Quantum Entropy and Information Measures

Introduction and context

Quantum information theory extends the ideas of uncertainty, information, correlation and distinguishability from classical probability to quantum states. These measures are essential in quantum communication, cryptography, error correction, sensing and computing. A quantum state is represented by a density operator ρ, which is a positive semidefinite operator satisfying Tr(ρ)=1. Pure states have the form ρ=|ψ⟩⟨ψ|, while mixed states represent classical uncertainty about a quantum preparation.

This chapter develops five central quantities:

• von Neumann entropy, measuring uncertainty in a quantum state;
• conditional entropy, measuring the remaining uncertainty about one system given another;
• quantum mutual information, measuring total correlation;
• quantum relative entropy, measuring distinguishability;
• fidelity, measuring closeness between states.

These quantities are linked by fundamental inequalities, including subadditivity, strong subadditivity and the data-processing inequality.

Learning objectives

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

1. Calculate and interpret von Neumann entropy for common density matrices.
2. Define quantum conditional entropy and mutual information.
3. Apply subadditivity and strong subadditivity.
4. Calculate relative entropy and fidelity, including important special cases.
5. Explain data processing and its significance for quantum channels.
6. Select suitable information measures for practical quantum-information tasks.

1. Density operators and spectra

The eigenvalues of a density operator are non-negative and sum to one. If

then the numbers λᵢ form a probability distribution over orthogonal eigenstates |i⟩. Consequently, many entropy calculations can be performed using the eigenvalues rather than the matrix entries.

For a qubit,

where I is the 2 × 2 identity matrix, σₓ, σᵧ and σ_z are Pauli matrices, and the Bloch vector r satisfies |r| ≤ 1. The eigenvalues are

A state is pure when |r| = 1 and maximally mixed when r = 0.

2. von Neumann entropy

The von Neumann entropy of a state ρ is

The logarithm base 2 measures information in bits. Using the spectral decomposition,

with the convention that 0 log₂0=0. Entropy is dimensionless when measured in bits.

Interpretation

• S(ρ)=0 if and only if ρ is pure.
• For a d-dimensional system, S(ρ)≤log₂d.
• The maximum entropy is reached by the maximally mixed state I/d.
• Entropy measures uncertainty in the state, not necessarily ignorance of an observer. A pure entangled state can have zero total entropy while its subsystems have nonzero entropy.

Worked example 1: a diagonal qubit

Consider

Its eigenvalues are 0.75 and 0.25. Therefore,

The state is mixed but not maximally mixed, since its entropy is below 1 bit.

Important special cases

For a pure state, S(|ψ⟩⟨ψ|)=0. For a maximally mixed qubit, ρ=I/2, so

For a classical-quantum state

the entropy is

where H(X) is the ordinary Shannon entropy of the classical distribution pₓ.

3. Composite systems and entropy relations

For a bipartite state ρ_AB, the reduced states are obtained by partial trace:

The joint entropy is S(AB) = S(ρ_AB), and the marginal entropies are S(A) = S(ρ_A) and S(B) = S(ρ_B).

Subadditivity

The subadditivity inequality states

Equality holds exactly when the state factorizes:

Thus, correlations cannot increase the joint uncertainty beyond the sum of separate uncertainties. Rearranging gives

which motivates quantum mutual information.

Strong subadditivity

For a tripartite state ρ_ABC,

Equivalently,

where conditional entropy is defined below. Additional conditioning cannot increase conditional entropy. Equality characterizes quantum Markov structure and is important in recovery-map theory.

4. Quantum conditional entropy

The conditional entropy of A given B is

Unlike classical conditional entropy, quantum conditional entropy can be negative. This occurs when A and B are entangled.

For a maximally entangled Bell state |Φ⁺⟩,

the joint state is pure, so S(AB) = 0. Each subsystem is maximally mixed, so S(B) = 1 bit. Hence

The negative value reflects quantum entanglement: possessing B gives a kind of quantum advantage concerning A that has no classical counterpart.

For a product state ρ_AB = ρ_A ⊗ ρ_B,

5. Quantum mutual information

Quantum mutual information is defined by

It can also be written as

Mutual information is always non-negative and equals zero if and only if ρ_AB = ρ_A ⊗ ρ_B. It measures total correlation, including both classical and quantum correlations. It is not, by itself, a measure solely of entanglement.

For a pure bipartite state |ψ⟩_AB, S(AB) = 0 and S(A) = S(B), so

Worked example 2: a classically correlated state

Let

The joint eigenvalues are 1/2 and 1/2, so S(AB) = 1 bit. Both reduced states equal I/2, giving S(A) = S(B) = 1 bit. Therefore,

The state contains one bit of shared correlation, but it is separable rather than entangled.

6. Quantum relative entropy

The quantum relative entropy of ρ with respect to σ is

It is defined when the support of ρ is contained in the support of σ. If this condition fails, D(ρ∥σ) is conventionally +∞. Relative entropy is not symmetric: generally D(ρ∥σ)≠D(σ∥ρ), and it is not a distance in the strict metric sense.

