Quantum Information Theory

Chapter 3
Quantum Dynamics, Measurement and Information Limits

Diploma in Quantum Computing
Quantum Information Theory   •   Diploma

Quantum Dynamics, Measurement and Information Limits

Introduction

Quantum information theory studies how information is represented, transformed, measured and restricted by the laws of quantum mechanics. Unlike classical information, quantum information is encoded in physical systems whose states may exist in superposition and may be correlated through entanglement. These features enable quantum algorithms, quantum communication and quantum sensing, but they also impose strict limits on what can be learned, copied or deleted.

This chapter develops four connected ideas. First, closed quantum systems evolve through unitary operators. Second, measurement converts quantum states into classical outcomes according to probability rules. Third, uncertainty and complementarity limit the simultaneous predictability of incompatible observables. Finally, no-cloning, no-deleting and no-broadcasting theorems show that quantum information cannot be manipulated as if it were an ordinary classical data file.

Learning Objectives

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

  1. Describe unitary evolution and calculate the state of a simple quantum system after an operation.
  2. Explain projective measurement and determine measurement probabilities and post-measurement states.
  3. Distinguish projective measurements from positive operator-valued measures (POVMs).
  4. Interpret expectation values, variance and uncertainty relations.
  5. Explain complementarity and its operational consequences.
  6. State and justify the no-cloning, no-deleting and no-broadcasting principles.
  7. Apply these ideas to quantum communication, error analysis and measurement design.

1. States and Dynamics of a Quantum System

A pure state of a qubit is represented by a unit vector in a two-dimensional complex Hilbert space. In the computational basis, a general state is

|ψ⟩ = α|0⟩ + β|1⟩

where α and β are complex probability amplitudes. Normalisation requires

|α|² + |β|² = 1

The quantities |α|² and |β|² are probabilities only when the qubit is measured in the computational basis. Amplitudes themselves are generally complex and can interfere constructively or destructively.

A more general mixed state is described by a density operator ρ. It satisfies

ρ = ρ†, ρ ≥ 0, Tr(ρ) = 1

Here ρ† is the conjugate transpose, ρ ≥ 0 means that all eigenvalues are non-negative, and Tr denotes the trace. A pure state has

ρ = |ψ⟩⟨ψ|

and obeys Tr(ρ²) = 1. A mixed state has Tr(ρ²) < 1. Density operators are essential when a system is prepared with classical uncertainty or is entangled with an inaccessible environment.

Unitary Evolution

For an isolated quantum system, time evolution is represented by a unitary operator U:

|ψ'⟩ = U|ψ⟩

A unitary operator satisfies

U†U = UU† = I

where I is the identity operator. For a density operator, the corresponding transformation is

ρ' = UρU†

Unitarity preserves inner products, norms and therefore total probability. If |ψ⟩ and |φ⟩ are two states, then

⟨ψ'|φ'⟩ = ⟨ψ|U†U|φ⟩ = ⟨ψ|φ⟩

This preservation is important: reversible quantum evolution cannot make two distinct pure states identical.

For a time-independent Hamiltonian H, the continuous-time evolution over an interval t is

U(t) = e^(−iHt/ℏ)

where ℏ is the reduced Planck constant, and H has units of energy. The operator exponential is defined through its power series or spectral decomposition. The state obeys the Schrödinger equation

iℏ d|ψ(t)⟩/dt = H|ψ(t)⟩

These equations apply to closed systems. Real devices are open systems and may experience noise, relaxation and decoherence. Their evolution is then generally described by a quantum channel rather than by one unitary operator acting on the system alone.

Worked Example: A Hadamard Operation

The Hadamard gate is

H = 1/√2 [[1, 1], [1, −1]]

Its action on |0⟩ is

H|0⟩ = (|0⟩ + |1⟩)/√2

If this state is measured in the computational basis, the probabilities are

P(0) = |1/√2|² = 1/2,
P(1) = |1/√2|² = 1/2

The gate has not created a classical 50–50 choice merely by randomising a bit. It has created a coherent superposition. Subsequent operations can make the two amplitudes interfere, which is impossible for an ordinary classical probability mixture.

