1. Mathematical and Quantum-Mechanical Foundations

Introduction and Context

Quantum computing is built upon a mathematical language that differs substantially from the binary abstractions of classical computation. Classical information is commonly represented by definite values, whereas quantum information is encoded in physical states described by vectors in complex vector spaces. Computation is represented by transformations of those vectors, and observation is represented by measurement operations that produce probabilistic classical outcomes.

This distinction is not merely terminological. Superposition, interference, and entanglement arise from the structure of complex Hilbert spaces and the rules of quantum mechanics. A quantum algorithm cannot be understood adequately by treating a qubit as an ordinary probabilistic bit. A probabilistic bit is in one classical state selected according to a probability distribution; a qubit is represented by a coherent complex-amplitude vector whose components can interfere before measurement.

The foundations developed in this chapter support the study of quantum circuits, quantum algorithms, quantum simulation, error correction, quantum communication, and advanced computational models. They also establish an important methodological principle: quantum theory predicts measurement statistics, not necessarily definite underlying trajectories. Consequently, a researcher must distinguish carefully among a state vector, an operator, a probability distribution, and an observed experimental result.

Learning Objectives

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

1. Explain vector spaces, inner products, norms, bases, and Hilbert spaces as they apply to quantum information.

2. Represent pure quantum states using complex amplitudes and apply normalization conditions.

3. Distinguish linear operators, unitary operators, Hermitian observables, and projectors.

4. Construct composite systems using tensor products and interpret separable and entangled states.

5. Explain superposition, interference, measurement, and the probabilistic Born rule.

6. Represent mixed states using density matrices and calculate expectation values and measurement probabilities.

7. Relate mathematical descriptions to quantum circuits, simulations, and professional applications.

8. Evaluate foundational assumptions, limitations, ethical issues, and emerging research directions.

The Mathematical Setting: Vector Spaces and Hilbert Spaces

A vector space consists of elements that can be added and multiplied by scalars while satisfying the usual algebraic rules. In quantum mechanics, the scalars are generally complex numbers. A complex number has the form

z = a + ib,

where a and b are real numbers and i² = −1. Its complex conjugate is z = a − ib, and its magnitude is |z| = √(zz).

A finite-dimensional complex vector space can be described by a basis. For a qubit, the computational basis is conventionally written as |0⟩ and |1⟩. A general pure qubit state is

|ψ⟩ = α|0⟩ + β|1⟩,

where α and β are complex amplitudes. Physical states must be normalized:

|α|² + |β|² = 1.

The coefficients are amplitudes, not direct probabilities. Their squared magnitudes determine measurement probabilities, while their relative phase can affect interference.

A Hilbert space is a complete inner-product vector space. In finite-dimensional quantum computing, completeness introduces no practical difficulty, but the Hilbert-space formulation is essential for infinite-dimensional systems such as harmonic oscillators, electromagnetic modes, and quantum fields. The inner product of vectors |φ⟩ and |ψ⟩ is denoted ⟨φ|ψ⟩. It is conjugate-linear in one argument and linear in the other, with

⟨φ|ψ⟩ = ⟨ψ|φ⟩*.

The norm of |ψ⟩ is ||ψ|| = √⟨ψ|ψ⟩. Orthogonal vectors satisfy ⟨φ|ψ⟩ = 0. An orthonormal basis satisfies ⟨i|j⟩ = δij, where δij is one when i = j and zero otherwise. Dirac notation is especially useful because it separates abstract states from their coordinate representations.

The dual vector associated with |ψ⟩ is written ⟨ψ|. If |ψ⟩ is represented as a column vector, then ⟨ψ| is its conjugate transpose. For example,

|ψ⟩ = [ α ; β ] implies ⟨ψ| = [ α β ].

The outer product |ψ⟩⟨φ| is an operator, whereas the inner product ⟨φ|ψ⟩ is a scalar. This distinction becomes central when constructing projectors and density matrices.

Qubits, Bloch Geometry, and Complex Amplitudes

The state space of one qubit is the two-dimensional complex Hilbert space C². A normalized state can be written as

|ψ⟩ = α|0⟩ + β|1⟩.

An overall phase has no observable consequence: |ψ⟩ and eiγ|ψ⟩ represent the same physical state. Relative phase does matter. The states

(|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2

have identical computational-basis probabilities, but they behave differently under later transformations because their relative phases differ.

