Quantum States and the Postulates of Quantum Mechanics

Introduction

Quantum computing is built on a physical theory whose rules differ fundamentally from those of classical computation. A classical bit is either 0 or 1. A quantum bit, or qubit, is described by a state that may combine the alternatives 0 and 1 through superposition. Measurement does not simply reveal a pre-existing classical value in every situation; instead, it produces outcomes probabilistically according to the state and the measurement being performed.

This chapter develops the mathematical and physical foundations required to describe a single qubit and small composite systems. The central ideas are state vectors, normalization, complex amplitudes, global and relative phase, superposition, unitary evolution, and measurement. These ideas are expressed through the postulates of quantum mechanics and applied to quantum-computing examples.

The discussion assumes familiarity with complex numbers, vectors, matrices, and basic probability. The objective is not to treat quantum mechanics as an abstract formalism only, but to show how its postulates support practical reasoning about quantum circuits and algorithms.

Learning objectives

By the end of this chapter, you should be able to:

• represent a pure quantum state using ket notation and column vectors;
• normalize state vectors and interpret amplitudes as probability amplitudes;
• distinguish global phase from physically significant relative phase;
• explain superposition and the state space of a qubit;
• apply unitary operators to predict state evolution;
• calculate measurement probabilities and post-measurement states;
• describe the postulates of quantum mechanics in a computing context;
• analyse a simple quantum-circuit application using these principles.

1. Mathematical language for quantum states

1.1 Kets and vectors

A quantum state is commonly written in Dirac, or bra–ket, notation. A ket is represented by a symbol such as |ψ⟩. For a qubit, the computational-basis states are

The superscript T denotes transpose. These two vectors form an orthonormal basis for the two-dimensional complex vector space associated with one qubit. Orthonormality means

where δᵢⱼ is 1 when i = j and 0 otherwise. The bra ⟨ψ| is the conjugate transpose of the ket |ψ⟩.

A general pure state of one qubit is

or, as a column vector,

Here α and β are complex numbers called probability amplitudes. They are not probabilities themselves. Their squared magnitudes determine measurement probabilities.

1.2 Normalization

The total probability of all possible outcomes must equal 1. Therefore every physically valid state vector satisfies the normalization condition

The notation |α|² means α multiplied by its complex conjugate. If α = a + ib, then |α|² = a² + b². Normalization is dimensionless because quantum amplitudes represent probability amplitudes rather than physical quantities with units.

If a nonzero vector is not normalized, it can be normalized by dividing by its norm:

This operation is applicable when ⟨ψ|ψ⟩ > 0. The zero vector cannot represent a quantum state.

#### Worked example: normalization

Suppose an unnormalized vector is

Its squared norm is

The normalized state is therefore

The probabilities of measuring 0 and 1 in the computational basis are both 1/2. The factor i still matters for interference, even though it does not change the individual magnitudes in this particular measurement.

1.3 Complex amplitudes and phase

A complex number can be written in polar form as

where r = |α| is the magnitude and φ is the phase. The magnitude influences measurement probabilities; phase influences how amplitudes combine during later operations.

For a normalized qubit, a useful form is

where 0 ≤ θ ≤ π and 0 ≤ φ < 2π. This representation separates the relative phase φ from the population weights determined by θ.

2. Global phase and relative phase

2.1 Global phase

Two state vectors that differ only by an overall phase factor represent the same physical pure state:

where γ is real. This is called global phase. For example, |ψ⟩ and -|ψ⟩ differ by the global phase e^iπ = -1. No measurement can distinguish these states when the phase multiplies the entire vector.

For a measurement described by an operator M, the probability involves an expression of the form ⟨ψ|M|ψ⟩. Replacing |ψ⟩ by e^iγ|ψ⟩ introduces a factor e^-iγ from the bra and e^iγ from the ket; these cancel.

2.2 Relative phase

Relative phase is the phase difference between components of a superposition. It is physically observable through interference. Consider

The minus sign is a relative phase of π, not merely an irrelevant global sign. Applying the Hadamard gate H gives

Thus the two states have identical computational-basis probabilities before the Hadamard operation, but produce different outcomes after it. This is the operational significance of relative phase.

3. Superposition and state spaces

3.1 Meaning of superposition

A state such as

is a coherent superposition of basis states when both amplitudes are nonzero. It does not mean that a measurement returns a fraction of a 0 and a fraction of a 1. A measurement returns one allowed outcome, while repeated preparation and measurement reveal a probability distribution.

Superposition differs from an ordinary classical uncertainty mixture. A classical system may be unknown to an observer but still possess one definite value. A quantum superposition contains phase information that can produce interference. This distinction is essential in quantum algorithms.