2. Measurement and the Born Rule

Measurement connects a quantum description to an observed classical result. The simplest measurement is a projective measurement with projectors {Pₘ}. They satisfy

PₘPₙ = δₘₙPₘ,
Pₘ† = Pₘ,
∑ₘ Pₘ = I

where δₘₙ is 1 when m = n and 0 otherwise. If the system is in state ρ, the probability of outcome m is given by the Born rule:

p(m) = Tr(Pₘρ)

For a pure state |ψ⟩ this becomes

p(m) = ⟨ψ|Pₘ|ψ⟩

If Pₘ projects onto a one-dimensional state |m⟩, then p(m) = |⟨m|ψ⟩|².

For an ideal projective measurement with outcome m, the conditional post-measurement state is

ρₘ = PₘρPₘ / Tr(Pₘρ)

provided p(m) ≠ 0. This is the Lüders update rule. It is a conditional state: it describes the system after learning the outcome m. If the outcome is ignored, the state becomes

ρ' = ∑ₘ PₘρPₘ

which generally loses coherence between different measurement subspaces.

Measurement in Different Bases

A qubit can be measured in the computational, or Z, basis with projectors

P₀ = |0⟩⟨0|, P₁ = |1⟩⟨1|

It can also be measured in the X basis, whose states are

|+⟩ = (|0⟩ + |1⟩)/√2,
|−⟩ = (|0⟩ − |1⟩)/√2

The same physical state may produce different probability distributions in different bases. For example, |0⟩ gives a certain outcome in the Z basis, but equal probabilities for + and − in the X basis. Measurement statistics therefore depend jointly on the prepared state and the chosen observable.

Expectation Values and Variance

An observable A is represented by a Hermitian operator, A = A†. Its expectation value in state ρ is

⟨A⟩ = Tr(ρA)

The variance is

(∆A)² = ⟨A²⟩ − ⟨A⟩²

where ∆A is the standard deviation. Expectation values describe long-run averages over repeated preparations, not necessarily the result of any single measurement.

3. POVMs: Generalised Measurements

Projective measurements are not the most general measurements permitted by quantum theory. A POVM is a collection of positive operators {Eₘ} satisfying

Eₘ ≥ 0,
∑ₘ Eₘ = I

The probability of outcome m is

p(m) = Tr(ρEₘ)

The operators Eₘ are called effects. Unlike projectors, they need not be orthogonal or idempotent. A POVM specifies outcome probabilities, but by itself does not always specify the physical state transformation after an outcome. For that, measurement operators Mₘ are required, with

Eₘ = Mₘ†Mₘ

and

ρₘ = MₘρMₘ† / Tr(MₘρMₘ†)

The completeness condition is

∑ₘ Mₘ†Mₘ = I

POVMs are useful when detectors are noisy, when outcomes are not perfectly distinguishable, or when an ancilla is measured after interacting with the system. Every POVM can be implemented, in principle, by attaching an auxiliary system, applying a unitary interaction and performing a projective measurement on a larger space. This is commonly associated with Naimark dilation.

Example: Unambiguous and Minimum-Error Discrimination

Suppose a device must distinguish two non-orthogonal states. Orthogonal states can be perfectly distinguished by a projective measurement, but non-orthogonal states cannot. A carefully designed POVM may either minimise the average error probability or sometimes identify a state without error while allowing an inconclusive outcome. The trade-off is fundamental: greater confidence in conclusive outcomes usually requires more inconclusive results.

For two known pure states |ψ₀⟩ and |ψ₁⟩ prepared with equal prior probabilities, the minimum-error probability is

Perror = 1/2(1 − √(1 − |⟨ψ₀|ψ₁⟩|²))

This Helstrom bound applies to the binary, equal-prior, minimum-error problem. It illustrates that state overlap directly limits discrimination performance.

