Qubits and Quantum State Representation

Introduction

Quantum information is encoded differently from classical information. A classical bit occupies one of two definite states, conventionally written 0 or 1. A quantum bit, or qubit, is a physical or abstract two-level quantum system. It may occupy either basis state, or a coherent superposition of both. This apparently small change creates the distinctive phenomena used in quantum algorithms, communication and sensing: interference, phase sensitivity, entanglement and measurement-dependent outcomes.

This chapter develops the mathematical language needed to describe qubits and small multi-qubit systems. The central representations are Dirac notation, state vectors and the Bloch sphere. Particular attention is given to normalization, global and relative phase, tensor products, measurement probabilities, separability and correlation. These ideas form the foundation for quantum circuits and quantum information theory.

Learning Objectives

After studying this chapter, you should be able to:

1. Explain the difference between a classical bit and a qubit.
2. Write pure quantum states using ket and bra notation.
3. Apply normalization and the Born rule to calculate measurement probabilities.
4. Distinguish global phase from physically observable relative phase.
5. Represent a single-qubit pure state as a state vector and on the Bloch sphere.
6. Construct multi-qubit states using tensor products.
7. Identify whether a two-qubit pure state is separable or entangled.
8. Interpret quantum-state representations in practical computing and communication contexts.

1. Classical Bits and Quantum Bits

A classical bit has a value 0 or 1. A probabilistic classical bit can be uncertain, but it is still described by a probability distribution: it is 0 with probability p and 1 with probability 1 − p. Uncertainty about a classical bit is not the same as quantum superposition.

A qubit has two orthonormal computational-basis states, written |0⟩ and |1⟩. The most general pure state of one qubit is

where α and β are complex probability amplitudes. For a valid state, the normalization condition is

Here |α|² and |β|² are real, non-negative numbers. If the qubit is measured in the computational basis, the Born rule states that the outcome 0 occurs with probability |α|² and outcome 1 occurs with probability |β|². The amplitudes themselves may be negative or complex; probabilities cannot be negative, but amplitudes can interfere.

For example, the state

produces 0 and 1 with equal probabilities. The state

has exactly the same computational-basis probabilities, but it is a different quantum state because its relative phase differs. Later operations can distinguish these states through interference.

A qubit is not simply “both 0 and 1 in the ordinary classical sense.” Before measurement, its state is a vector in a complex vector space. Measurement produces one classical outcome, and the state generally changes as a consequence.

2. Dirac Notation and State Vectors

2.1 Kets and bras

A ket |ψ⟩ denotes a column vector representing a state. In the computational basis,

Thus

The corresponding bra, written ⟨ψ|, is the conjugate transpose of |ψ⟩:

The symbol α* denotes the complex conjugate of α. The inner product of two states |φ⟩ and |ψ⟩ is ⟨φ|ψ⟩. Two states are orthogonal when ⟨φ|ψ⟩ = 0. The norm of |ψ⟩ is

A physical state vector must have norm 1. State vectors are therefore elements of a two-dimensional complex Hilbert space, commonly denoted ℂ².

2.2 Basis changes

The computational basis is not the only possible basis. The Hadamard basis is

and

These states are orthonormal. A state that is certain to be |+⟩ in the Hadamard basis gives equal probabilities when measured in the computational basis. Consequently, statements about a qubit being “definite” or “in superposition” are meaningful only relative to a chosen measurement basis.

2.3 Global and relative phase

Multiplying a state by a common phase factor does not change any measurement prediction. If θ is real, then |ψ⟩ and

represent the same physical state for all measurement purposes. This is called global phase.

By contrast, the phase between components is physically relevant. The states

and

have a relative phase difference of π. Although their computational-basis probabilities are identical, a Hadamard operation maps |ψ₁⟩ to |0⟩ and |ψ₂⟩ to |1⟩. Relative phase is therefore a resource for interference.

