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.

Unless stated otherwise, all systems here are finite dimensional and all logarithms are base $2$, so entropies are in bits. These assumptions keep the entropy differences below finite.

Helpful background. Density operators explains spectra and partial traces; classical information introduces Shannon entropy and mutual information.

Von Neumann entropy and the spectrum

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 $I_d/d$.

Unitary changes and measurement uncertainty

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. For components with positive probabilities, the lower equality holds exactly when all $\rho_x$ are identical. The upper equality holds exactly when their supports are mutually orthogonal, so a measurement can identify the label perfectly. The entropy increase here compares the unflagged state with the average conditional entropy $\sum_xp_xS(\rho_x)$, not with the entropy of the larger flagged state $XB$.

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

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

For example, $S_2(\rho)=-\log\operatorname{Tr}\rho^2$ converts purity into an entropy. Increasing $\alpha$ puts greater weight on the largest eigenvalues. These are spectral entropies of one state. Quantum Rényi relative and conditional entropies have several inequivalent definitions; their data-processing theorems cannot be inferred from this formula alone.

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 states and entanglement

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 the classical mixture $(|00\rangle\langle00|+|11\rangle\langle11|)/2$, all three entropies $S(A),S(B),S(AB)$ equal $1$, so $I(A:B)=1$. A Bell pair has the same marginals but $S(AB)=0$, giving $I(A:B)=2$ and $S(A|B)=-1$. The value $2$ does not mean that separate measurements can read two shared classical bits from one pair.

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.

Relative entropy compares two states using the same ordering of its arguments. In particular, comparing a mixed state with a pure reference usually gives infinity because the support condition fails. 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 and conditioning

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. Equivalently, $S(A|BC)\le S(A|B)$: supplying the additional system $C$ cannot increase this conditional entropy. The inequality holds for arbitrary correlations, including entanglement; it is not a weak-coupling approximation. The proof is the strong-subadditivity theorem of Lieb and Ruskai.

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.

Worked example: two nonorthogonal signals

A sender chooses $|0\rangle$ or $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ with equal probability. Each signal is pure, but their average is

$$\bar\rho=\begin{pmatrix}3/4&1/4\\1/4&1/4\end{pmatrix}, \qquad \lambda_\pm=\frac{1\pm1/\sqrt2}{2}.$$

Thus $\chi=h_2(\lambda_+)\simeq0.6009$ bits, below the sender’s one-bit label entropy. Suppose the receiver measures in the computational basis. The outcome distribution is $(3/4,1/4)$, while the conditional outcome entropy is $0$ for $|0\rangle$ and $1$ for $|+\rangle$. Consequently,

$$I(X:Y)=H(Y)-H(Y|X)=h_2(1/4)-\frac12\simeq0.3113.$$

This particular measurement falls strictly below the Holevo bound. Choosing a better measurement can improve the recovered information, but cannot overcome the bound. The optimization is a different task from simply diagonalizing $\bar\rho$.

Continuity and Infinite-Dimensional Cautions

Let

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

For $d\ge2$, Audenaert’s sharp bound holds at every exact trace distance $0\le T\le1$:

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

For example, $\sigma=|0\rangle\langle0|$ and the diagonal state with probabilities $(1-T,T/(d-1),\ldots,T/(d-1))$ attain it. If only an upper bound $T\le\varepsilon$ is known, substituting $\varepsilon$ is justified while $\varepsilon\le1-1/d$, where $f_d$ increases. For larger $\varepsilon$, use $\log d$; $f_d$ decreases beyond its maximum. Confusing an exact distance with a distance bound can therefore underestimate the entropy uncertainty.

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.

Exercises

1. Compare $I(A:C|B)$ for the pure GHZ state $(|000\rangle+|111\rangle)/\sqrt2$ and the classical mixture $(|000\rangle\langle000|+|111\rangle\langle111|)/2$. What does conditioning on $B$ remove in the classical case?

Solution

Both states have the same one-party and two-party reduced states, each of entropy $1$. The GHZ joint entropy is $0$, whereas the mixture has joint entropy $1$. Hence

$$I(A:C|B)_{\mathrm{GHZ}}=1+1-1-0=1, \qquad I(A:C|B)_{\mathrm{mixture}}=1+1-1-1=0.$$

For the mixture, reading the classical bit $B$ determines both $A$ and $C$, leaving no conditional correlation. Quantum conditioning in the GHZ expression is the entropy definition above; it does not mean measuring $B$ first and replacing the state by its postmeasurement mixture.

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.

References

• K. M. R. Audenaert, “A Sharp Continuity Estimate for the von Neumann Entropy”, Journal of Physics A: Mathematical and Theoretical 40, 8127–8136 (2007). Author preprint.
• A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed., De Gruyter, 2019.
• E. H. Lieb and M. B. Ruskai, “Proof of the Strong Subadditivity of Quantum-Mechanical Entropy”, Journal of Mathematical Physics 14, 1938–1941 (1973).
• 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, Chapter 5.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.