Quantum Mathematics begins with an unusual but precise idea: the state of a quantum system is represented mathematically before a measurement turns that state into an ordinary outcome.
This guide develops the mathematics behind that sentence. It is not a Singapore school syllabus topic and it is not a shortcut into quantum physics. It is a bridge from familiar Mathematics—algebra, vectors, complex numbers and probability—into the formal language used in introductory quantum mechanics and quantum information.
The central objects in this first guide are complex numbers, vectors, probability amplitudes, phase, normalisation and inner products. The aim is to see why quantum theory needs them, how they fit together and how to calculate with them carefully.
Classical probability assigns probabilities directly. Quantum theory assigns amplitudes first, then converts amplitudes into probabilities.
1. Why ordinary real numbers are not enough
In school Mathematics, most quantities are represented by real numbers. Length, time, gradient, probability and coordinates on a number line can all be handled using real values. Quantum mechanics goes further because it must keep track not only of magnitude but also of phase. Complex numbers provide the smallest standard number system that can carry both pieces of information cleanly while remaining compatible with linear algebra.
A complex number has the form z = a + bi, where a and b are real numbers and i² = -1. We call a the real part and b the imaginary part. Although the word “imaginary” sounds non-physical, the mathematics is completely precise. Complex numbers can be added, multiplied, divided, conjugated and represented geometrically on the complex plane.
The complex conjugate of z = a + bi is z* = a - bi. Multiplying a complex number by its conjugate gives a non-negative real number:
zz* = (a + bi)(a - bi) = a² + b².
This quantity is the square of the magnitude of the complex number. We write |z| = √(a²+b²), so |z|² = zz*. That operation—multiply an amplitude by its complex conjugate—is one of the most important moves in elementary quantum mathematics because it converts a complex amplitude into a real, non-negative probability weight.
Worked example: magnitude of a complex amplitude
Suppose z = (3 + 4i)/5. Then |z|² = (3/5)² + (4/5)² = 9/25 + 16/25 = 1. Therefore |z| = 1. This number lies on the unit circle in the complex plane.
That unit-circle idea matters because a complex number can change its phase while keeping its magnitude fixed. For quantum states, phase can change how amplitudes later combine even when the individual probability weights initially look unchanged.
2. Polar form and phase
A non-zero complex number can be written in polar form as z = r(cos θ + i sin θ). Using Euler’s relation, this becomes z = re^{iθ}. Here r = |z| is the magnitude and θ is the phase angle.
Two amplitudes can therefore have the same magnitude but different phases. For example, 1/√2 and i/√2 both have magnitude 1/√2, so each contributes probability weight 1/2 when considered alone. Yet they are not mathematically identical, because one points along the real direction and the other along the imaginary direction.
This is the first major difference from ordinary probability. Classical probabilities are already final non-negative weights. Quantum amplitudes can be positive, negative or complex, and their relative phases matter before the final probability is calculated.
3. From numbers to vectors
A quantum state is usually represented as a vector in a complex vector space. For a two-state system, we can choose basis vectors
|0⟩ = [1, 0]ᵀ and |1⟩ = [0, 1]ᵀ.
A general state can then be written
|ψ⟩ = α|0⟩ + β|1⟩ = [α, β]ᵀ,
where α and β are complex numbers. These coefficients are called probability amplitudes.
The notation |ψ⟩ is called a ket, part of Dirac notation. It is simply a compact way to speak about vectors and the spaces they live in. The notation can feel exotic at first, but the underlying linear algebra is familiar: basis vectors, coefficients, vector addition and scalar multiplication.
4. Normalisation: the state must carry total probability one
If |ψ⟩ = α|0⟩ + β|1⟩, then the probabilities of obtaining the basis outcomes 0 and 1 are |α|² and |β|². Because one of the outcomes must occur when this measurement is made, the probabilities must add to one:
|α|² + |β|² = 1.
This is the normalisation condition. In vector language, the state vector has norm one.
Worked example: checking a state
Take |ψ⟩ = (1/√5)|0⟩ + (2i/√5)|1⟩. The first amplitude has squared magnitude 1/5. The second has squared magnitude |2i/√5|² = 4/5. Their sum is 1, so the state is normalised.
If we measure in the |0⟩, |1⟩ basis, the probability of outcome 0 is 1/5 and the probability of outcome 1 is 4/5.
Worked example: normalising an unnormalised vector
Suppose a vector is given as v = [1+i, 2]ᵀ. Its squared norm is
|1+i|² + |2|² = 2 + 4 = 6.
So its norm is √6. The corresponding normalised state is
|ψ⟩ = (1/√6)[1+i, 2]ᵀ.
The resulting measurement probabilities in the computational basis are 2/6 = 1/3 and 4/6 = 2/3.
5. The Born rule: amplitudes become probabilities
The standard rule connecting the quantum state to measurement probabilities is the Born rule. In the simplest basis-state setting, if the amplitude attached to an outcome is α, the probability of obtaining that outcome is |α|².
This is easy to state but conceptually important. The amplitude itself is not a probability. A negative amplitude is allowed. A complex amplitude is allowed. What becomes a probability is the squared magnitude.