3. Measurement and the Born Rule

Suppose a normalized state is measured in an orthonormal basis {|u₀⟩, |u₁⟩}. The probability of obtaining the result associated with |uᵢ⟩ is

The quantity ⟨uᵢ|ψ⟩ is the amplitude for that outcome. The probabilities sum to one because the state is normalized and the basis is complete.

For computational-basis measurement,

and

Measurement is not merely the revelation of a pre-existing classical value. It is an interaction between a quantum system and a measurement apparatus. In the ideal projective-measurement model, after outcome 0 the state becomes |0⟩; after outcome 1 it becomes |1⟩. Repeating the same measurement immediately gives the same result, assuming no intervening disturbance.

Worked Example 1: Normalization and measurement

Consider

The amplitudes are α = 3/5 and β = 4i/5. Since |4i/5|² = 16/25,

The state is normalized. A computational-basis measurement gives

and

The phase i on the second amplitude does not affect this particular measurement, but it can affect later interference.

4. The Bloch-Sphere Representation

Every pure single-qubit state can be written, up to global phase, as

where 0 ≤ θ ≤ π is the polar angle and 0 ≤ φ < 2π is the azimuthal angle. The associated point on the unit Bloch sphere has Cartesian coordinates

The north pole represents |0⟩, and the south pole represents |1⟩. Points on the equator represent equal-magnitude superpositions. For example, |+⟩ lies at (1, 0, 0), while |−⟩ lies at (−1, 0, 0). The states (|0⟩ + i|1⟩)/√2 and (|0⟩ − i|1⟩)/√2 lie at (0, 1, 0) and (0, −1, 0), respectively.

The Bloch sphere is a complete representation of pure single-qubit states, apart from the physically irrelevant global phase. It provides geometric intuition for rotations caused by quantum gates. A general mixed state is represented not by the surface but by a point inside the sphere. Pure states have Bloch-vector length 1; mixed states have length less than 1.

Figure: A Bloch sphere showing the computational-basis poles, phase-basis equator and an arbitrary pure state.

The Bloch representation is powerful but limited. It is exact for one qubit, whereas no comparably simple three-dimensional sphere represents an arbitrary multi-qubit state. Also, a point inside the sphere represents a mixed state and cannot be interpreted as a pure superposition with a smaller amplitude vector.

5. Quantum Gates as State Transformations

A closed quantum system evolves through a unitary operator U. A unitary operator satisfies

where U† is the conjugate transpose and I is the identity matrix. Unitary evolution preserves normalization and is reversible.

Important one-qubit gates include the Pauli operators and the Hadamard gate:

The X gate exchanges |0⟩ and |1⟩. The Z gate leaves |0⟩ unchanged and changes the phase of |1⟩. The Hadamard gate maps

and, because H is its own inverse, H|+⟩ = |0⟩ and H|−⟩ = |1⟩.

These transformations explain why relative phase matters. Applying H to the two states |+⟩ and |−⟩ converts an invisible computational-basis phase difference into different deterministic measurement outcomes.

6. Multi-Qubit State Representation

6.1 Tensor products

A system of n qubits has a state space of dimension 2ⁿ. The joint state of two qubits is written using the tensor product, denoted ⊗. For basis states,

The ordering convention must be stated. In the common convention, |q₁q₀⟩ means q₁ is the first or most significant qubit and q₀ is the second or least significant qubit.

The computational basis for two qubits is

and a general pure state is

with normalization

In vector form, using the stated basis order,

For n qubits, there are 2ⁿ computational-basis states and 2ⁿ complex amplitudes, subject to normalization and an irrelevant global phase. This exponential growth is central to quantum computing, but it does not mean that all amplitudes can be directly read out from one measurement.

6.2 Product and separable states

A two-qubit state is a product state if it can be written as

for single-qubit states |u⟩ and |v⟩. A pure state that can be written in this way is called separable. If it cannot, it is entangled.