4. Uncertainty and Complementarity

Two observables may be individually well-defined but impossible to predict simultaneously with arbitrary precision. For observables A and B, the Robertson uncertainty relation is

∆A∆B ≥ 1/2 |⟨[A,B]⟩|

where [A,B] = AB − BA is the commutator. The relation assumes a normalised state with finite variances. A stronger Schrödinger form is

(∆A)²(∆B)² ≥ (1/4)|⟨[A,B]⟩|² + (1/4)|⟨{∆A,∆B}⟩|²

where ∆A = A − ⟨A⟩I and {X,Y} = XY + YX is the anticommutator. The simpler Robertson relation is often sufficient for introductory analysis.

For the Pauli X and Z operators, [X,Z] = −2iY. A state that is an eigenstate of Z has zero Z uncertainty but nonzero X uncertainty. This is not merely a limitation of an imperfect instrument. It follows from the noncommutativity of the observables.

Entropic Uncertainty

For measurements in bases X and Z, an information-oriented uncertainty relation is

H(X) + H(Z) ≥ −2 log₂ c

where H is the Shannon entropy of the measurement distribution and c = max₍ᵢ,ⱼ₎ |⟨xᵢ|zⱼ⟩|. For mutually unbiased qubit bases, c = 1/√2, so the lower bound is 1 bit. Entropic relations are useful in quantum cryptography because they quantify uncertainty as missing classical information rather than only as a physical variance.

Complementarity

Complementarity refers to the fact that different experimental arrangements reveal different, mutually exclusive aspects of a quantum system. Wave-like interference and particle-like path information are familiar examples. In a two-path interferometer, obtaining reliable which-path information destroys or reduces interference. The loss is not caused simply by a careless observer; information becomes correlated with the path and makes the alternatives distinguishable.

A practical distinction is useful:

  • Uncertainty concerns the spread of outcomes for observables, often expressed through variances or entropies.
  • Complementarity concerns the incompatibility of experimental questions and the trade-off between observing different aspects of a system.
  • Disturbance concerns the change caused by a measurement or interaction.

These ideas are related but are not identical. A measurement can be statistically uncertain without strongly disturbing a state, and a measurement can disturb a state even when its outcome is highly predictable.

5. Information Limits and Impossibility Theorems

Quantum information is not simply classical information stored in a smaller device. The linearity and reversibility of quantum mechanics impose special restrictions.

No-Cloning Theorem

The no-cloning theorem states that no physical operation can perfectly copy an arbitrary unknown quantum state. Suppose a unitary U and a fixed blank state |b⟩ could clone every state:

U(|ψ⟩|b⟩) = |ψ⟩|ψ⟩

For two states |ψ⟩ and |φ⟩, preservation of inner products would require

⟨ψ|φ⟩ = ⟨ψ|φ⟩²

This equation holds for all pairs only when the states are identical or orthogonal. Therefore arbitrary, generally non-orthogonal states cannot be cloned. Known states from a mutually orthogonal set can be copied because they can first be distinguished without error.

The theorem does not prohibit all copying of quantum information. It prohibits a universal device that copies every unknown state perfectly. Classical information encoded in orthogonal quantum states can be copied.

No-Deleting Theorem

The no-deleting theorem is a time-reversed analogue of no-cloning. Given two identical unknown states, no unitary operation can delete one copy while leaving the other unchanged and a fixed blank state:

U(|ψ⟩|ψ⟩) = |ψ⟩|b⟩

for every |ψ⟩. Inner-product preservation again rules out this transformation for arbitrary non-orthogonal states. Quantum information can be moved, transformed or transferred, but it cannot be universally erased from one copy while another remains through a reversible closed-system process.

This theorem should not be confused with ordinary reset operations. A physical system can be reset by coupling it to an environment, but the information and entropy are transferred to that environment rather than destroyed without a trace.

No-Broadcasting Theorem

Broadcasting is weaker than cloning. A channel broadcasts a set of states if the two output systems each have the same reduced state as the input, even if the joint output is not two identical copies. The no-broadcasting theorem states that a set of quantum states can be broadcast if and only if all states in the set commute.

Commuting density operators share an eigenbasis and can therefore be interpreted as classical probability distributions over that basis. Noncommuting states contain genuinely quantum information and cannot be broadcast. Thus the theorem identifies commutativity as the boundary between information that behaves classically and information that remains intrinsically quantum.

