Qubits and Multi-Qubit Systems

Introduction

A qubit is the fundamental unit of quantum information. Like a classical bit, it is associated with two computational-basis states, usually written |0⟩ and |1⟩. Unlike a classical bit, however, a qubit may occupy a coherent superposition of these basis states. This difference is not merely a larger choice of symbols: it changes how information is represented, manipulated, measured, and correlated.

A single qubit provides the foundation for quantum algorithms, quantum communication, sensing, and simulation. Real quantum processors, however, require many qubits. The state space of a multi-qubit system grows exponentially with the number of qubits, and the resulting systems may exhibit entanglement—correlations that cannot be explained by assigning independent states to the individual qubits.

This chapter develops the mathematical and physical language needed to describe single-qubit and multi-qubit systems. It also distinguishes pure and mixed states, explains separability and entanglement, and shows how quantum correlations differ from ordinary classical correlations.

Learning Objectives

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

1. Represent a qubit using ket notation and column vectors.
2. Calculate measurement probabilities from a qubit state.
3. Interpret a qubit geometrically using the Bloch sphere.
4. Distinguish pure states from mixed states using state vectors and density matrices.
5. Construct composite systems using tensor products.
6. Determine whether a two-qubit state is separable or entangled.
7. Explain quantum correlations through measurement examples.
8. Relate these concepts to the design and interpretation of quantum circuits.

1. The State of a Single Qubit

The computational basis for one qubit is

A general pure qubit state is

where α and β are complex probability amplitudes. The normalization condition is

The quantities |α|² and |β|² are real probabilities. If the qubit is measured in the computational basis, the Born rule gives

The normalization condition ensures that the probabilities sum to one. It is important that α and β are amplitudes, not probabilities. Amplitudes may be negative or complex, whereas measurement probabilities must be non-negative real numbers.

The relative phase between α and β can influence later interference effects. A global phase does not affect measurement predictions: the states |ψ⟩ and e^{iγ}|ψ⟩ describe the same physical state for any real γ. A relative phase, by contrast, is physically meaningful. For example, the states

have identical computational-basis measurement probabilities but behave differently under subsequent quantum operations.

Worked Example 1: Measurement Probabilities

Consider

The state is normalized because

Therefore,

Over many identical preparations, approximately 75% of measurements produce 0 and 25% produce 1. A single measurement produces only one outcome; it does not reveal both amplitudes simultaneously.

2. The Bloch-Sphere Representation

Every pure single-qubit state can be written, apart from an irrelevant global phase, as

where 0 ≤ θ ≤ π is a polar angle and 0 ≤ φ < 2π is an azimuthal angle. The corresponding point lies on the surface of a unit sphere. The north pole represents |0⟩, and the south pole represents |1⟩.

The equatorial states include

at φ = 0 and

at φ = π. The states (|0⟩ + i|1⟩)/√2 and (|0⟩ − i|1⟩)/√2 lie at opposite points on the y-axis.

Figure: The Bloch sphere maps pure single-qubit states to points on its surface; the vector from the centre identifies the state.

The Bloch sphere is a visualization, not a claim that the qubit is literally a small spinning object. Quantum gates act as rotations or combinations of rotations on this representation. For example, a bit-flip operation rotates the state through π radians about the x-axis, while a phase operation changes the azimuthal angle.

A general single-qubit density matrix can be written as

where I is the 2 × 2 identity matrix, σₓ, σᵧ, and σ_z are the Pauli matrices, and r = (rₓ, rᵧ, r_z) is the Bloch vector. Physical states satisfy |r| ≤ 1. Pure states have |r| = 1 and lie on the sphere’s surface. Mixed states have |r| < 1 and lie inside the sphere. The completely mixed state has r = 0 and is located at the centre.

3. Pure and Mixed States

A pure state represents maximal knowledge of the quantum state, subject to the limits imposed by quantum mechanics. It may be represented by a normalized state vector |ψ⟩. Its density matrix is

A mixed state represents a classical statistical ensemble of possible pure states. If state |ψᵢ⟩ occurs with probability pᵢ, then

The density matrix is Hermitian, has trace one, and is positive semidefinite. For a density matrix ρ, the probability of obtaining an outcome associated with projector P is

where Tr denotes the trace. For a computational-basis measurement, the probabilities are the diagonal entries of ρ.

A useful test for a finite-dimensional state is purity:

For a pure state, Tr(ρ²) = 1. For a mixed state, Tr(ρ²) < 1. This criterion assumes that ρ is a valid normalized density matrix.