Amplitude → squared magnitude → probability.
IBM Quantum’s introductory material uses exactly this vector-and-amplitude framework: quantum states are complex vectors, measurements produce classical outcomes according to squared magnitudes, and unitary matrices describe allowed closed-system operations. For a formal course-level treatment, see IBM Quantum Learning: Single Systems.
6. Why phase matters: interference
Suppose two mathematical routes contribute amplitudes a and b to the same final outcome. Quantum theory combines the amplitudes first. The probability is then based on |a+b|², not merely on |a|² + |b|².
Expand the squared magnitude:
|a+b|² = (a+b)(a*+b*) = |a|² + |b|² + ab* + a*b.
The final two terms are interference terms. They depend on the relative phase between the amplitudes.
Worked example: constructive interference
Let a = 1/2 and b = 1/2. Then a+b = 1, so |a+b|² = 1. The amplitudes reinforce each other.
Worked example: destructive interference
Let a = 1/2 and b = -1/2. Each amplitude separately has squared magnitude 1/4, but together a+b = 0, so the final probability contribution is zero. The difference is phase, represented here by the sign.
This is why replacing quantum amplitudes with ordinary probabilities too early destroys essential information. The probabilities alone cannot remember how the amplitudes should interfere later.
7. Global phase and relative phase
Quantum mathematics distinguishes global phase from relative phase. Multiplying an entire state vector by the same unit-magnitude complex number e^{iφ} does not change measurement probabilities. The states |ψ⟩ and e^{iφ}|ψ⟩ represent the same physical pure state.
Relative phase is different. Compare
|+⟩ = (|0⟩ + |1⟩)/√2
with
|−⟩ = (|0⟩ - |1⟩)/√2.
Both give probability 1/2 for 0 and 1/2 for 1 when measured directly in that basis. Yet they behave differently under later operations because the sign between the amplitudes changes the interference structure.
8. Inner products: comparing quantum vectors
For real Euclidean vectors, the dot product measures alignment. Quantum mathematics uses a complex analogue called the inner product.
If |ψ⟩ = [α, β]ᵀ, its corresponding bra is ⟨ψ| = [α*, β*]. The conjugation is essential. The inner product of |φ⟩ with |ψ⟩ is written ⟨φ|ψ⟩.
The normalisation condition can now be written compactly as ⟨ψ|ψ⟩ = 1.
Worked example: inner product
Let |ψ⟩ = [1/√2, i/√2]ᵀ and |φ⟩ = [1, 0]ᵀ. Then ⟨φ|ψ⟩ = 1/√2. The squared magnitude is 1/2, matching the probability of finding |ψ⟩ in the |0⟩ basis state.
This generalises the simple “read the coefficient” rule. In an arbitrary orthonormal basis, the amplitude for basis state |φ⟩ is the inner product ⟨φ|ψ⟩, and the corresponding probability is |⟨φ|ψ⟩|².
9. Orthogonality and distinguishable basis states
Two vectors are orthogonal when their inner product is zero. For the standard basis, ⟨0|1⟩ = 0. Orthogonal basis states play the role of perfectly distinguishable alternatives in the ideal mathematical model.
An orthonormal basis consists of vectors that are all mutually orthogonal and each have norm one. Once a basis is chosen, any state in the space can be expanded as a linear combination of those basis vectors.
This is closely related to familiar coordinate geometry. The same point can have different coordinates under different axes. Likewise, the same quantum state can have different coefficient lists in different bases. The state is the object; the coordinates depend on the representation.
Do not confuse a state with one particular list of coordinates used to describe it.
10. Basis change as a mathematical idea
School Mathematics already prepares students for basis changes, even if the phrase is unfamiliar. A geometric vector can be described relative to different coordinate axes. A polynomial can be described by coefficients in different polynomial bases. A function can be expanded in different function families.
Quantum mathematics makes this central. Measurement is not simply “look at the hidden value.” Instead, the chosen measurement basis determines which set of outcomes is being resolved. That is why the same state can produce deterministic behaviour in one basis and probabilistic behaviour in another.
Worked example: the plus state in two bases
In the computational basis, |+⟩ = (|0⟩+|1⟩)/√2, so measuring in that basis gives 0 and 1 with equal probabilities.
But if we instead use the basis |+⟩, |−⟩, the same state has coordinates [1,0]ᵀ. In that basis, the outcome associated with |+⟩ occurs with probability one.
This is not a contradiction. It is a reminder that a probability statement is incomplete unless the measurement being discussed is specified.
11. Classical probability vectors versus quantum state vectors
A classical probability vector for two outcomes might be [p, 1-p], where entries are real, non-negative and sum to one. A quantum state vector might be [α, β]ᵀ, where entries are complex and their squared magnitudes sum to one.
- Classical: entries are probabilities.
- Quantum: entries are amplitudes.
- Classical normalisation: entries sum to one.
- Quantum normalisation: squared magnitudes sum to one.
- Classical mixing: combines probability weights.
- Quantum superposition: combines amplitudes and therefore preserves phase information.
The distinction is mathematical before it is philosophical. Different state spaces imply different allowable transformations and different ways probabilities emerge.
