Quantum Entropy

Quantum entropy measures spectral uncertainty and information content of a density operator. Its central form is the von Neumann entropy

$$S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$

This is not merely the Shannon entropy of whichever measurement happens to be performed. It is basis independent and depends only on the eigenvalues of $\rho$. Different entropy expressions answer different operational questions, so the state, subsystems, logarithm base, and asymptotic setting must be stated.

Conventions and Spectral Form

Let

$$\rho=\sum_i\lambda_i|i\rangle\langle i|, \qquad \lambda_i\ge0, \qquad \sum_i\lambda_i=1.$$

Functional calculus gives

$$S(\rho)=-\sum_i\lambda_i\log\lambda_i,$$

with $0\log0=0$ by continuity. This page uses base-$2$ logarithms when entropy is reported in bits. Natural logarithms give nats and multiply every value by $\ln2$ relative to bits:

$$S_{\ln}(\rho)=(\ln2)S_{\log_2}(\rho).$$

For a $d$-dimensional state,

$$0\le S(\rho)\le\log d.$$

The lower value occurs exactly for a pure state; the upper value occurs exactly for the maximally mixed state $I_d/d$.

Why the Spectrum Is the Right Object

If $\rho'=U\rho U^\dagger$, then $\rho'$ has the same eigenvalues as $\rho$, so

$$S(U\rho U^\dagger)=S(\rho).$$

Matrix off-diagonal entries are basis dependent, while entropy is not. A pure coherent superposition can have a broad probability distribution in a chosen measurement basis and still have zero von Neumann entropy.

For a qubit with eigenvalues $p$ and $1-p$,

$$S(\rho)=h_2(p) =-p\log_2p-(1-p)\log_2(1-p).$$

Equivalently, if $\rho=(I+\mathbf r\cdot\boldsymbol\sigma)/2$, then $p=(1+\lVert\mathbf r\rVert)/2$.

Products, Classical Flags, and Mixtures

Entropy is additive on tensor products:

$$S(\rho_A\otimes\sigma_B)=S(\rho_A)+S(\sigma_B).$$

For a state with an orthogonal classical flag,

$$\rho_{XB} =\sum_xp_x|x\rangle\langle x|\otimes\rho_B^x,$$

one has the exact block-diagonal identity

$$S(XB)=H(p)+\sum_xp_xS(\rho_B^x).$$

For an unflagged mixture $\bar\rho=\sum_xp_x\rho_x$, concavity gives

$$S(\bar\rho)\ge\sum_xp_xS(\rho_x).$$

The companion upper bound

$$S(\bar\rho) \le H(p)+\sum_xp_xS(\rho_x)$$

shows how much uncertainty can be added by forgetting a classical label. Equality conditions depend on the supports of the component states.

Rényi Entropies

For $\alpha\gt0$, $\alpha\ne1$, the quantum Rényi entropy is

$$S_\alpha(\rho) =\frac{1}{1-\alpha} \log\operatorname{Tr}(\rho^\alpha).$$

Important limits are

$$\lim_{\alpha\to1}S_\alpha(\rho)=S(\rho),$$ $$S_0(\rho)=\log\operatorname{rank}\rho, \qquad S_\infty(\rho)=-\log\lambda_{\max}(\rho).$$

Rényi entropies emphasize different parts of the spectrum. Their data processing and operational properties depend on which quantum Rényi divergence is used; formulas from distinct conventions should not be mixed.

Bipartite Entropy and Conditional Entropy

For a state $\rho_{AB}$, define

$$S(A)=S(\rho_A), \qquad S(B)=S(\rho_B), \qquad S(AB)=S(\rho_{AB}).$$

The conditional entropy is

$$S(A|B)=S(AB)-S(B).$$

Unlike classical conditional entropy, it can be negative. Negative $S(A|B)$ is a quantum correlation signature with operational roles in state merging; it is not a negative number of classical outcomes.

Subadditivity and the Araki–Lieb inequality state

$$S(AB)\le S(A)+S(B),$$ $$|S(A)-S(B)|\le S(AB).$$

Together they bound the possible entropy triples of a bipartite state.