3.2 The qubit state space

The state space of a qubit is a two-dimensional complex Hilbert space, denoted ℂ², with the inner product defined by the conjugate transpose. Physical states are normalized vectors, with vectors differing by global phase identified as physically equivalent.

The Bloch-sphere representation provides a geometric picture. Every pure qubit can be written as

The corresponding Cartesian coordinates are

These satisfy x² + y² + z² = 1, which is the surface of the unit Bloch sphere. The north and south poles represent |0⟩ and |1⟩. The equator contains equal-magnitude superpositions, including |+⟩ and |−⟩.

The Bloch sphere is a visualization for one pure qubit. It does not mean that a qubit is literally a tiny sphere, and it does not directly represent arbitrary multi-qubit states.

4. The postulates of quantum mechanics

The following formulation is especially useful for quantum computing.

Postulate 1: State description

The state of an isolated quantum system is represented by a normalized vector in a complex Hilbert space. For a qubit, the state is an element of ℂ² satisfying ⟨ψ|ψ⟩ = 1.

A state vector is a complete description for predicting the probabilities of all measurements permitted by the model. This does not necessarily mean that all physical properties have definite values before measurement.

Postulate 2: Evolution of an isolated system

The evolution of an isolated quantum system is described by a unitary operator U:

where

Here U† is the conjugate transpose and I is the identity operator. Unitarity preserves inner products and therefore preserves normalization:

Quantum gates are finite-dimensional unitary operators. Examples include the Pauli-X gate, which exchanges computational-basis states,

and the Hadamard gate,

The matrix notation uses rows separated by semicolons. The Hadamard gate acts as

Postulate 3: Measurement

A quantum measurement is represented by a collection of measurement operators {M_m}, indexed by possible outcomes m, satisfying

If the pre-measurement state is |ψ⟩, the probability of outcome m is

If outcome m occurs, the normalized post-measurement state is

provided p(m) > 0. This is the general measurement rule.

For computational-basis measurement of a qubit, the projectors are

For |ψ⟩ = α|0⟩ + β|1⟩,

After observing 0, the state becomes |0⟩; after observing 1, it becomes |1⟩. These probabilities sum to one because the state was normalized.

Postulate 4: Composite systems

The state space of a composite system is the tensor product of the component spaces. Two qubits occupy

The computational basis is

A general pure two-qubit state is

with

Some states can be written as a tensor product of single-qubit states. For example,

Other states cannot be factored. The Bell state

is entangled. Entanglement is a correlation structure that cannot be reproduced by assigning independent single-qubit state vectors to the two systems.

5. Measurement in different bases

Measurement probabilities depend on the measurement basis. A state that is uncertain in one basis may be definite in another. For example, |+⟩ is a 50–50 state in the computational basis, but measuring it in the Hadamard basis {|+⟩, |−⟩} produces the |+⟩ outcome with probability 1.

To measure a qubit in a basis whose basis vectors are columns of a unitary matrix V, one practical method is to apply V† and then measure in the computational basis. This works because V† transforms the chosen basis into the computational basis.

#### Worked example: Hadamard interference

Start with |0⟩. Apply H:

A computational-basis measurement now gives 0 and 1 with probabilities 1/2 each. Apply H again:

The two paths interfere constructively for |0⟩ and destructively for |1⟩. The final measurement therefore returns 0 with probability 1. This example shows why amplitudes, including their signs and phases, must be tracked rather than replaced immediately by probabilities.

6. Expectation values and observables

An observable is represented by a Hermitian operator A satisfying A† = A. Its possible measurement values are the eigenvalues of A. For a state |ψ⟩, the expected value is

This is an average over many identically prepared trials, not necessarily the result of a single measurement. For example, the Pauli-Z operator is

For |ψ⟩ = α|0⟩ + β|1⟩,

The eigenvalues are +1 and −1. If the state is |+⟩, the expectation value is zero, although individual measurements still return either +1 or −1.

7. Worked application: a one-qubit sensing circuit

Consider an illustrative phase-sensitive circuit used to demonstrate the operating principle of interference-based quantum sensing. The input is |0⟩. The circuit applies H, then a phase gate

and finally H.

Analysis

After the first H,

After S,

The phase gate has not changed the computational-basis probabilities: both remain 1/2. However, it has changed the relative phase between the two components.

Applying H gives

The outcome probabilities are

Thus the final measurement is balanced, even though the intermediate phase was meaningful. If a different phase gate had been used, the final probabilities would generally change. The circuit illustrates a practical principle: phase information becomes observable when a later operation causes amplitudes to interfere.