Example: Same Measurement Statistics, Different States

The pure state |+⟩ has density matrix

The equally weighted mixture of |0⟩ and |1⟩ has density matrix

Both produce 50% zeros and 50% ones in a computational-basis measurement. They are nevertheless different physical states. The off-diagonal entries of ρ₊ encode coherence, whereas ρ_mix has no such coherence. A measurement in the |+⟩, |−⟩ basis distinguishes them: |+⟩ is always obtained from ρ₊, while the mixture gives each outcome with probability 1/2.

This distinction is central in practice. Noise, uncontrolled environmental interactions, imperfect preparation, and uncertainty about preparation procedures may all lead to mixed states. A statistical mixture is not the same as a coherent superposition.

4. Composite Systems and Tensor Products

To describe two or more qubits, individual state spaces are combined using the tensor product. If qubit A is in |a⟩ and qubit B is in |b⟩, the joint state is

The computational basis for two qubits is

Using the ordering |00⟩, |01⟩, |10⟩, |11⟩, these vectors are

A general pure two-qubit state is

with

For n qubits, the computational basis contains 2ⁿ basis states, and a general pure state requires 2ⁿ amplitudes subject to normalization and an irrelevant global phase. This exponential state-space growth is a major source of quantum-computational power and a major challenge for classical simulation.

Tensor-Product Calculation

Suppose

Then

The first qubit is in a superposition, while the second is definitely 1. The joint probabilities are P(01) = 1/2 and P(11) = 1/2.

For density matrices, the corresponding composition is

when the systems are independently prepared. More general joint states need not factor in this way.

5. Separability and Entanglement

A pure two-qubit state is separable, or a product state, if it can be written as

If no such factorization exists, the state is entangled. For example,

is entangled. It cannot be expressed as a product of two single-qubit states.

A convenient test for a two-qubit pure state is to arrange its amplitudes in a coefficient matrix

The state is separable exactly when det(C) = 0. Thus, for |Φ⁺⟩, the coefficient matrix is

and its determinant is 1/2, not zero; the state is therefore entangled. This determinant test applies to pure two-qubit states, not arbitrary mixed states.

The four Bell states are maximally entangled two-qubit states:

In |Φ⁺⟩, measuring both qubits in the computational basis always gives equal results: 00 or 11. Each result occurs with probability 1/2. Yet neither individual qubit has a definite value before measurement. The reduced state of either qubit is maximally mixed:

Here Tr_B is the partial trace over subsystem B. The joint state is pure, while each subsystem considered alone is mixed—a characteristic feature of entanglement.

6. Quantum Correlations and Classical Correlations

Correlation does not automatically mean entanglement. Consider the mixed state

Measurements in the computational basis are perfectly correlated, just as for |Φ⁺⟩. However, ρ_classical contains no coherent terms linking |00⟩ and |11⟩. It is a classical mixture of product states and is therefore separable as a mixed state.

The distinction becomes visible in other measurement bases. The Bell state |Φ⁺⟩ exhibits phase-sensitive correlations that the classical mixture does not. Entanglement can produce correlations that cannot be reproduced by any classical model based on pre-existing local values. Bell inequalities provide experimental tests of this distinction.

For diploma-level work, the essential conceptual separation is:

Entanglement does not permit controllable faster-than-light communication. Although measurement outcomes can be strongly correlated, each local outcome is intrinsically random, and classical communication is required to compare results.

7. Measuring Parts of a Multi-Qubit System

When only one subsystem is measured or observed, the appropriate description is obtained through a partial trace. For a joint density matrix ρ_AB, the reduced state of A is

and similarly ρ_B = Tr_A(ρ_AB). The reduced state contains all measurement predictions for subsystem A alone.

For |Φ⁺⟩, the reduced state is I/2. Consequently, measuring A alone gives 0 or 1 with equal probability. If B is then measured in the same computational basis, the results agree. The correlation exists in the joint state even though neither local subsystem has a definite pure state.

This idea is operationally important. Quantum processors often involve many qubits, but calibration and readout may concern only selected qubits. Reduced density matrices allow engineers and researchers to analyse local behaviour without representing every measurement explicitly.

8. Creating Entanglement with Quantum Gates

A common circuit for creating a Bell state begins with |00⟩. Apply a Hadamard gate H to the first qubit:

Then apply a controlled-NOT gate with the first qubit as control and the second as target. The component |00⟩ remains |00⟩, while |10⟩ becomes |11⟩:

The controlled operation creates entanglement because its action cannot be represented as independent operations on the two qubits. In hardware, the exact implementation may use cross-resonance, controlled-phase interactions, or another native entangling mechanism, but the abstract circuit has the same logical purpose.

A useful practical principle is that single-qubit gates alone cannot create entanglement from an initially separable state. At least one genuinely multi-qubit interaction is required.