Its central properties are:

• D(ρ∥σ)≥0;
• D(ρ∥σ) = 0 if and only if ρ = σ;
• it is invariant under a common unitary transformation;
• it does not increase under a quantum channel.

A quantum channel is a completely positive trace-preserving map Λ. The data-processing inequality states

Noise, measurement and discarding systems cannot make two states more distinguishable.

If ρ and σ commute and have eigenvalue distributions pᵢ and qᵢ, then relative entropy reduces to classical Kullback–Leibler divergence:

Worked example 3: classical relative entropy

For p = (0.8, 0.2) and q = (0.5, 0.5),

The positive value indicates that p and q are different. The result is not obtained by simply subtracting probabilities, and reversing the arguments would produce a different value.

Relative entropy also connects to mutual information:

Thus, mutual information measures how distinguishable the joint state is from an uncorrelated state with the same marginals.

7. Fidelity

Fidelity quantifies how similar two quantum states are. A widely used definition is

This convention gives 0 ≤ F ≤ 1, with F = 1 if and only if ρ = σ. Some texts define the unsquared root fidelity; users must check the convention before comparing numerical results.

For pure states ρ = |ψ⟩⟨ψ| and σ = |φ⟩⟨φ|,

For a pure state and an arbitrary state,

Fidelity is symmetric, unlike relative entropy. It is useful for assessing state-preparation quality, gate performance, quantum teleportation output and error-correction recovery.

Worked example 4: pure-state fidelity

Let

The overlap is cos(θ), so

At θ = π/4, F = 1/2. The states are neither identical nor orthogonal.

Fidelity and trace distance are related by the Fuchs–van de Graaf bounds. If trace distance is defined as

then

Here ||X||₁ = Tr√(X†X) is the trace norm. These inequalities provide operational bounds even when exact state discrimination is difficult.

8. Data processing and quantum channels

A physical process that transforms states is represented by a completely positive trace-preserving map Λ. Examples include unitary evolution, noisy gates, measurements when the outcome is ignored, and partial trace.

Data processing means that post-processing cannot create useful distinguishability or correlation from nothing. Important forms include:

and, for a channel acting on B,

The latter says that local processing of B cannot increase its total correlation with A. Equality indicates that the processing has not lost the relevant information; in more advanced settings this is connected with recovery maps.

Data processing is central to quantum cryptography. An eavesdropper’s noisy interaction cannot improve distinguishability beyond what was present before the noise. It also constrains machine-learning and benchmarking claims: a data-analysis pipeline cannot legitimately claim to extract more physical information than was available in its input state.

9. Applied case: evaluating a noisy quantum communication link

Background

This illustrative case considers a laboratory transmitting a qubit through a depolarizing channel. The channel is modeled as

where 0 ≤ p ≤ 1 is the noise parameter. The output is a mixture of the intended state and the maximally mixed state.

Problem

The laboratory must compare the ideal input |0⟩ with the received state and determine whether additional processing can improve the fundamental information content.

Analysis

For input ρ = |0⟩⟨0|, the output is diagonal with eigenvalues

At p = 0.2, these are 0.9 and 0.1. The output entropy is therefore

The fidelity with the intended pure state is

Approach and outcome

The engineer can use fidelity as a direct quality measure and entropy as a measure of uncertainty introduced by the channel. A subsequent physical channel cannot increase the relative distinguishability of two possible input states, by data processing. Therefore, post-processing may improve a chosen operational estimate or correct structured errors, but it cannot violate the fundamental information inequalities.

Learning points

1. Fidelity is intuitive for state-preparation quality.
2. Entropy quantifies noise-induced uncertainty.
3. The channel model and fidelity convention must be reported.
4. Data processing prevents unsupported claims that processing created information.

10. Professional practice, limitations and developments

When calculating these measures, first diagonalize the density matrix or use a numerically stable eigensolver. Small negative eigenvalues caused by floating-point error should be treated carefully, not silently accepted as physical probabilities. Probabilities should be normalized, logarithm bases should be stated, and zero eigenvalues should be handled using the limiting convention.

Entropy and mutual information depend on the state description and subsystem partition. They do not automatically identify the source of correlation or distinguish classical correlation from entanglement. Fidelity depends on convention, while relative entropy is asymmetric and can diverge. Finite experimental data also produce estimation uncertainty; confidence intervals and calibration errors should accompany reported values.

In quantum technologies, these measures support randomized benchmarking, tomography, quantum-network analysis, cryptographic security proofs and variational algorithm evaluation. Current research connects strong subadditivity with approximate recovery, one-shot information theory and resource theories. Practical systems increasingly use certified bounds rather than exact tomography because reconstructing a large density matrix is experimentally expensive.

Chapter Summary

Von Neumann entropy S(ρ)=−Tr(ρ log₂ρ) measures uncertainty in a quantum state and is determined by its eigenvalues. Joint, marginal and conditional entropies describe composite systems, with quantum conditional entropy capable of being negative for entangled states. Subadditivity states that S(AB)≤S(A)+S(B), while strong subadditivity states that S(ABC)+S(B)≤S(AB)+S(BC).