Professional relevance

This type of reasoning appears in quantum phase estimation, interferometry, calibration of quantum gates, and error diagnosis. Engineers must distinguish a change in population probability from a change in phase, because a device may preserve populations while still accumulating damaging phase errors.

8. Case study: diagnosing a phase error in a teaching quantum processor

Background

This illustrative case concerns a diploma-level laboratory using a two-gate calibration circuit. Students prepare |+⟩ with H, apply a nominal identity operation, and then apply H before measurement. Ideally, the result is always 0.

Problem

The laboratory instead records approximately 50% zeros and 50% ones. The hardware team initially suspects that the first Hadamard gate is producing the wrong populations.

Analysis

The correct ideal sequence is

Suppose the nominal identity operation introduces a relative phase of π, effectively applying Z to |+⟩. Then

and the final Hadamard produces

A phase error can therefore change a deterministic final result without changing the 50–50 computational-basis distribution observed in the middle of the circuit.

Approach and solution

The team runs three tests:

1. measure immediately after the first H to verify the populations;
2. perform the full circuit to detect interference failure;
3. vary the duration of the nominal identity operation to determine whether the error is a systematic phase accumulation.

The first test gives the expected 50–50 distribution. The second test gives an excess of 1 outcomes. The third test shows oscillation as the duration changes, identifying a phase-accumulation mechanism rather than a preparation-population error.

Outcome

The calibration procedure is revised to include phase-sensitive benchmark circuits. The device is not judged solely by individual gate output probabilities; complete circuits are used to test coherent evolution.

Learning points

• Relative phase is operationally important even when immediate measurement probabilities are unchanged.
• Interference is a diagnostic resource, not only an algorithmic resource.
• Quantum-device validation requires repeated trials and statistical interpretation.
• A circuit-level error may arise from a physically small phase error accumulated over time.

9. Limitations and practical considerations

9.1 Measurement is probabilistic

A single result cannot generally establish the underlying probability exactly. If an experiment is repeated N times and outcome 0 occurs n₀ times, the empirical frequency is f₀ = n₀/N. This estimates the theoretical probability but fluctuates because of finite sampling.

9.2 State-vector descriptions scale exponentially

An n-qubit pure state generally requires 2ⁿ complex amplitudes:

with ∑ₓ|αₓ|² = 1. This exponential representation is one reason classical simulation becomes difficult as n grows, although special states and structured algorithms can often be simulated efficiently.

9.3 Noise and open systems

The unitary postulate applies exactly to isolated systems. Real quantum processors interact with their environments. Noise can cause decoherence, relaxation, and gate errors. More general descriptions use density operators and quantum channels, which extend the pure-state framework introduced here. For the present chapter, the state-vector model is sufficient for ideal circuits and foundational calculations.

9.4 Interpretation and responsible communication

Quantum superposition should not be presented as a license for unsupported claims about consciousness, unlimited parallelism, or instantaneous communication. Quantum computing has precise operational rules. Measurements produce ordinary classical data, and useful computational advantage depends on algorithm design, hardware quality, error management, and problem structure.

Best practices for solving state-evolution problems

1. Write the initial state clearly in the computational basis.
2. Check that the state is normalized.
3. Apply gates in the order they occur; matrix products act right to left on column vectors.
4. Preserve complex phases until the final measurement calculation.
5. Use unitarity as a check that normalization is preserved.
6. Compute probabilities as squared magnitudes, not ordinary squares of complex numbers.
7. Normalize the post-measurement state when using general measurement operators.
8. State the measurement basis explicitly.
9. For repeated experiments, distinguish theoretical probability from observed frequency.
10. In multi-qubit systems, maintain a consistent basis ordering such as |00⟩, |01⟩, |10⟩, |11⟩.

Chapter Summary

A pure quantum state is represented by a normalized vector in a complex Hilbert space. A qubit has the general form |ψ⟩ = α|0⟩ + β|1⟩, with |α|² + |β|² = 1. Amplitudes are complex, and their magnitudes determine measurement probabilities while their relative phases determine interference. A global phase multiplying the entire state has no observable effect, whereas relative phase can change later measurement results.

The postulates provide a predictive framework: states describe systems, unitary operators describe isolated evolution, measurement operators determine probabilities and state updates, and tensor products describe composite systems. The Hadamard gate demonstrates the creation of superposition and the conversion of phase differences into observable interference. These foundations support quantum-circuit analysis, device calibration, quantum sensing, and the study of larger algorithms.

Key Terms

Pure state: A state represented by a normalized vector in Hilbert space.

Qubit: The basic two-level quantum information unit.

Ket: A vector written in Dirac notation, such as |ψ⟩.

Bra: The conjugate transpose of a ket, such as ⟨ψ|.

Amplitude: A complex coefficient associated with a basis state.