No-Communication and Measurement

Entanglement produces correlations that cannot be explained by local classical variables, but it cannot be used by itself to send a controllable message faster than light. Local measurement outcomes are random, and the statistics available to one party do not depend on which measurement another distant party chooses. Classical communication is required to compare results.

6. Worked Application: Measuring a Qubit with a POVM

Consider the state

|ψ⟩ = √3/2 |0⟩ + 1/2 |1⟩

The state is normalised because 3/4 + 1/4 = 1. A projective Z measurement gives

P(0) = 3/4,
P(1) = 1/4

Now consider a three-outcome POVM with effects

E₀ = 1/2 |0⟩⟨0|,
E₁ = 1/2 |1⟩⟨1|,
E_? = 1/2 I

These effects are positive and sum to I. The probabilities are

p(0) = ⟨ψ|E₀|ψ⟩ = 3/8,
p(1) = ⟨ψ|E₁|ψ⟩ = 1/8,
p(?) = ⟨ψ|E_?|ψ⟩ = 1/2

They sum to 1. The question-mark outcome is inconclusive. This example demonstrates that a generalised measurement can deliberately distribute information among conclusive and inconclusive outcomes. The POVM alone specifies the probabilities; a physical implementation must additionally specify measurement operators if the post-measurement state is required.

7. Case Study: Quantum Key Distribution and Information Limits

Background

In the BB84 protocol, a sender prepares qubits randomly in one of two incompatible bases, commonly the Z basis or the X basis. A receiver independently chooses a measurement basis. After transmission, they publicly compare bases but not the measured bit values, retaining only events where their bases agree.

Problem

An eavesdropper wishes to learn the key without being detected. If the eavesdropper does not know the preparation basis, a measurement in the wrong basis can disturb the state.

Analysis

Suppose the eavesdropper measures every qubit in a randomly chosen basis and resends the result. When the eavesdropper chooses the wrong basis, the measurement outcome is incompatible with the sender’s preparation. On later measurement by the receiver in the correct basis, the resent state produces a random result. Under the idealised intercept-resend attack, the eavesdropper uses the wrong basis with probability 1/2, and conditional on that event introduces an error with probability 1/2. Thus the expected error rate in the sifted key is

Perror = 1/2 × 1/2 = 1/4

or 25 percent, assuming ideal devices, independent choices and a full intercept-resend attack on every transmitted qubit.

Approach and Outcome

The legitimate users estimate the error rate by publicly comparing a sample of sifted bits. A significant error rate indicates noise or eavesdropping. In a complete protocol, error correction and privacy amplification are then used to produce a shorter but more secure key. The security reasoning relies on non-orthogonal-state discrimination, measurement disturbance and the impossibility of obtaining complete information without consequences.

Learning Points

  1. Measurement basis is part of the physical information-access strategy.
  2. Noncommuting measurements prevent an eavesdropper from learning both complementary descriptions perfectly.
  3. Security is statistical: practical systems must account for losses, detector imperfections and finite sample sizes.
  4. The idealised 25 percent figure is not a universal error rate; it applies to the stated attack and assumptions.

8. Practical Implementation and Limitations

Quantum devices are open systems. Unitary circuit descriptions remain valuable, but real experiments require calibration and noise models. Common limitations include:

  • State-preparation and measurement errors.
  • Relaxation, in which an excited state loses energy.
  • Dephasing, in which relative phase information is lost.
  • Crosstalk between nearby qubits.
  • Detector inefficiency and dark counts.
  • Readout bias and finite sampling error.

A useful experimental workflow is to repeat a preparation-measurement procedure many times, estimate frequencies, compare them with Born-rule predictions and report statistical confidence. For an outcome with true probability p observed over N independent trials, the empirical frequency has standard deviation approximately

σf = √(p(1 − p)/N)

This binomial approximation assumes independent identically distributed trials. It describes sampling uncertainty, not necessarily physical device noise.

Measurement design must also distinguish an ideal mathematical POVM from the instrument implementing it. Detector calibration, assignment matrices and tomography can be used to estimate how actual outcomes differ from ideal predictions. In professional settings, claims about quantum advantage or security should therefore specify the state-preparation model, measurement model, noise assumptions and statistical method.