Pure Bipartite States and Entanglement Entropy

If $|\psi\rangle_{AB}$ is pure, its Schmidt decomposition implies that $\rho_A$ and $\rho_B$ have the same nonzero eigenvalues. Therefore

$$S(AB)=0, \qquad S(A)=S(B).$$

The shared value is the entropy of entanglement for a pure bipartite state. It vanishes exactly for a product state. In this case,

$$S(A|B)=-S(A), \qquad S(B|A)=-S(B).$$

For mixed states, subsystem entropy alone is not an entanglement measure; classical mixing also contributes.

Mutual Information

Quantum mutual information is

$$I(A:B) =S(A)+S(B)-S(AB).$$

It is nonnegative, vanishes exactly for a product state, and measures total correlation rather than entanglement alone. It also has the relative-entropy form

$$I(A:B) =D\!\left(\rho_{AB}\middle\|\rho_A\otimes\rho_B\right).$$

For a pure bipartite state,

$$I(A:B)=2S(A).$$

For a perfectly correlated classical bit pair, $I(A:B)=1$ bit, whereas a Bell pair has $I(A:B)=2$ bits. The extra bit reflects quantum coherence and negative conditional entropy, not two independent classical shared bits.

Relative Entropy and Data Processing

The quantum relative entropy is

$$D(\rho\|\sigma) =\operatorname{Tr}\rho(\log\rho-\log\sigma),$$

provided $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$; otherwise it is $+\infty$. It is not symmetric and is not a metric.

Klein’s inequality gives

$$D(\rho\|\sigma)\ge0,$$

with equality exactly when $\rho=\sigma$. For a quantum channel $\mathcal N$, data processing gives

$$D(\mathcal N(\rho)\|\mathcal N(\sigma)) \le D(\rho\|\sigma).$$

The same principle implies that a local channel cannot increase mutual information:

$$I(A:B)_{(\mathcal N_A\otimes\mathcal M_B)(\rho)} \le I(A:B)_\rho.$$

This is a statement about discarded distinguishability, not a claim that every entropy of every subsystem must increase under every channel.

Strong Subadditivity

For every tripartite state,

$$S(ABC)+S(B)\le S(AB)+S(BC).$$

Equivalently, the conditional mutual information

$$I(A:C|B) =S(AB)+S(BC)-S(B)-S(ABC)$$

is nonnegative. Strong subadditivity underlies monotonicity of mutual information, consistency of conditional entropy, and the structure of quantum Markov chains. It is a theorem, not an assumption about weak correlations.

Classical–Quantum States and the Holevo Quantity

For the classical–quantum state

$$\rho_{XB}=\sum_xp_x|x\rangle\langle x|\otimes\rho_x,$$

the mutual information is the Holevo quantity

$$I(X:B)=\chi =S\!\left(\sum_xp_x\rho_x\right) -\sum_xp_xS(\rho_x).$$

The Holevo bound limits how much classical information about $X$ can be recovered from a measurement on $B$. It does not say that every measurement achieves $\chi$, nor does it convert nonorthogonal signal states into perfectly distinguishable classical symbols.

Continuity and Infinite-Dimensional Cautions

Let

$$T=D(\rho,\sigma)=\frac12\lVert\rho-\sigma\rVert_1.$$

In dimension $d$, for $0\le T\le1-1/d$, the sharp continuity bound is

$$|S(\rho)-S(\sigma)| \le T\log(d-1)+h_2(T).$$

Outside that parameter range the trivial bound $\log d$ remains safe. The dimension dependence is essential: entropy is not uniformly continuous on the unrestricted infinite-dimensional state space.

In infinite dimensions, finite trace does not guarantee finite entropy. Thermal or energy-constrained continuity statements require explicit Hamiltonian and energy assumptions. Expressions such as $\log d$ have no meaning when $d=\infty$ unless a cutoff or effective support is part of the model.

Thermodynamic Cautions

Information entropy is dimensionless once a logarithm base is fixed. Thermodynamic entropy often includes Boltzmann’s constant:

$$S_{\mathrm{therm}}=k_BS_{\ln}(\rho).$$

The equality between von Neumann entropy and a thermodynamic entropy is context dependent: the state, coarse graining, ensemble, conserved quantities, and equilibrium assumptions matter. A unitary evolution preserves the fine-grained entropy of a closed system even while a coarse-grained or subsystem entropy can increase.