Every pure qubit state can be expressed, up to a global phase, as

|ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩.

This parameterization gives the Bloch sphere representation. The polar angle θ determines the relative magnitudes of the amplitudes, while φ determines their relative phase. The north and south poles correspond to |0⟩ and |1⟩. Points on the equator represent equal-magnitude superpositions with varying phase.

The Bloch sphere is a visualization of one-qubit pure states, not a replacement for the Hilbert-space formalism. Mixed states occupy the interior of the sphere, while pure states lie on its surface. A useful warning is that the Bloch sphere does not generalize directly to an equally intuitive geometric picture for many-qubit systems.

Linear Operators and Quantum Transformations

An operator A maps vectors to vectors. In a selected basis it is represented by a matrix. The action of an operator on a state is written A|ψ⟩. Operators can be added and multiplied, but matrix multiplication is generally noncommutative: AB need not equal BA.

A unitary operator U satisfies

U†U = UU† = I,

where U† is the conjugate transpose and I is the identity operator. Unitary transformations preserve inner products, norms, and therefore total probability. Closed-system quantum evolution is modeled by unitary operators. Common single-qubit gates include

X = [0 1; 1 0], Z = [1 0; 0 −1],

and the Hadamard operator

H = (1/√2)[1 1; 1 −1].

The X operator exchanges |0⟩ and |1⟩. The Z operator changes the phase of |1⟩. The Hadamard operator creates or removes particular superpositions. For example,

H|0⟩ = (|0⟩ + |1⟩)/√2,

whereas

H[(|0⟩ + |1⟩)/√2] = |0⟩.

This second relation illustrates interference: amplitudes combine algebraically, and negative or complex phases can cause cancellation.

A Hermitian operator A satisfies A† = A. Hermitian operators represent observables because their eigenvalues are real. If

A|a⟩ = a|a⟩,

then a is a possible measurement value and |a⟩ is the associated eigenstate. Projectors are operators P satisfying P² = P and P† = P. A projective measurement is described by a collection of orthogonal projectors whose sum is the identity.

The expectation value of an observable A in state |ψ⟩ is

⟨A⟩ψ = ⟨ψ|A|ψ⟩.

It is the average value obtained from many identically prepared experiments, not necessarily the result of any single measurement.

Measurement and the Born Rule

Suppose a measurement has projectors Pm associated with possible outcomes m. For a normalized state |ψ⟩, the probability of outcome m is

p(m) = ⟨ψ|Pm|ψ⟩.

For a measurement in the computational basis, P0 = |0⟩⟨0| and P1 = |1⟩⟨1|. Thus, for |ψ⟩ = α|0⟩ + β|1⟩,

p(0) = |α|², p(1) = |β|².

After an ideal projective measurement yielding m, the state is updated to

Pm|ψ⟩ / √p(m),

provided p(m) is nonzero. This state-update rule is sometimes called wave-function collapse. In operational terms, it describes the state assigned after conditioning on a measurement result. Interpretations of quantum mechanics differ over the ontological meaning of collapse, but the mathematical prediction is shared by standard formulations.

Measurement is basis-dependent. A state that is uncertain in the computational basis may be definite in another basis. For example, |+⟩ = (|0⟩ + |1⟩)/√2 is an eigenstate of X with eigenvalue +1. Measuring |+⟩ in the X basis produces a definite result, whereas measuring it in the computational basis produces 0 and 1 with equal probability.

More general measurements are represented by positive operator-valued measures, or POVMs. A POVM is a set of positive operators {Em} satisfying Σm Em = I, with probabilities p(m) = ⟨ψ|Em|ψ⟩. POVMs are important in quantum communication, state discrimination, noisy experiments, and quantum sensing because they describe measurements beyond ideal orthogonal projections.

Superposition and Interference

Superposition is the closure of a quantum state space under linear combination. It does not mean that a system possesses two ordinary classical values simultaneously in the same everyday sense. Rather, the state contains amplitudes for possible measurement outcomes, and those amplitudes evolve coherently.

Interference results from the addition of amplitudes before probabilities are calculated. If two computational paths contribute amplitudes a and b, the resulting probability is |a + b|², which includes the cross terms ab + ab. These terms may increase or reduce the probability. Classical probabilities would instead be added without such phase-sensitive cross terms.