Quantum mutual information I(A:B)=S(A)+S(B)−S(AB) measures total correlation and equals a relative entropy from the product of marginals. Relative entropy measures distinguishability, is non-negative, and cannot increase under a quantum channel. Fidelity measures state similarity, ranges from zero to one under the squared convention, and reduces to squared overlap for pure states. These measures provide a common language for evaluating uncertainty, correlations, noise, communication and quantum-device performance.

Key Terms

Von Neumann entropy: Entropy of a density operator, measured from its eigenvalues.

Density operator: Positive semidefinite, unit-trace operator representing a quantum state.

Pure state: A state represented by a one-dimensional projector with zero entropy.

Mixed state: A statistical mixture of quantum states.

Partial trace: Operation producing a subsystem state from a composite state.

Subadditivity: Inequality S(AB)≤S(A)+S(B).

Strong subadditivity: Three-system entropy inequality underlying conditional-information monotonicity.

Conditional entropy: S(A|B)=S(AB)−S(B).

Quantum mutual information: Total correlation I(A:B)=S(A)+S(B)−S(AB).

Relative entropy: Asymmetric distinguishability measure D(ρ∥σ).

Support: Subspace spanned by the eigenvectors having nonzero eigenvalues.

Quantum channel: Completely positive trace-preserving state transformation.

Data processing inequality: Principle that physical processing cannot increase distinguishability.

Fidelity: Similarity measure between two quantum states.

Trace norm: Norm ||X||₁=Tr√(X†X).

Trace distance: Operational distinguishability measure T=½||ρ−σ||₁.

Entanglement: Nonclassical correlation that cannot be produced by local preparation.

Maximally mixed state: State I/d in dimension d, having maximum entropy.

Quantum Markov structure: Correlation structure associated with equality in strong subadditivity.

Review Questions

Short Answer Questions

1. What is the von Neumann entropy of a pure quantum state?
2. Why can quantum conditional entropy be negative?
3. State the quantum mutual-information formula.
4. Is quantum relative entropy symmetric? Explain briefly.
5. What does the data-processing inequality say about a quantum channel?

Long Answer Questions

1. Explain how the eigenvalues of a density operator determine its von Neumann entropy, including the pure and maximally mixed cases.
2. Derive the relationship between mutual information and conditional entropy.
3. Explain subadditivity and strong subadditivity and discuss their significance for correlations.
4. Compare relative entropy and fidelity in terms of symmetry, interpretation and applications.
5. Explain why data processing is important in quantum communication and cryptography.

Analytical / Application Questions

1. Calculate the entropy and mutual information of the classically correlated state ρ_AB=½|00⟩⟨00|+½|11⟩⟨11|. Interpret the result.
2. A pure qubit output has overlap squared 0.81 with the intended state. Calculate its fidelity and explain what the value means.
3. For a depolarizing channel with p = 0.4 and input |0⟩, calculate the output eigenvalues and entropy. Explain how increasing p affects state quality and uncertainty.

Multiple Choice Questions with Answers

Question 1

What is the von Neumann entropy of a pure state?

A. 0 bits

B. 1 bit

C. log₂d bits for every dimension

D. It is undefined

Correct Answer: A. 0 bits

Question 2

What is the maximum entropy of a d-dimensional quantum system?

A. d bits

B. log₂d bits

C. d² bits

D. 1/d bits

Correct Answer: B. log₂d bits

Question 3

For a maximally entangled Bell state, what is S(AB)?

A. 0 bits

B. 1 bit

C. 2 bits

D. ½ bit

Correct Answer: A. 0 bits

Question 4

Which expression defines quantum mutual information?

A. S(AB)−S(A)−S(B)

B. S(A)+S(B)−S(AB)

C. S(A)S(B)−S(AB)

D. S(A|B)+S(B|A)

Correct Answer: B. S(A)+S(B)−S(AB)

Question 5

Which statement about quantum relative entropy is correct?

A. It is always symmetric

B. It can be negative

C. It is non-negative and generally asymmetric

D. It is always equal to fidelity

Correct Answer: C. It is non-negative and generally asymmetric

Question 6

Under the squared fidelity convention, what is the fidelity between |0⟩ and (|0⟩+|1⟩)/√2?

A. 0

B. ¼

C. ½

D. 1

Correct Answer: C. ½

Question 7

What does subadditivity state?

A. S(AB) ≥ S(A)+S(B)

B. S(AB) ≤ S(A)+S(B)

C. S(A)=S(B) always

D. S(AB)=0 always

Correct Answer: B. S(AB) ≤ S(A)+S(B)

Question 8

If ρAB is a product state, what is I(A:B)?

A. 0

B. 1

C. S(A)+S(B)

D. It must be negative

Correct Answer: A. 0

Question 9

What is the effect of a quantum channel on relative entropy according to data processing?

A. It always increases it

B. It leaves it undefined

C. It cannot increase it

D. It makes it symmetric

Correct Answer: C. It cannot increase it

Question 10

For a pure state |ψ⟩ and arbitrary σ, which expression gives fidelity?

A. Tr(σ)

B. ⟨ψ|σ|ψ⟩

C. S(σ)

D. D(|ψ⟩⟨ψ|∥σ)

Correct Answer: B. ⟨ψ|σ|ψ⟩