9. Applied Case: Diagnosing Correlations in a Two-Qubit Processor

Illustrative case. A laboratory team prepares two superconducting qubits and reports that measurements produce 00 and 11 with approximately equal frequency. The team claims that the qubits have been entangled.

Background and Problem

Equal outcomes suggest correlation, but they do not prove entanglement. The observed data could result from the Bell state |Φ⁺⟩ or from the classical mixture ρ_classical.

Analysis

The team measures only in the computational basis. This measurement cannot distinguish the two candidate states because both produce 00 and 11 with probability 1/2. The team therefore performs measurements in additional bases, including the X basis associated with |+⟩ and |−⟩. For |Φ⁺⟩, the outcomes show characteristic correlations in multiple complementary bases. The classical mixture does not reproduce all of these correlations.

The team also performs state tomography, reconstructing an estimated density matrix. Significant off-diagonal coherence between |00⟩ and |11⟩ supports the Bell-state interpretation. Statistical uncertainty, readout errors, gate errors, and decoherence must be included in the analysis; a small apparent correlation may arise from experimental bias.

Approach and Outcome

The correct conclusion is not based on one histogram. It requires measurements in suitable bases, calibration of readout errors, estimation of uncertainty, and—when appropriate—a Bell-test or entanglement witness. The experiment finds correlations consistent with |Φ⁺⟩, but the strength of the claim depends on the quality of the reconstructed state and the statistical confidence of the test.

Learning Points

1. Correlation in one basis does not establish entanglement.
2. Coherence and phase-sensitive measurements are essential.
3. Density matrices provide a practical language for noisy laboratory states.
4. Experimental claims must account for noise, finite samples, and measurement imperfections.

10. Challenges, Limitations, and Best Practices

Quantum states are difficult to characterize because direct observation generally disturbs the state. State tomography becomes increasingly expensive as the number of qubits grows: a general density matrix for n qubits has dimensions 2ⁿ × 2ⁿ. Consequently, large systems are often analysed using reduced models, randomized measurements, classical shadows, or carefully selected observables.

Decoherence reduces useful quantum coherence through interactions with the environment. It can transform an intended pure state into a mixed state and can destroy entanglement. Noise models should therefore be stated explicitly when interpreting experiments or simulations.

Good practice includes:

• maintaining normalized state vectors and trace-one density matrices;
• distinguishing amplitudes from probabilities;
• specifying basis ordering for tensor products;
• distinguishing global phase from relative phase;
• testing correlations in more than one measurement basis;
• stating whether a separability test applies to pure or mixed states;
• reporting uncertainty and experimental assumptions;
• avoiding the claim that entanglement enables faster-than-light signalling.

Quantum software frameworks represent multi-qubit states according to a defined qubit-order convention. An incorrect assumption about whether the first tensor factor is the most- or least-significant bit can produce apparently incorrect results. Professional practice therefore requires checking documentation, basis ordering, and measurement conventions before interpreting output.

Contemporary Developments

Current quantum technologies increasingly use error mitigation, entanglement benchmarking, and device-independent ideas to assess the quality of quantum states. Variational algorithms and quantum machine-learning proposals also depend on preparing and controlling multi-qubit states, although practical advantage is not guaranteed merely by using entanglement. In quantum communication, entangled pairs support teleportation protocols and foundational tests, but classical communication remains necessary in teleportation and no-signalling constraints remain valid.

For emerging professionals, the most transferable skill is not memorising named states. It is the ability to move correctly among state-vector notation, density matrices, tensor products, measurement probabilities, and experimentally meaningful claims.

Chapter Summary

A qubit is a normalized superposition α|0⟩ + β|1⟩, with computational-basis probabilities |α|² and |β|². The Bloch sphere represents pure single-qubit states on its surface and mixed states inside its volume. Pure states can be represented by state vectors, whereas mixed states require density matrices. The density-matrix formalism also supports measurement calculations and descriptions of noisy systems.

Composite systems are formed with tensor products. Two qubits have four computational-basis states, and n qubits have 2ⁿ basis states. A pure composite state is separable if it factors into individual states; otherwise it is entangled. Entangled states such as the Bell states exhibit correlations that cannot be reduced to independent local states. Classical mixtures may also be correlated, so correlation in one measurement basis is not sufficient evidence of entanglement. Partial traces describe subsystems, and multi-qubit gates such as controlled-NOT operations can create entanglement from separable inputs.

Key Terms

Qubit: The basic unit of quantum information.

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

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

Normalization: The requirement that the total probability equals one.

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

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

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

Pure state: A quantum state represented by a state vector or rank-one density matrix.