Let

and

Then

Therefore, if the coefficient vector is (a, b, c, d), a necessary and sufficient condition for separability of a pure two-qubit state is

This condition follows because a product state has a = αγ, b = αδ, c = βγ and d = βδ, giving ad − bc = αβγδ − αβδγ = 0. Conversely, a zero determinant means the coefficient matrix [[a, b], [c, d]] has rank one and can be factored into two vectors.

6.3 Entangled states

The Bell state

is not separable because a = 1/√2, d = 1/√2, and b = c = 0, so ad − bc = 1/2 ≠ 0. If both qubits are measured in the computational basis, the results are 00 or 11, each with probability 1/2. The individual outcomes are random, but they are perfectly correlated.

Correlation alone does not always imply entanglement. The separable state

also describes a joint system, but its measurement probabilities factor into independent single-qubit probabilities. Entanglement is a structural property of the quantum state, not merely the observation that two recorded data streams happen to agree.

7. Worked Multi-Qubit Examples

Worked Example 2: Constructing a product state

Let

and

Then

A measurement in the computational basis gives 01 with probability 1/2 and 11 with probability 1/2. The first qubit is random, while the second qubit is certainly 1. The state is separable because it was explicitly constructed as a tensor product.

Worked Example 3: Testing separability

Consider

The coefficients are a = 1/2, b = 1/2, c = 1/2 and d = −1/2. Hence

Since this is nonzero, the state is entangled. Notice that all four computational-basis outcomes have probability 1/4. Equal marginal probabilities do not establish separability; the relative signs also matter.

8. Case Study: Diagnosing a Two-Qubit Preparation

Background

A laboratory team is testing a two-qubit processor. State tomography, after calibration and reconstruction, reports the idealized state

The team wants to determine whether it has prepared two independent superposition states or a genuinely correlated state.

Problem

A computational-basis measurement produces approximately 50% 00 and 50% 11, with almost no 01 or 10 results. Does this pattern prove that each qubit independently occupies the state |+⟩?

Analysis

If the two qubits independently occupied |+⟩, their joint state would be

This state predicts all four outcomes with probability 1/4. It therefore cannot explain the observed absence of 01 and 10. The reconstructed state instead has coefficient determinant 1/2, which is nonzero. It is entangled.

The team should also measure in other bases. Measurements in the computational basis reveal strong correlation, but computational-basis data alone do not fully characterize phase relationships. A complete diagnostic uses several measurement settings, estimates uncertainties and checks for device errors such as readout bias, relaxation and gate infidelity.

Approach and outcome

The team compares observed frequencies with predictions for candidate states, performs measurements in the X and Z bases, and uses tomography or a more efficient verification protocol. Agreement with the Bell-state correlations supports the conclusion that an entangled state was produced. In a real experiment, statistical confidence and error mitigation are essential; a finite data sample can never establish an exact state with certainty.

Learning points

1. A joint distribution is not enough to identify the full quantum state.
2. Product-state predictions factor into predictions for the component systems.
3. Relative phase can be revealed by changing the measurement basis.
4. Experimental claims require calibration, uncertainty analysis and noise assessment.

9. Practical Applications and Professional Implications

Qubit representation is used throughout quantum technology. In quantum algorithms, state vectors allow developers to simulate small circuits and verify gate sequences. In quantum communication, phase and entanglement determine whether information can be encoded, transmitted or verified securely. In quantum sensing, small changes in phase can encode information about time, magnetic fields or acceleration.

For software and hardware professionals, conventions matter. A circuit description must specify qubit ordering, basis conventions, endianness, measurement mapping and whether amplitudes are stored as complex numbers. Confusing |01⟩ with |10⟩ can produce apparently inexplicable results even when the underlying algorithm is correct.

State-vector simulation also has a scaling limitation. An n-qubit state vector contains 2ⁿ complex amplitudes. For example, 30 qubits require over one billion complex amplitude entries. Memory requirements, precision and numerical error therefore become important in classical simulation. Quantum hardware does not provide direct access to all amplitudes; algorithms must extract useful information through carefully designed measurements.

