Renyi Entropies

Renyi entropies are a one-parameter family of entropy measures built from powers of a density operator. For a density operator $\rho$ and a parameter $\alpha>0$ with $\alpha\ne1$,

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

The limit $\alpha\to1$ gives the von Neumann entropy,

$$\lim_{\alpha\to1}S_\alpha(\rho) = S(\rho) = -\operatorname{Tr}(\rho\log\rho).$$

For a pure bipartite state, applying $S_\alpha$ to a reduced state gives a family of pure-state entanglement measures:

$$S_\alpha(A) = S_\alpha(\rho_A), \qquad \rho_A=\operatorname{Tr}_B\lvert\Psi\rangle\langle\Psi\rvert.$$

Renyi entropies are especially useful because $S_2$ is determined by the purity $\operatorname{Tr}(\rho^2)$, integer orders connect to replica methods, and different values of $\alpha$ probe different parts of the eigenvalue spectrum.

Definition

Let $\rho$ have spectral decomposition

$$\rho = \sum_k p_k \lvert k\rangle\langle k\rvert, \qquad p_k\ge0, \qquad \sum_k p_k=1.$$

Then

$$\operatorname{Tr}(\rho^\alpha) = \sum_k p_k^\alpha,$$

so

$$S_\alpha(\rho) = \frac{1}{1-\alpha} \log \left( \sum_k p_k^\alpha \right).$$

The logarithm base fixes the units. Base $2$ gives bits; natural logarithms give nats. This page states finite-dimensional examples in bits when numerical values are given.

Renyi entropy depends only on the eigenvalues of $\rho$. It is invariant under unitary conjugation:

$$S_\alpha(U\rho U^\dagger) = S_\alpha(\rho).$$

For a pure state, all Renyi entropies vanish:

$$S_\alpha(\lvert\psi\rangle\langle\psi\rvert)=0.$$

For a maximally mixed state $\rho_\ast=I/d$,

$$S_\alpha(\rho_\ast) = \log d$$

for every $\alpha>0$.

Pure Bipartite Entanglement

Let a normalized bipartite pure state have Schmidt decomposition

$$\lvert\Psi\rangle = \sum_r s_r \lvert u_r\rangle_A \lvert v_r\rangle_B.$$

Define Schmidt probabilities

$$p_r=s_r^2.$$

The reduced density operator on $A$ has nonzero eigenvalues $p_r$, so

$$S_\alpha(A) = \frac{1}{1-\alpha} \log \left( \sum_r p_r^\alpha \right).$$

The same value is obtained from $\rho_B$:

$$S_\alpha(A)=S_\alpha(B)$$

for pure bipartite states. Thus Renyi entropies form a family of pure-state entanglement measures across the chosen split.

The case $\alpha\to1$ is the usual Entanglement Entropy:

$$S_1(A) = -\sum_r p_r\log p_r.$$

The case $\alpha\to0$ gives the logarithm of the Schmidt Rank:

$$S_0(A) = \log \operatorname{SR}(\Psi),$$

provided the support is finite-dimensional.

Limit to von Neumann Entropy

Starting from

$$S_\alpha(\rho) = \frac{\log f(\alpha)}{1-\alpha}, \qquad f(\alpha) = \operatorname{Tr}(\rho^\alpha) = \sum_k p_k^\alpha,$$

we have $f(1)=1$. The limit $\alpha\to1$ has the indeterminate form $0/0$. Differentiating numerator and denominator gives

$$\lim_{\alpha\to1} \frac{\log f(\alpha)}{1-\alpha} = - \frac{f'(1)}{f(1)}.$$

Since

$$f'(\alpha) = \sum_k p_k^\alpha\log p_k,$$

the limit is

$$\lim_{\alpha\to1}S_\alpha(\rho) = - \sum_k p_k\log p_k = S(\rho).$$

This is why von Neumann entropy is often denoted $S_1$ in contexts where Renyi entropies are being discussed.

Purity and Second Renyi Entropy

The second Renyi entropy is

$$S_2(\rho) = -\log \operatorname{Tr}(\rho^2).$$

The quantity

$$\operatorname{Tr}(\rho^2)$$

is the purity. It equals $1$ for pure states and is less than $1$ for mixed states.

For a bipartite pure state, $S_2(\rho_A)$ is a simple entanglement measure. In terms of Schmidt probabilities,

$$S_2(A) = -\log \left( \sum_r p_r^2 \right).$$

For a Bell state, $p_1=p_2=1/2$, so

$$S_2(A) = -\log_2 \left( \frac14+\frac14 \right) = 1$$

bit.

For

$$\lvert\Psi_\theta\rangle = \cos\theta\,\lvert00\rangle + \sin\theta\,\lvert11\rangle,$$

the second Renyi entropy in bits is

$$S_2(A) = -\log_2 \bigl( \cos^4\theta+\sin^4\theta \bigr).$$

It vanishes at product endpoints and reaches one bit at $\theta=\pi/4$.

Other Useful Orders

The order $\alpha=0$ is the rank entropy:

$$S_0(\rho) = \log \operatorname{rank}\rho,$$

where the rank is the number of nonzero eigenvalues. It is sensitive to arbitrarily small nonzero eigenvalues, so it is mathematically clean but not robust under small perturbations.

The limit $\alpha\to\infty$ gives the min-entropy:

$$S_\infty(\rho) = -\log p_{\max},$$

where $p_{\max}$ is the largest eigenvalue of $\rho$.

For a fixed density operator, Renyi entropies are nonincreasing as $\alpha$ increases:

$$\alpha<\beta \quad \Longrightarrow \quad S_\alpha(\rho)\ge S_\beta(\rho).$$

Small $\alpha$ gives more weight to the number of populated eigenvalues. Large $\alpha$ emphasizes the largest eigenvalues.