9. Best Practices for Quantum Information Analysis

  1. State the basis and measurement apparatus explicitly; probabilities have no meaning without a measurement context.
  2. Check normalisation and positivity when manipulating states and density operators.
  3. Use unitary evolution only for closed-system evolution or as part of a larger system model.
  4. Separate the probability rule from the state-update rule.
  5. Identify whether a measurement is projective or a general POVM.
  6. Distinguish an unknown state from a known state selected from an orthogonal alphabet when discussing copying.
  7. State assumptions behind numerical error rates and security claims.
  8. Treat noise and finite data as engineering realities rather than defects in the quantum formalism.
  9. Do not interpret a single measurement outcome as an expectation value.
  10. Use tomography or calibrated benchmarking when experimental evidence is required.

Contemporary Developments

Modern quantum technologies use increasingly general measurement strategies. Adaptive measurements choose later measurement settings based on earlier outcomes. Weak and continuous measurements monitor a system while disturbing it gradually rather than producing one immediate projective result. Error-correcting codes use repeated syndrome measurements to obtain information about errors without directly measuring the encoded logical state. Quantum machine learning and quantum sensing also depend on carefully selecting measurements that extract useful information while preserving relevant coherence.

These developments do not remove the fundamental limits discussed in this chapter. They optimise the trade-offs imposed by quantum mechanics. Better algorithms and hardware can improve estimation, control and error tolerance, but they cannot provide a universal copier for unknown quantum states or broadcast noncommuting information.

Chapter Summary

Closed quantum systems evolve reversibly through unitary operators that preserve inner products and total probability. Pure states are described by state vectors, while mixed states are represented by density operators. The Born rule gives measurement probabilities, and projective measurements provide a basic model in which projectors determine both outcomes and state updates.

POVMs generalise projective measurements and are essential for realistic detectors, state discrimination and quantum communication. Observables that do not commute obey uncertainty relations, while complementarity expresses the incompatibility of different experimental descriptions. Measurement can reveal information while disturbing the system.

The no-cloning theorem rules out perfect copying of arbitrary unknown states; the no-deleting theorem rules out universal deletion of one copy while retaining another; and the no-broadcasting theorem shows that only mutually commuting sets of states can be broadcast. These principles support the security of quantum communication and define the boundary between classical and quantum information processing.

Key Terms

Quantum state: Mathematical description of the condition of a quantum system.

Qubit: A two-level quantum system represented by a vector in a two-dimensional Hilbert space.

Density operator: Operator ρ representing a pure or mixed quantum state.

Unitary operator: Operator U satisfying U†U = I and describing reversible closed-system evolution.

Hamiltonian: Hermitian operator representing the energy and generating continuous-time evolution.

Observable: Hermitian operator associated with a measurable physical quantity.

Born rule: Rule assigning p(m) = Tr(ρEₘ) to a measurement outcome.

Projector: Hermitian idempotent operator representing a subspace measurement outcome.

Projective measurement: Measurement described by mutually orthogonal projectors summing to identity.

POVM: Positive operator-valued measure whose effects sum to identity.

Effect: Positive POVM element that determines an outcome probability.

Expectation value: Long-run average ⟨A⟩ = Tr(ρA) of an observable.

Variance: Statistical spread (∆A)² = ⟨A²⟩ − ⟨A⟩².

Commutator: [A,B] = AB − BA, measuring operator noncommutativity.

Uncertainty relation: Bound limiting the simultaneous precision of incompatible observables.

Complementarity: Principle that certain experimental descriptions cannot be jointly obtained completely.

No-cloning theorem: Impossibility of perfectly copying an arbitrary unknown quantum state.

No-deleting theorem: Impossibility of universally deleting one of two identical unknown copies.

No-broadcasting theorem: Impossibility of broadcasting noncommuting quantum states.

Quantum channel: Completely positive, trace-preserving map describing general physical evolution.