Quantum algorithms exploit this mechanism by arranging transformations so that amplitudes of desirable states reinforce and amplitudes of undesirable states cancel. However, superposition alone does not guarantee computational advantage. The algorithm must preserve coherence and engineer useful interference under realistic resource constraints.

Composite Systems and Tensor Products

The state space of a composite system is the tensor product of the component spaces. If system A has state space HA and system B has state space HB, then the joint system has state space HA ⊗ HB. For two qubits, the computational basis is

|00⟩, |01⟩, |10⟩, |11⟩.

A general pure two-qubit state is

|ψ⟩ = α00|00⟩ + α01|01⟩ + α10|10⟩ + α11|11⟩,

with Σ|αij|² = 1.

If |a⟩ = a0|0⟩ + a1|1⟩ and |b⟩ = b0|0⟩ + b1|1⟩, their product state is

|a⟩⊗|b⟩ = a0b0|00⟩ + a0b1|01⟩ + a1b0|10⟩ + a1b1|11⟩.

Operators also combine through tensor products. A gate acting on the first qubit alone is represented by U ⊗ I. Tensor-product dimensions grow exponentially: n qubits require a state vector with 2ⁿ complex amplitudes in a general pure-state simulation. This exponential scaling is both a source of quantum computational capacity and a major challenge for classical simulation.

Entanglement and Nonlocal Correlations

A bipartite state is separable if it can be written as |a⟩⊗|b⟩. If no such factorization exists, the state is entangled. The Bell state

|Φ+⟩ = (|00⟩ + |11⟩)/√2

is entangled. Measuring the first qubit in the computational basis yields 0 or 1 with equal probability; conditional on that result, the second qubit is found in the same value. Neither individual qubit has a definite computational-basis value before measurement, yet the joint correlations are exact.

Entanglement is not equivalent to ordinary classical correlation. Classical correlation can be represented by a probability distribution over pre-existing alternatives. Entangled states contain correlations that cannot generally be reproduced by local hidden-variable models satisfying the assumptions tested by Bell-type inequalities.

Entanglement is a resource in teleportation, quantum error correction, distributed quantum computing, and several quantum algorithms. It is not, however, a means of transmitting classical information instantaneously. Operational constraints, including the need for classical communication in teleportation, preserve compatibility with relativistic causality.

Density Matrices and Mixed States

A pure state may be represented by the density operator

ρ = |ψ⟩⟨ψ|.

For |ψ⟩ = α|0⟩ + β|1⟩,

ρ = [ |α|² αβ ; αβ |β|² ].

A density matrix is Hermitian, positive semidefinite, and has unit trace. A state is pure if Tr(ρ²) = 1. It is mixed if Tr(ρ²) < 1. A mixed state represents classical uncertainty about preparation, quantum entanglement with an unobserved environment, or both.

For an ensemble of pure states |ψk⟩ prepared with probabilities pk,

ρ = Σk pk|ψk⟩⟨ψk|.

Different ensembles can produce the same density matrix. Therefore, the density matrix captures all operationally accessible information about the system, rather than a unique story about how the state was prepared.

Measurement probabilities and expectation values become

p(m) = Tr(ρPm),

and

⟨A⟩ = Tr(ρA).

A unitary transformation acts as ρ′ = UρU†. For a composite state ρAB, the state of subsystem A is obtained by the partial trace:

ρA = TrB(ρAB).

For the Bell state |Φ+⟩, the joint state is pure, but each subsystem separately has density matrix I/2. This fact demonstrates that a subsystem can be maximally mixed even when the total system is in a pure state.

Open-system evolution may be described by quantum channels. A channel is a completely positive, trace-preserving map, often written in Kraus form:

E(ρ) = Σk EkρEk†,

where Σk Ek†Ek = I. Noise processes such as bit flips, phase flips, amplitude damping, and depolarization can be modeled in this framework. The density-matrix formalism is therefore indispensable for hardware characterization, error modeling, and realistic simulation.

