maps them to |0> and |1> respectively. Relative phase affects later interference, so it can become observable after a basis change. ## Practical Lab Treat this lab as a fluency exercise with state preparation and prediction. - Prepare several one-qubit states and inspect their amplitudes in a simulator. - Compare two states that differ only by global phase and verify that measurement statistics do not change. - Prepare two states with different relative phase and use a later gate to make that difference observable in output counts. ## Homework Use the homework to move comfortably between notation and prediction. - Write a normalized one-qubit state in ket notation and verify its norm. - Compute its measurement probabilities in the computational basis. - Explain the difference between global phase and relative phase without using vague phrases like "one matters and one does not." ## References - Scott Aaronson, [Introduction to Quantum Information Science](https://www.scottaaronson.com/qclec.pdf), Lecture 3. - John Watrous, [Quantum Computation lecture notes](https://cs.uwaterloo.ca/~watrous/QC-notes/). - IBM Quantum Learning, [qubit states and measurement](https://quantum.cloud.ibm.com/learning/en). - Microsoft Learn, [Azure Quantum documentation](https://learn.microsoft.com/en-us/azure/quantum/).