Mixed state: A statistical ensemble of possible pure states.

Density matrix: An operator used to represent pure or mixed quantum states.

Trace: The sum of diagonal entries of a matrix, used in normalization and probability calculations.

Tensor product: The mathematical operation used to combine quantum systems.

Composite system: A system formed from two or more subsystems.

Separable state: A state that can be expressed as a product of subsystem states, or a mixture of such products.

Entanglement: A nonseparable form of quantum correlation.

Bell state: One of four maximally entangled two-qubit states.

Reduced state: The state of a subsystem obtained using a partial trace.

Partial trace: An operation that removes one subsystem from a joint density matrix.

Decoherence: Loss of quantum coherence through environmental interaction.

Quantum correlation: A statistical relationship between outcomes of quantum measurements.

Review Questions

5 Short Answer Questions

1. What is the normalization condition for a single-qubit state α|0⟩ + β|1⟩?
2. What physical information is represented by |α|²?
3. How does a pure state appear on the Bloch sphere?
4. What mathematical operation combines the state spaces of two qubits?
5. Why does correlation in one measurement basis not prove entanglement?

5 Long Answer Questions

1. Explain the Bloch-sphere representation and distinguish global phase from relative phase.
2. Compare pure and mixed states using state vectors, density matrices, and measurement predictions.
3. Construct the computational basis of a two-qubit system and explain the exponential growth of the state space.
4. Explain separability and entanglement using a Bell state and a classical correlated mixture.
5. Describe how a Hadamard gate followed by a controlled-NOT gate creates a Bell state from |00⟩.

3 Analytical / Application Questions

1. A two-qubit device produces only 00 and 11 in equal proportions. Design a measurement strategy that could distinguish a Bell state from a classical mixture.
2. Determine whether the pure state (|00⟩ + |01⟩ + |10⟩ + |11⟩)/2 is separable. Show the factorization or explain why none exists.
3. A researcher reconstructs a density matrix with Tr(ρ²) = 0.82. Interpret this result and state what additional information would be needed before diagnosing the source of the mixedness.

Multiple Choice Questions with Answers

Question 1

For the normalized state |ψ⟩ = (√3/2)|0⟩ + (1/2)|1⟩, what is the probability of measuring 1 in the computational basis?

A. 1/4

B. 1/2

C. 3/4

D. 1

Correct Answer: A. 1/4

Question 2

Which statement about the Bloch sphere is correct?

A. All mixed states lie on its surface.

B. Pure single-qubit states lie on its surface.

C. It represents only classical bits.

D. The centre represents a definite |0⟩ state.

Correct Answer: B. Pure single-qubit states lie on its surface.

Question 3

What is the density matrix of the completely mixed single-qubit state?

A. |0⟩⟨0|

B. |1⟩⟨1|

C. I/2

D. |+⟩⟨+|

Correct Answer: C. I/2

Question 4

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

A. n

B. n²

C. 2n

D. 2ⁿ

Correct Answer: D. 2ⁿ

Question 5

Which state is entangled?

A. |0⟩⊗|1⟩

B. (|0⟩ + |1⟩)⊗|0⟩

C. (|00⟩ + |11⟩)/√2

D. |1⟩⊗|1⟩

Correct Answer: C. (|00⟩ + |11⟩)/√2

Question 6

For a pure two-qubit state with coefficient matrix C, which condition indicates separability?

A. det(C) = 0

B. det(C) = 1

C. Tr(C) = 0

D. C must be diagonal

Correct Answer: A. det(C) = 0

Question 7

What does the partial trace over subsystem B produce from ρ_AB?

A. A measurement outcome for B

B. The reduced state ρ_A

C. The tensor product of A and B

D. The inverse of ρ_AB

Correct Answer: B. The reduced state ρ_A

Question 8

Which operation can create entanglement from an initially separable two-qubit state?

A. A global phase operation only

B. A measurement of one qubit only

C. A genuinely two-qubit controlled operation

D. Changing notation from vectors to matrices

Correct Answer: C. A genuinely two-qubit controlled operation

Question 9

The states |+⟩ and the equal mixture of |0⟩ and |1⟩ have the same computational-basis probabilities. How can they be distinguished?

A. They cannot be distinguished by any measurement.

B. By measuring in the |+⟩, |−⟩ basis.

C. Only by changing the normalization condition.

D. Only by applying a global phase.

Correct Answer: B. By measuring in the |+⟩, |−⟩ basis.

Question 10

For a valid density matrix, what does Tr(ρ²) = 1 indicate?

A. The state is maximally mixed.

B. The state is pure.

C. The state has trace zero.

D. The state is necessarily entangled.

Correct Answer: B. The state is pure.