Review Questions

Short Answer Questions

  1. What condition makes an operator unitary, and why is unitarity important for probability conservation?
  2. State the Born rule for a POVM and define each symbol.
  3. What is the difference between a projective measurement and a POVM?
  4. What does it mean for two observables to be complementary?
  5. Why does the no-cloning theorem not prohibit copying classical bits encoded in orthogonal quantum states?

Long Answer Questions

  1. Explain pure and mixed states using state vectors and density operators. Include the conditions a valid density operator must satisfy.
  2. Derive the inner-product argument that rules out a universal quantum cloning machine.
  3. Explain projective measurement, conditional state update and the non-selective post-measurement state.
  4. Discuss uncertainty, complementarity and measurement disturbance, explaining why these concepts should not be treated as identical.
  5. Compare no-cloning, no-deleting and no-broadcasting, including the role of orthogonality and commutativity.

Analytical / Application Questions

  1. A qubit is prepared as |ψ⟩ = √3/2|0⟩ + 1/2|1⟩. Calculate the probabilities of Z-basis outcomes and explain the post-measurement state for each outcome.
  2. A communication system uses the intercept-resend attack described in the BB84 case study. Calculate the expected sifted-key error rate and state the assumptions required for the calculation.
  3. A proposed device claims to copy every unknown qubit perfectly. Design a conceptual test using two non-orthogonal input states and explain which property of quantum evolution invalidates the claim.

Multiple Choice Questions with Answers

Question 1

Which equation describes unitary evolution of a density operator?

A. ρ' = UρU†

B. ρ' = UρU

C. ρ' = ρ + U

D. ρ' = U†ρU†

Correct Answer: A. ρ' = UρU†

Question 2

For a POVM with effect Eₘ, the probability of outcome m for state ρ is:

A. Tr(Eₘ + ρ)

B. Tr(ρEₘ)

C. Tr(ρ)Tr(Eₘ) only

D. EₘρEₘ

Correct Answer: B. Tr(ρEₘ)

Question 3

What is required of the effects in a POVM?

A. They must all be mutually orthogonal projectors

B. They must all have trace zero

C. They must be positive and sum to the identity

D. They must all be unitary

Correct Answer: C. They must be positive and sum to the identity

Question 4

A qubit in state |0⟩ is measured in the X basis {|+⟩, |−⟩}. What are the outcome probabilities?

A. 1 and 0

B. 3/4 and 1/4

C. 1/2 and 1/2

D. 0 and 0

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

Question 5

The Robertson uncertainty relation contains which operator expression?

A. The commutator [A,B]

B. The determinant AB

C. The trace A + B only

D. The inverse A⁻¹B⁻¹

Correct Answer: A. The commutator [A,B]

Question 6

Why can arbitrary unknown quantum states not be perfectly cloned?

A. Quantum states cannot be measured at all

B. Unitary evolution preserves inner products

C. All quantum states are orthogonal

D. Classical probabilities are always zero

Correct Answer: B. Unitary evolution preserves inner products

Question 7

The no-broadcasting theorem permits broadcasting a set of states when the states:

A. Are all non-orthogonal

B. Have different dimensions

C. Commute with one another

D. Have unknown phases only

Correct Answer: C. Commute with one another

Question 8

In the illustrative BB84 intercept-resend attack, the wrong-basis probability is 1/2 and the conditional error probability is 1/2. What is the resulting error rate?

A. 10%

B. 25%

C. 50%

D. 75%

Correct Answer: B. 25%

Question 9

For a pure state density operator ρ = |ψ⟩⟨ψ|, which equality holds?

A. Tr(ρ²) = 0

B. Tr(ρ²) = 1

C. Tr(ρ²) = 2

D. Tr(ρ²) is always negative

Correct Answer: B. Tr(ρ²) = 1

Question 10

Which statement best distinguishes a POVM from its physical implementation?

A. A POVM gives outcome probabilities, while measurement operators also specify state transformations

B. A POVM is only valid for classical systems

C. Measurement operators cannot be positive

D. A POVM never has an inconclusive outcome

Correct Answer: A. A POVM gives outcome probabilities, while measurement operators also specify state transformations
Quantum Information Theory
Chapter 3 of 15