10. Limitations, Noise and Mixed States

The pure-state formalism describes maximum knowledge of an isolated quantum system. Real systems interact with their environment and may suffer decoherence, thermal noise, leakage and measurement error. Such situations are described by a density operator ρ rather than a single ket. A valid density operator satisfies

A pure state has ρ = |ψ⟩⟨ψ|. A probabilistic mixture of states |ψᵢ⟩ with classical probabilities pᵢ is

where pᵢ ≥ 0 and ∑ᵢ pᵢ = 1. Different mixtures can produce the same measurement statistics in one basis but differ in another. Consequently, state preparation, tomography and benchmarking must be designed across suitable measurement settings.

For this diploma-level foundation, the distinction is essential: a pure superposition is a coherent state vector, whereas a mixed state reflects classical uncertainty, entanglement with an unobserved environment, or both. Treating a mixture as though it were a pure ket leads to incorrect predictions about interference.

11. Best Practices for Representing Qubits

• State the basis order before writing a multi-qubit vector.
• Check normalization after every state-preparation calculation.
• Distinguish amplitudes from probabilities; probabilities are squared magnitudes.
• Track relative phase even when it is invisible in the current measurement basis.
• Ignore only global phase, not phase differences between components.
• Use tensor products explicitly when constructing composite systems.
• Test pure two-qubit separability with ad − bc = 0 when the coefficient order is |00⟩, |01⟩, |10⟩, |11⟩.
• Treat experimental frequencies as estimates with statistical and systematic uncertainty.
• Do not infer entanglement from correlation alone without considering alternative separable explanations and measurement errors.
• Document software conventions, qubit ordering and gate-matrix definitions.

Chapter Summary

A qubit is a normalized vector in a two-dimensional complex Hilbert space. Its general pure state is α|0⟩ + β|1⟩, with |α|² + |β|² = 1. The Born rule converts amplitudes into measurement probabilities. Dirac notation provides a compact language for states, inner products and transformations, while state vectors support numerical calculation.

Global phase has no observable consequence, but relative phase affects interference. The Bloch sphere represents every pure single-qubit state geometrically and provides an intuitive view of one-qubit transformations. Multi-qubit systems are constructed with tensor products and require 2ⁿ computational-basis amplitudes for n qubits. A pure two-qubit state is separable when it factors into single-qubit states; otherwise it is entangled. The determinant test ad − bc = 0 provides a convenient criterion under the standard two-qubit basis ordering. These representations support quantum-circuit design, simulation, communication, sensing and experimental diagnosis, while real devices require density operators and explicit treatment of noise.

Key Terms

Qubit: A two-level quantum information system.

Computational basis: The basis consisting of |0⟩ and |1⟩.

Ket: A column-vector representation of a quantum state, written |ψ⟩.

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

State vector: A vector containing the complex amplitudes of a pure state.

Amplitude: A complex coefficient whose squared magnitude gives a measurement probability.

Normalization: The requirement that the total probability equals one.

Born rule: The rule P(i) = |⟨uᵢ|ψ⟩|² for measurement probabilities.

Superposition: A linear combination of basis states.

Global phase: A common phase factor that does not alter physical predictions.

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

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

Unitary operator: A reversible norm-preserving quantum transformation.

Tensor product: The operation used to combine state spaces of separate systems.

Product state: A composite state expressible as a tensor product of subsystem states.

Separable state: A pure composite state that is a product state.

Entanglement: A nonseparable relationship between quantum subsystems.

Bell state: A maximally entangled two-qubit state such as (|00⟩ + |11⟩)/√2.

Density operator: A mathematical representation of pure or mixed quantum states.

Decoherence: Loss of quantum coherence through interaction with the environment.