A Conceptual Framework for Quantum Computation

The mathematical workflow of a quantum computation can be summarized as follows:

1. Specify the initial state, often |0⟩⊗n or a density matrix ρ.

2. Apply a sequence of unitary operators or noisy quantum channels.

3. Represent the intended measurement using projectors or POVM elements.

4. Calculate outcome probabilities using amplitudes or traces.

5. Repeat the experiment to estimate a probability distribution from finite samples.

6. Interpret the distribution in relation to the computational problem.

This framework separates four issues often confused by beginners: state preparation, coherent transformation, observation, and statistical inference. A simulator may calculate exact probabilities, whereas a quantum processor returns samples affected by finite-shot variation and hardware noise.

Illustrative Case Study: Entanglement-Based State Analysis

Background

Consider an illustrative research laboratory investigating whether a two-qubit processor can prepare and analyze a Bell state. The objective is educational and methodological rather than a report of a specific real experiment.

Situation and Problem

The team prepares |00⟩, applies a Hadamard gate to the first qubit, and then applies a controlled-NOT gate with the first qubit as control and the second as target. Ideally, the resulting state is

(H⊗I)|00⟩ = (|00⟩ + |10⟩)/√2,

followed by CNOT, producing

|Φ+⟩ = (|00⟩ + |11⟩)/√2.

The team must determine whether the observed data support entanglement rather than merely classical correlation.

Analysis

Computational-basis measurements should yield approximately 00 and 11 with equal frequencies, while 01 and 10 should be strongly suppressed. This observation establishes correlation but does not by itself establish entanglement. A classical mixture

ρclassical = 1/2|00⟩⟨00| + 1/2|11⟩⟨11|

produces the same computational-basis distribution.

The distinction lies in coherence. The Bell state contains off-diagonal terms |00⟩⟨11| and |11⟩⟨00| in its density matrix. The classical mixture does not. Measurements in additional bases, together with suitable correlation statistics, can test whether these coherence terms are present. Experimental analysis must account for readout errors, gate errors, drift, and finite sample sizes.

Approach and Outcome

The team performs measurements in several bases, estimates expectation values, and reconstructs either selected correlators or a two-qubit density matrix through quantum state tomography. If the measured correlations agree with the Bell-state predictions within uncertainty and violate an appropriately designed classical bound under the relevant assumptions, the evidence supports entanglement. If not, the result may reflect noise, imperfect calibration, or an inadequate measurement design.

Learning Points

The case illustrates four principles. First, a state vector predicts probabilities only after a measurement basis is specified. Second, a single probability distribution may be compatible with multiple physical explanations. Third, density matrices expose coherence and mixedness that are invisible in one measurement basis. Fourth, claims about entanglement require statistical analysis and careful control of experimental assumptions.

Applications in Research and Industry

These foundations support quantum simulation of molecular Hamiltonians, materials, and many-body systems; quantum cryptographic protocols; quantum sensing; optimization heuristics; and error-corrected computation. In software engineering, tensor representations and circuit simulators are used to validate algorithms before hardware execution. In hardware development, operators and density matrices model calibration procedures, crosstalk, decoherence, and measurement error.

Researchers also use linear algebra in quantum machine learning, where parameterized unitary circuits generate feature maps or variational models. However, the presence of quantum notation does not automatically imply a computational advantage. Claims must be evaluated through complexity analysis, data-loading costs, training stability, noise sensitivity, and comparisons with strong classical baselines.

For organizational decision-makers, the relevant question is not simply whether a system uses qubits. It is whether the physical resources, error rates, connectivity, control electronics, software stack, and algorithmic structure jointly support a valuable task. A mathematically correct circuit may still be operationally unsuitable because of depth, sampling overhead, or noise.

Challenges, Limitations, and Common Errors

A frequent error is to interpret amplitudes as probabilities. Amplitudes may be negative or complex; probabilities are nonnegative real numbers obtained by squaring magnitudes or applying the density-matrix rule. Another error is to assume that measurement merely reveals a pre-existing classical value. In general, measurement changes the state and depends on the chosen basis.

It is also incorrect to identify every superposition with entanglement. Superposition concerns linear combinations within a state space; entanglement concerns the impossibility of factoring a composite state into subsystem states. Likewise, correlation in one basis is insufficient evidence of entanglement.

Practical limitations include decoherence, imperfect gates, measurement noise, leakage outside the intended computational subspace, limited connectivity, and the exponential classical memory required for generic state-vector simulation. Density-matrix simulation is even more demanding because an n-qubit density matrix has 4ⁿ complex entries before exploiting structure.