12. Common misconception: “a qubit is both 0 and 1”
The phrase is popular but mathematically weak. A better statement is that a qubit pure state is a normalised vector in a two-dimensional complex vector space and can be written as a superposition α|0⟩ + β|1⟩ relative to a chosen basis.
That sentence sounds less dramatic, but it is much more useful. It tells us exactly what can be calculated: amplitudes, norms, inner products, basis changes and measurement probabilities.
13. Common misconception: amplitudes are hidden probabilities
They are not. If amplitudes were merely disguised probabilities, their phases would be irrelevant. But phases affect interference, and interference changes later measurement statistics. Two states can have the same probabilities in one basis and still be different states because their relative phases differ.
14. Common misconception: complex numbers are an optional notation choice
For standard quantum mechanics, complex structure is not decorative. It supports phase, interference and unitary time evolution in a natural linear framework. MIT OpenCourseWare’s Quantum Physics I notes explicitly treat complex numbers and linearity as foundational mathematical features of the theory. See MIT OpenCourseWare: Quantum Physics I lecture notes.
15. A compact mathematical checklist
Given a proposed two-state pure state |ψ⟩ = α|0⟩ + β|1⟩, check:
- Are
αandβallowed to be complex? Yes. - Is the state normalised? Verify
|α|² + |β|² = 1. - What measurement basis is being used?
- What is the amplitude for each outcome?
- What is the squared magnitude of each amplitude?
- Do the resulting probabilities sum to one?
- Is there relative phase information that may matter after a later transformation?
16. Worked synthesis problem
Consider
|ψ⟩ = (1/2)|0⟩ + ((√3)i/2)|1⟩.
Step 1: Normalisation. The squared magnitudes are 1/4 and 3/4. Their sum is one.
Step 2: Computational-basis probabilities. Outcome 0 has probability 1/4; outcome 1 has probability 3/4.
Step 3: Relative phase. The first amplitude is real and positive. The second has an additional phase of π/2. This phase does not alter the two direct probabilities above, but it can alter probabilities after a basis-changing operation.
Step 4: Inner product with |0⟩. We obtain ⟨0|ψ⟩ = 1/2. Squaring its magnitude again gives 1/4.
Step 5: Interpret carefully. The state is not a classical lottery that secretly chose 0 with probability one-quarter and 1 with probability three-quarters before measurement. The vector carries additional phase structure that can affect later interference.
17. What a Secondary or JC Mathematics student already knows
The jump into quantum mathematics is smaller when we identify the pieces already present in school Mathematics:
- Algebra: manipulating symbolic expressions without changing their value.
- Complex numbers: magnitude, argument, conjugation and polar form.
- Vectors: components, bases, length and geometric interpretation.
- Probability: normalisation, conditional reasoning and outcome spaces.
- Functions: one representation acting on another under clear rules.
Quantum theory does not discard these ideas. It composes them into a stricter linear-algebraic system.
18. What changes at university level
The main increase is not merely harder arithmetic. It is structural density. Vectors become elements of complex inner-product spaces. Functions can become vectors. Physical observables become linear operators. Eigenvectors identify states with definite values for particular observables. Composite systems use tensor products. Time evolution is generated by a Hamiltonian through the Schrödinger equation.
Each of those steps will be developed in this four-part series. The important learning move is to preserve the chain rather than memorise the vocabulary separately.
Complex numbers carry phase. Vectors carry state. Inner products compare states. Squared magnitudes produce probabilities. Linear operators will carry the state forward.
19. Practice set
- Find the complex conjugate and squared magnitude of
3-4i. - Show that
(1+i)/√2has magnitude one. - Check whether
[1/2, √3/2]ᵀis normalised. - Normalise
[1, i]ᵀ. - For
|ψ⟩=(|0⟩+i|1⟩)/√2, find the computational-basis probabilities. - Compare the direct basis probabilities of
|+⟩and|−⟩. - Explain why equal direct probabilities do not imply identical quantum states.
- Compute
⟨0|ψ⟩for|ψ⟩=(3|0⟩+4i|1⟩)/5. - Show that
⟨ψ|ψ⟩=1for the same state. - Give one reason ordinary probability vectors cannot preserve quantum phase information.
Answers
3+4i; squared magnitude25.|(1+i)/√2|²=(1+1)/2=1.- Yes:
1/4+3/4=1. [1/√2, i/√2]ᵀ.1/2and1/2.- Both give
1/2,1/2in the computational basis. - The relative phase differs and can alter later interference.
3/5.9/25+16/25=1.- Probabilities discard sign/complex phase before amplitudes have finished interfering.
20. Series navigation
- Guide 1: Complex Numbers, State Vectors and Probability Amplitudes
- Guide 2: Matrices, Operators, Eigenvalues and Measurement
- Guide 3: Tensor Products, Qubits and Entanglement
- Guide 4: Unitary Evolution, Schrödinger’s Equation and Quantum Dynamics
- Return to the BTT Mathematics Learning Hub
Educational note: this series explains mathematical foundations used in quantum mechanics and quantum information. It does not replace a university physics course, laboratory training or a formal linear algebra course.