Review Questions

Short Answer Questions

1. What condition must the amplitudes α and β satisfy for a single-qubit state to be valid?
2. What is the difference between global phase and relative phase?
3. State the Born-rule probability of obtaining |1⟩ from α|0⟩ + β|1⟩.
4. What does the tensor product accomplish in a multi-qubit system?
5. Why is the Bell state (|00⟩ + |11⟩)/√2 not separable?

Long Answer Questions

1. Explain the state-vector and Bloch-sphere representations of a pure qubit and discuss the limitations of the Bloch sphere.
2. Describe projective measurement in the computational basis, including probabilities and post-measurement states.
3. Explain how the Hadamard gate demonstrates the physical significance of relative phase.
4. Derive the separability condition ad − bc = 0 for a pure two-qubit state.
5. Compare pure-state and density-operator descriptions, including situations in which each is appropriate.

Analytical / Application Questions

1. A qubit is prepared as |ψ⟩ = (1/√3)|0⟩ + √(2/3)e^{iπ/4}|1⟩. Verify normalization and calculate computational-basis measurement probabilities. Explain whether the phase affects those immediate probabilities.
2. Determine whether |χ⟩ = (2|00⟩ + |01⟩ + 2|10⟩ + |11⟩)/3 is separable. Show the normalization check and apply the determinant criterion.
3. A laboratory observes only 00 and 11 in a two-qubit measurement. Propose at least two additional measurements or analyses needed before claiming that the source produces an entangled Bell state.

Multiple Choice Questions with Answers

Question 1

Which condition makes the single-qubit state |ψ⟩ = α|0⟩ + β|1⟩ normalized?

A. |α| + |β| = 1

B. |α|² + |β|² = 1

C. α + β = 1

D. αβ = 1

Correct Answer: B. |α|² + |β|² = 1

Question 2

For the state |ψ⟩ = (3/5)|0⟩ + (4i/5)|1⟩, what is the probability of measuring 1 in the computational basis?

A. 0.16

B. 0.36

C. 0.64

D. 1.00

Correct Answer: C. 0.64

Question 3

Which statement about global phase is correct?

A. It changes all measurement probabilities.

B. It is physically observable in every experiment.

C. It has no effect on physical measurement predictions.

D. It must always be equal to zero.

Correct Answer: C. It has no effect on physical measurement predictions.

Question 4

Where is the state |+⟩ = (|0⟩ + |1⟩)/√2 located on the Bloch sphere?

A. At the north pole

B. At the south pole

C. On the positive x-axis

D. At the centre only

Correct Answer: C. On the positive x-axis

Question 5

How many computational-basis states does an n-qubit system have?

A. n

B. 2n

C. n²

D. 2ⁿ

Correct Answer: D. 2ⁿ

Question 6

Which operation combines the states of two separate quantum systems?

A. Ordinary addition only

B. Tensor product

C. Classical sorting

D. Normalization alone

Correct Answer: B. Tensor product

Question 7

For a two-qubit state with coefficients a, b, c and d in the order |00⟩, |01⟩, |10⟩, |11⟩, which condition indicates separability?

A. ad + bc = 1

B. a + b + c + d = 0

C. ad − bc = 0

D. |a|² = |d|²

Correct Answer: C. ad − bc = 0

Question 8

What are the computational-basis probabilities for the Bell state (|00⟩ + |11⟩)/√2?

A. 00 and 11 each have probability 1/2

B. All four outcomes have probability 1/4

C. Only 01 occurs

D. 00 has probability 1 and 11 has probability 0

Correct Answer: A. 00 and 11 each have probability 1/2

Question 9

Which gate maps |+⟩ to |0⟩?

A. X

B. Z

C. H

D. I only after measurement

Correct Answer: C. H

Question 10

A point strictly inside the Bloch sphere generally represents what kind of single-qubit state?

A. A normalized computational basis state only

B. A mixed state

C. An impossible state

D. A state with non-unitary evolution only

Correct Answer: B. A mixed state