Numerical simulations introduce additional issues: floating-point rounding, truncation, sampling error, and inaccurate noise models. Reproducible research therefore requires documented initial states, gate conventions, basis ordering, random seeds where appropriate, calibration information, and uncertainty estimates.

Ethical, Governance, and Professional Considerations

Quantum technologies may influence cybersecurity, finance, medicine, defense, and scientific discovery. Researchers should avoid overstating near-term capabilities, particularly claims that a prototype has solved a useful problem merely because it demonstrates a quantum effect. Security claims must distinguish theoretical protocol security from implementation security, including side channels and faulty random sources.

Responsible professional practice includes transparent reporting of benchmark definitions, hardware conditions, error mitigation methods, failed experiments, and classical comparison methods. Quantum computing education and research should also recognize unequal access to specialized hardware and avoid presenting speculative commercial forecasts as established evidence.

Current Developments and Emerging Directions

Current research increasingly focuses on noisy intermediate-scale devices, error mitigation, fault-tolerant architectures, and logical qubits encoded across many physical qubits. The density-matrix and quantum-channel formalisms are central to these efforts because realistic devices are open systems rather than perfectly isolated mathematical objects.

Hybrid quantum–classical algorithms use classical optimization around parameterized quantum circuits. Their performance depends on trainability, noise, measurement cost, and problem structure. Quantum simulation is also expanding through tensor-network methods, stabilizer techniques, Monte Carlo approaches, and specialized hardware emulators. These methods exploit structure to avoid representing every amplitude explicitly, although no single simulation technique is efficient for all quantum states.

A continuing research challenge is to identify computational tasks for which quantum resources provide demonstrable, scalable, and practically meaningful benefit. Addressing this challenge requires mathematical rigor, experimental discipline, and fair comparison with classical methods.

Best Practices for Mathematical and Quantum Modeling

A sound workflow begins with explicit notation and dimensional checks. Researchers should verify normalization after state preparation, confirm that proposed gates are unitary when modeling closed-system evolution, and use density matrices when uncertainty or environmental coupling matters. Tensor-product ordering should be stated clearly, since inconsistent qubit ordering can produce apparently mysterious errors.

Measurement claims should be expressed probabilistically and supported by repeated trials. When comparing theory with hardware, one should separate ideal predictions, noisy simulations, raw observations, and mitigated estimates. Finally, every claimed advantage should be tied to a defined metric such as runtime, sample complexity, energy use, accuracy, or scalability.

Chapter Summary

Quantum computing is formulated in complex Hilbert spaces. Pure states are normalized vectors, while mixed states are represented by density matrices. Inner products define orthogonality and probabilities; unitary operators describe reversible closed-system evolution; Hermitian operators describe observables; and projectors or POVMs describe measurements. The Born rule converts amplitudes or density matrices into outcome probabilities.

Tensor products describe composite systems and reveal the exponential growth of general quantum-state representations. Entanglement occurs when a composite state cannot be factored into subsystem states, while interference arises from coherent addition of complex amplitudes. Density matrices extend the formalism to statistical mixtures, subsystems, and noisy open-system dynamics. These foundations support quantum algorithms, simulation, hardware analysis, communication, sensing, and error correction. Their responsible use requires attention to noise, statistical evidence, scalability, reproducibility, and realistic comparisons with classical computation.

Key Terms

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

Qubit: A two-level quantum system represented by a normalized vector in C².

Complex amplitude: A complex coefficient whose squared magnitude contributes to a measurement probability.

Inner product: A conjugate-linear, linear mapping that defines overlap, orthogonality, and norm.

Orthonormal basis: A set of mutually orthogonal unit vectors spanning a vector space.

Global phase: An overall complex phase that does not affect observable predictions.

Relative phase: The phase difference between amplitudes that can affect interference.

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

Hermitian operator: An operator equal to its conjugate transpose and representing an observable.

Projector: An operator P satisfying P² = P, commonly used to represent a measurement outcome.

Born rule: The rule assigning probabilities from amplitudes or density operators.

Superposition: A linear combination of allowed quantum states.

Interference: Phase-sensitive combination of amplitudes that reinforces or cancels probabilities.

Tensor product: The mathematical operation constructing the state space of a composite system.