Diagnostic Examples

Product state

For $\rho_{AB}=\rho_A\otimes\rho_B$,

$$S(AB)=S(A)+S(B), \qquad I(A:B)=0.$$

Bell pair

For

$$|\Phi^+\rangle =\frac{|00\rangle+|11\rangle}{\sqrt2},$$

the joint state is pure while each marginal is maximally mixed:

$$S(AB)=0, \qquad S(A)=S(B)=1,$$

so

$$S(A|B)=-1, \qquad I(A:B)=2.$$

Perfectly correlated classical bits

For

$$\rho_{AB} =\frac12|00\rangle\langle00| +\frac12|11\rangle\langle11|,$$

one has

$$S(A)=S(B)=S(AB)=1,$$

and therefore $S(A|B)=0$ and $I(A:B)=1$.

Common Mistakes

• Omitting the logarithm base when reporting a numerical entropy.
• Treating von Neumann entropy as the entropy of one arbitrary measurement distribution.
• Assuming conditional entropy must be nonnegative.
• Calling mutual information an entanglement measure for mixed states.
• Using $D(\rho\|\sigma)$ without checking the support condition.
• Assuming every channel increases entropy; reset and cooling channels give immediate counterexamples.
• Applying finite-dimensional continuity bounds with no dimension or energy constraint.
• Interpreting fine-grained unitary entropy conservation as a denial of thermodynamic entropy production.

Where to Go Next

• Information-Theoretic Foundations routes entropy, correlation, distinguishability, and resource questions to their canonical owners; Mutual Information owns the structural bipartite examples, while Relative Entropy owns the classical KL support and statistical baseline.
• Classical Information Review owns classical source, coding, secrecy, and comparator baselines; this page owns von Neumann and Rényi entropies, quantum relative entropy, data processing, and Holevo quantities.
• Resource Theories uses contractive divergences as monotones only after the free set, free transformations, task, and regime are fixed; this page owns quantum relative entropy and data processing.
• Entanglement Entropy owns the bipartite entanglement treatment.
• Density Operators for Quantum Information supplies the state language.
• Entropy Identities is the compact convention-aware lookup card.
• Von Neumann Entropy is the single-formula reference.
• Trace Distance supplies the continuity parameter used above.

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
• E. H. Lieb and M. B. Ruskai, ‘Proof of the Strong Subadditivity of Quantum-Mechanical Entropy,’ Journal of Mathematical Physics 14, 1938-1941 (1973).
• K. M. R. Audenaert, ‘A Sharp Continuity Estimate for the von Neumann Entropy,’ Journal of Physics A 40, 8127-8136 (2007).
• A. S. Holevo, Quantum Systems, Channels, Information, 2nd ed., De Gruyter, 2019.

Exercises

1. Compute $S(A)$, $S(B)$, $S(AB)$, $S(A|B)$, and $I(A:B)$ for a product of two maximally mixed qubits.

Solution

Each marginal is $I_2/2$, so $S(A)=S(B)=1$ bit. Their product is $I_4/4$, so $S(AB)=2$ bits. Hence

$$S(A|B)=2-1=1, \qquad I(A:B)=1+1-2=0.$$

2. Let $|\psi\rangle_{ABC}$ be pure. Prove the conditional-entropy duality $S(A|B)=-S(A|C)$.

Solution

Purity gives $S(AB)=S(C)$ and $S(AC)=S(B)$. Therefore

$$S(A|B)=S(AB)-S(B)=S(C)-S(AC)=-S(A|C).$$

3. An ensemble contains mutually orthogonal pure states $|x\rangle$ with probabilities $p_x$. Compute its Holevo quantity.

Solution

Each signal state has zero entropy. The average state is diagonal with eigenvalues $p_x$, so

$$\chi=S\!\left(\sum_xp_x|x\rangle\langle x|\right) -\sum_xp_xS(|x\rangle\langle x|) =H(p).$$

An orthogonal measurement can attain this value.

4. Let a local channel act on $B$. Use data processing to show that $I(A:B)$ cannot increase.

Solution

Write mutual information as relative entropy:

$$I(A:B)_\rho =D(\rho_{AB}\|\rho_A\otimes\rho_B).$$

Apply $\operatorname{id}_A\otimes\mathcal N_B$ to both arguments and use data processing. The first becomes the output joint state and the second becomes the product of its marginals, so the resulting relative entropy is the output mutual information and cannot exceed the input value.