Normalization: The condition that total probability equals 1.

Inner product: A complex-valued product used to determine lengths and overlaps.

Global phase: An overall phase factor that does not affect physical predictions.

Relative phase: A phase difference between components of a superposition.

Superposition: A coherent linear combination of basis states.

Hilbert space: A complex inner-product vector space used to represent quantum states.

Unitary operator: An operator satisfying U†U = I and preserving normalization.

Quantum gate: A unitary operation used to transform quantum states.

Measurement: A physical process producing an outcome according to quantum probabilities.

Projector: An operator selecting a subspace or measurement outcome.

Observable: A Hermitian operator representing a measurable physical quantity.

Tensor product: The operation used to construct composite-system state spaces.

Entanglement: Correlation that cannot be represented as a product of subsystem states.

Interference: The combination of amplitudes in which relative phase affects probabilities.

Bloch sphere: A geometric representation of pure single-qubit states.

Review Questions

5 Short Answer Questions

1. What is the normalization condition for a general pure qubit state?
2. What is the difference between a probability amplitude and a probability?
3. Why is global phase not observable in a pure state?
4. State the mathematical condition that defines a unitary operator.
5. What is the computational-basis probability of measuring 1 in |ψ⟩ = α|0⟩ + β|1⟩?

5 Long Answer Questions

1. Explain the role of complex amplitudes, phase, and normalization in representing a qubit.
2. Describe the main postulates of quantum mechanics used in quantum computing.
3. Explain, with equations, how the Hadamard gate creates and later reveals interference.
4. Compare a quantum superposition with a classical probabilistic mixture.
5. Explain how tensor products describe two-qubit systems and why entangled states are significant.

3 Analytical / Application Questions

1. A qubit is prepared in |ψ⟩ = (√3/2)|0⟩ + (i/2)|1⟩. Calculate the computational-basis measurement probabilities and the expectation value of Z.
2. Analyse the circuit HZH applied to |0⟩. Determine the final state and explain the role of the relative phase.
3. A calibration experiment produces the correct intermediate populations but incorrect final interference. Propose two tests that distinguish a phase error from a state-preparation error, and justify your choices.

Multiple Choice Questions with Answers

Question 1

Which condition must a valid pure qubit state satisfy?

A. |α|² + |β|² = 0

B. |α|² + |β|² = 1

C. α + β = 1 for all states

D. αβ = 1 for all states

Correct Answer: B. |α|² + |β|² = 1

Question 2

What is the computational-basis probability of measuring 1 from |ψ⟩ = α|0⟩ + β|1⟩?

A. |α|²

B. α + β

C. |β|²

D. β² without taking magnitude

Correct Answer: C. |β|²

Question 3

Which statement about global phase is correct?

A. It changes every measurement probability

B. It is physically observable in every experiment

C. It has no effect on physical predictions for a pure state

D. It must always be zero

Correct Answer: C. It has no effect on physical predictions for a pure state

Question 4

Which property defines a unitary operator U?

A. U†U = 0

B. U†U = I

C. U† = 0

D. U² = I in every case

Correct Answer: B. U†U = I

Question 5

What state results from applying H to |0⟩?

A. |1⟩

B. (|0⟩ − |1⟩)/√2

C. (|0⟩ + |1⟩)/√2

D. 0|0⟩ + 1|1⟩ with probability 1

Correct Answer: C. (|0⟩ + |1⟩)/√2

Question 6

Why can relative phase be physically important?

A. It changes the dimension of the Hilbert space

B. It can affect interference after later operations

C. It eliminates the need for normalization

D. It turns a qubit into a classical bit

Correct Answer: B. It can affect interference after later operations

Question 7

For the state |+⟩ = (|0⟩ + |1⟩)/√2, what is the result of applying H?

A. |0⟩

B. |1⟩

C. |+⟩

D. A non-normalized vector

Correct Answer: A. |0⟩

Question 8

What mathematical construction describes the state space of two qubits?

A. Ordinary addition of two real numbers

B. The tensor product of two single-qubit spaces

C. The intersection of two classical probability spaces

D. A one-dimensional vector space

Correct Answer: B. The tensor product of two single-qubit spaces

Question 9

Which operator represents an observable in the standard formalism?

A. A Hermitian operator

B. Any non-square matrix

C. Only the identity operator

D. A non-normalized ket

Correct Answer: A. A Hermitian operator

Question 10

For |ψ⟩ = (|0⟩ + i|1⟩)/√2, what are the computational-basis probabilities?

A. 0 and 1

B. 1 and 0

C. 1/2 and 1/2

D. 1/4 and 3/4

Correct Answer: C. 1/2 and 1/2