Entanglement: A nonseparable property of a composite quantum state.

Density matrix: A positive, unit-trace operator representing pure or mixed states.

Mixed state: A state represented by a density matrix with Tr(ρ²) less than one.

Partial trace: An operation that obtains a subsystem state from a composite density matrix.

POVM: A set of positive measurement operators summing to the identity.

Quantum channel: A completely positive, trace-preserving map describing general quantum evolution.

Review Questions

Short Answer Questions

1. Why are quantum amplitudes not themselves probabilities?

2. What condition must a qubit state satisfy to be normalized?

3. What is the difference between a unitary operator and a Hermitian operator?

4. Define entanglement in terms of factorization.

5. What information does the partial trace provide?

Long Answer Questions

1. Explain the role of Hilbert spaces, inner products, and orthonormal bases in quantum-state representation.

2. Derive the computational-basis measurement probabilities for a general qubit and explain the role of relative phase.

3. Describe how tensor products represent composite systems and why their dimensions create classical simulation challenges.

4. Compare pure states, mixed states, and density matrices, including their mathematical properties.

5. Explain superposition and interference using the action of the Hadamard operator.

Analytical / Application Questions

1. A qubit is prepared as (|0⟩ + i|1⟩)/√2. Determine its computational-basis probabilities and explain whether the phase i is observable in that measurement basis.

2. A two-qubit experiment produces only 00 and 11 in equal proportions. Explain why this evidence alone does not establish entanglement and propose a suitable follow-up analysis.

3. A noisy quantum processor is modeled by a density matrix rather than a state vector. Explain what information this choice captures and identify two practical sources of mixedness or noise.

22. Multiple Choice Questions with Answers

Question 1

A qubit is in the state α|0⟩ + β|1⟩. Which condition is required for normalization?

A. α + β = 1

B. |α|² + |β|² = 1

C. |α + β|² = 0

D. αβ = 1

Correct Answer: B. |α|² + |β|² = 1

Question 2

What is the principal physical significance of a global phase multiplying a quantum state?

A. It changes all measurement probabilities

B. It changes the dimension of the state space

C. It has no observable consequence by itself

D. It converts a pure state into a mixed state

Correct Answer: C. It has no observable consequence by itself

Question 3

Which property defines a unitary operator U?

A. U = U†

B. U² = U

C. U†U = I

D. Tr(U) = 1

Correct Answer: C. U†U = I

Question 4

For a computational-basis measurement of |ψ⟩ = α|0⟩ + β|1⟩, what is the probability of obtaining 1?

A. β

B. β² without regard to phase

C. |β|²

D. |α|²

Correct Answer: C. |β|²

Question 5

Which state is entangled?

A. |0⟩⊗|1⟩

B. (|0⟩ + |1⟩)⊗|0⟩/√2

C. (|00⟩ + |11⟩)/√2

D. |+⟩⊗|−⟩

Correct Answer: C. (|00⟩ + |11⟩)/√2

Question 6

Why can the states (|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2 behave differently under later gates?

A. They have different normalization constants

B. Their relative phases differ

C. One is necessarily mixed

D. They belong to different Hilbert-space dimensions

Correct Answer: B. Their relative phases differ

Question 7

Which expression represents the expectation value of observable A for density matrix ρ?

A. Tr(ρ + A)

B. Tr(ρA)

C. ρAρ

D. det(ρA)

Correct Answer: B. Tr(ρA)

Question 8

A density matrix represents a pure state when which condition holds?

A. Tr(ρ²) = 0

B. Tr(ρ²) = 1

C. Tr(ρ) = 0

D. det(ρ) = 1

Correct Answer: B. Tr(ρ²) = 1

Question 9

What does the partial trace of ρAB over subsystem B produce?

A. A unitary operator on B

B. The joint state with B removed physically

C. The reduced state ρA

D. The measurement outcome of B

Correct Answer: C. The reduced state ρA

Question 10

Why is observing only equal frequencies of 00 and 11 insufficient to prove that a two-qubit state is entangled?

A. Those outcomes violate normalization

B. A classical mixture can produce the same computational-basis distribution

C. Entangled states cannot produce 00

D. Measurement probabilities do not depend on the state

Correct Answer: B. A classical mixture can produce the same computational-basis distribution