Entropy and Trace in Infinite Dimensions

A density operator can have infinite von Neumann entropy, and states can converge in trace norm while their entropies diverge. These failures concern the unbounded number of available levels, not a failure of probability normalization. Entropy formulas in infinite dimensions therefore need explicit finiteness or energy assumptions beyond the density-operator conditions.

Helpful background. Entropy overview supplies the statistical interpretation; density operators and trace norms supply the analytic setting.

Entropy as an extended positive trace

For $\rho=\sum_np_n|u_n\rangle\langle u_n|$, define, with natural logarithms,

$$S(\rho)=\sum_n(-p_n\log p_n)\in[0,\infty], \qquad 0\log0=0.$$

These are nonnegative terms because $0\leq p_n\leq1$. There is no conditional-convergence ambiguity. To use bits, divide by $\log2$.

The continuous function $f(x)=-x\log x$, extended by $f(0)=0$, defines a bounded positive operator $f(\rho)$: $f$ is bounded on $[0,1]$. Entropy is its extended trace. Finiteness of $S(\rho)$ is exactly the assertion that $f(\rho)$ is trace class. Although $\log\rho$ may be unbounded on the support and is problematic on the kernel, this functional-calculus definition avoids treating $\rho\log\rho$ as an unrestricted product.

The eigenvalues of $\rho$ are summable; the entropy eigenvalues $-p_n\log p_n$ need not be. Trace class is stable under many operator products, but it does not imply closure under every continuous function vanishing at zero.

A normalized state with infinite entropy

On a basis indexed by $n\geq2$, let

$$p_n=\frac{C}{n(\log n)^2}, \qquad C^{-1}=\sum_{n=2}^{\infty}\frac1{n(\log n)^2}.$$

The normalization sum converges by the integral test. For large $n$,

$$\log(1/p_n)=\log n+2\log\log n-\log C,$$

so $p_n\log(1/p_n)$ is asymptotic to $C/[n\log n]$. Its sum diverges. The diagonal operator with eigenvalues $p_n$ is a perfectly valid normal state with $S(\rho)=\infty$.

The example also rules out the inference that a discrete spectral decomposition makes entropy finite. Countably many small probabilities can contain an infinite entropy tail, even though their total probability is one.

Trace-norm convergence does not control entropy

Let $|0\rangle,|1\rangle,\ldots$ be orthonormal. For $N\geq3$, set $\epsilon_N=1/\sqrt{\log N}$ and

$$\rho_N=(1-\epsilon_N)|0\rangle\langle0| +\frac{\epsilon_N}{N}\sum_{n=1}^{N}|n\rangle\langle n|.$$

The trace distance to $P_0$ is $\epsilon_N\to0$, yet

$$S(\rho_N)=h_2(\epsilon_N)+\epsilon_N\log N =h_2(\epsilon_N)+\sqrt{\log N}\longrightarrow\infty,$$

where $h_2(x)=-x\log x-(1-x)\log(1-x)$. A vanishing probability is spread over an ever larger subspace. Every bounded measurement changes little, while entropy is sensitive to the number of occupied levels.

Von Neumann entropy is lower semicontinuous in trace norm: $S(\rho)\leq\liminf_nS(\rho_n)$ when $\rho_n\to\rho$. The example respects this inequality, since the limiting entropy is zero. Lower semicontinuity prevents a finite limiting sequence from suddenly hiding a larger entropy at the limit; it does not supply an upper error bound.

Energy controls can make entropy manageable

Suppose $H\geq0$ and $Z(\beta)=\operatorname{Tr}e^{-\beta H}<\infty$ for a given $\beta>0$. The Gibbs variational bound gives, for finite-energy states,

$$S(\rho)\leq\beta\operatorname{Tr}(\rho H)+\log Z(\beta).$$

The parameter $\beta$ has reciprocal-energy units. One obtains the bound from nonnegativity of relative entropy with respect to $\gamma_\beta=e^{-\beta H}/Z(\beta)$, using finite cutoffs to justify the argument before passing to the infinite limit. It implies finite entropy on any fixed finite-energy set under this partition-function hypothesis.

For uniform continuity on an energy-bounded set, a useful sufficient assumption is $Z(\beta)<\infty$ for every $\beta>0$. This stronger growth condition controls entropy carried by small high-energy tails. Winter establishes quantitative energy-constrained continuity bounds, including oscillator systems. A finite energy bound without an appropriate Hamiltonian growth condition is not a universal entropy-continuity theorem.

Relative entropy avoids some subtraction traps

Quantum relative entropy is conventionally written

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

In finite dimensions this formula is direct when the support condition is met. In infinite dimensions it denotes the standard extended nonnegative relative-entropy functional, with a careful spectral or limiting definition. It is not permission to subtract two separately infinite traces.

If $\operatorname{supp}\rho$ is not contained in $\operatorname{supp}\sigma$, the relative entropy is infinite. Support containment is necessary for finiteness but not sufficient: a sufficiently thin tail of $\sigma$ can still cause divergence. Lower semicontinuity and data processing make relative entropy useful in limits, while naive expressions such as $S(AB)-S(B)$ can become the undefined difference $\infty-\infty$.

Exercises

1. Show that a geometric state with $p_n=(1-q)q^n$ has finite entropy for $0<q<1$.

Solution

Use $\sum_np_n=1$ and $\sum_nnp_n=q/(1-q)$ to obtain $S=-\log(1-q)-[q/(1-q)]\log q$. Both terms are finite. This is the single-mode thermal example underlying the entropy overview.

2. In the discontinuity example, replace $\epsilon_N$ by $1/\log N$. What entropy limit results?

Solution

The trace distance still tends to zero, but $S(\rho_N)=h_2(1/\log N)+1\to1$ nat. The same limiting pure state can be approached with a nonzero finite entropy limit or with divergent entropy.

References

• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019, Chapters 7 and 11 — entropy and constrained infinite systems. Publisher.
• Winter, Andreas, “Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints,” Communications in Mathematical Physics 347, 291–313, 2016 — continuity bounds under dimension and energy constraints. Preprint; DOI.