Uses in Numerics and Experiments

Renyi entropies appear often in many-body physics, quantum information, and quantum simulation.

In numerical work, $S_2$ is useful because it only requires $\operatorname{Tr}(\rho^2)$. This is simpler than reconstructing the entire eigenvalue spectrum of $\rho$ when the Hilbert space is large.

In tensor-network and many-body settings, Renyi entropies summarize how entanglement is distributed across a spatial or site bipartition. They can diagnose area-law behavior, critical scaling, and finite-size trends. The interpretation depends on the physical split and on whether the total state is pure.

For spatial Rényi entropies, critical scaling, finite-size diagnostics, and numerical uses, see Entanglement Entropy in Many-Body Systems.

In experimental settings, purity and second Renyi entropy can sometimes be estimated from two-copy interference, randomized measurements, or statistical correlations across repeated preparations. The experimental protocol is not part of this page; the composite-systems point is that powers $\operatorname{Tr}(\rho^\alpha)$ can be more directly accessible than the full density matrix in some settings.

Replica Trick Preview

For an integer $n>1$,

$$S_n(\rho_A) = \frac{1}{1-n} \log \operatorname{Tr}(\rho_A^n).$$

The quantity $\operatorname{Tr}(\rho_A^n)$ can sometimes be computed by considering $n$ copies of the system. In statistical mechanics and QFT, this idea leads to replica methods: compute for integer $n$, then analytically continue and take the limit $n\to1$ to obtain von Neumann entropy.

This is only a preview. In QFT, entanglement entropies of spatial regions can depend on regulators, boundary geometry, and continuum limits. The finite-dimensional Renyi formulas here are the conceptual entry point; Entanglement in QFT Preview explains the wider field-theory setting.

Mixed-State Caution

For a mixed joint state $\rho_{AB}$, the Renyi entropy of $\rho_A$ measures local mixedness, not entanglement by itself. The same warning that applies to von Neumann subsystem entropy applies to every $S_\alpha(\rho_A)$.

For example,

$$\rho_{\mathrm{cc}} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert$$

has $\rho_A=I/2$, so every Renyi entropy of $A$ is one bit. The state is separable. Its local entropy reflects classical correlation with $B$, not pure-state entanglement.

By contrast, for a Bell pure state, the same one-qubit reduced state $I/2$ arises from entanglement with the other qubit. The joint-state context determines the interpretation.

Common Mistakes

• Forgetting that $\alpha=1$ is a limit, not a direct substitution into the formula with $1-\alpha$ in the denominator.
• Confusing $S_2(\rho)$ with the square of $S(\rho)$.
• Treating $S_\alpha(\rho_A)$ as a mixed-state entanglement measure without knowing that $\rho_{AB}$ is pure.
• Comparing numerical values without stating the logarithm base.
• Assuming full Schmidt rank implies large Renyi entropy for every order.
• Ignoring zero eigenvalues when discussing $S_0$.
• Assuming the replica trick is automatic in continuum QFT; analytic continuation and regularization are separate issues.

Cross-Links

• Subsystem Entropy
• Entanglement Entropy
• Mutual Information
• Concurrence for Two Qubits
• Schmidt Rank
• Schmidt Decomposition
• Reduced Density Operators
• Purification
• Bell States
• Entanglement Entropy in Many-Body Systems
• Entanglement in Many-Body Physics
• Entanglement in QFT Preview
• Pure Versus Mixed States
• Formula Sheet

References

• A. Renyi, “On Measures of Entropy and Information,” in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 547-561, University of California Press, 1961.
• A. Wehrl, “General Properties of Entropy,” Reviews of Modern Physics 50, 221-260, 1978.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• P. Calabrese and J. Cardy, “Entanglement Entropy and Quantum Field Theory,” Journal of Statistical Mechanics P06002, 2004.
• J. Eisert, M. Cramer, and M. B. Plenio, “Area Laws for the Entanglement Entropy,” Reviews of Modern Physics 82, 277-306, 2010.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

Exercises

1. Compute $S_\alpha$ for a pure state.

Solution

A pure state has eigenvalues $1,0,0,\ldots$. Therefore

$$\operatorname{Tr}(\rho^\alpha) = 1^\alpha = 1,$$

so

$$S_\alpha(\rho) = \frac{1}{1-\alpha}\log1 = 0.$$

2. Compute $S_\alpha$ for the maximally mixed state $I/d$.

Solution

The eigenvalues are all $1/d$. Hence

$$\operatorname{Tr}(\rho^\alpha) = d\left(\frac1d\right)^\alpha = d^{1-\alpha}.$$

Thus

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

3. Compute the second Renyi entropy in bits for a Bell state reduction.

Solution

For a Bell state, $\rho_A=I/2$. Thus

$$\operatorname{Tr}(\rho_A^2) = \operatorname{Tr}\left(\frac14 I\right) = \frac12.$$

Therefore

$$S_2(\rho_A) = -\log_2\frac12 = 1$$

bit.

4. Show that the limit $\alpha\to1$ gives the von Neumann entropy for eigenvalues $p_k$.

Solution

Let

$$f(\alpha)=\sum_k p_k^\alpha.$$

Then $f(1)=1$, and

$$S_\alpha = \frac{\log f(\alpha)}{1-\alpha}.$$

Using L’Hospital’s rule,

$$\lim_{\alpha\to1}S_\alpha = - f'(1) = - \sum_k p_k\log p_k.$$

5. For Schmidt probabilities $p,1-p$, write $S_2$ in bits.

Solution

The second Renyi entropy is

$$S_2 = -\log_2 \bigl( p^2+(1-p)^2 \bigr).$$

It is zero at $p=0$ and $p=1$, and equals one bit at $p=1/2$.