Rényi and Sandwiched Rényi Divergences

Shannon and von Neumann entropies are first moments. They report the average behaviour of long independent-and-identically-distributed (i.i.d.) sequences, and through the coding and channel theorems they fix the asymptotic capacities of information theory. But the sharpest operational questions are not about averages: how well can a finite block of $n$ qubits be compressed, how fast does the error probability decay with $n$ , and — the question that organises this article — how violently does the probability of success collapse once you push a rate above capacity, the phenomenon called the strong converse? Answering these needs a finer resolution of a state's spectrum, or of the relationship between two states, than a single average can carry. The Rényi entropies and relative entropies supply that resolution through a tunable order $\alpha$ . Classically the recipe is clean: $H_\alpha(p)=\frac{1}{1-\alpha}\log\sum_x p_x^\alpha$ re-weights the distribution's tail and $D_\alpha(p\|q)=\frac{1}{\alpha-1}\log\sum_x p_x^\alpha q_x^{1-\alpha}$ does the same for a pair. Quantumly there is a fork with no classical shadow: $\rho^\alpha$ and $\sigma^{1-\alpha}$ do not commute, and there is no canonical way to order them. That ambiguity spawns a family of admissible quantum Rényi divergences, of which two matter — the Petz divergence $D_\alpha$ and the sandwiched divergence $\widetilde D_\alpha$ . The sandwiched family, introduced in 2013 by Müller-Lennert–Dupuis–Szehr–Fehr–Tomamichel and independently by Wilde–Winter–Yang, closed the modern strong-converse programme, because for $\alpha>1$ it delivers exponentially tight bounds on the failure of impossible tasks. This article builds both families from the commuting case, proves what must hold, and shows why their validity windows — $[0,2]$ for Petz, $[\tfrac12,\infty]$ for sandwiched — are the whole story.

Key takeaways

The quantum Rényi entropy $H_\alpha(\rho)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho^\alpha$ is a one-knob family interpolating the max-entropy ( $\alpha\to0$ ), von Neumann entropy ( $\alpha\to1$ ), collision entropy ( $\alpha=2$ ) and min-entropy ( $\alpha\to\infty$ ); it is non-increasing in $\alpha$ .
Non-commutativity forces two inequivalent quantum lifts of the relative entropy: the Petz $D_\alpha=\frac{1}{\alpha-1}\log\operatorname{Tr}[\rho^\alpha\sigma^{1-\alpha}]$ and the sandwiched $\widetilde D_\alpha=\frac{1}{\alpha-1}\log\operatorname{Tr}[(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\,\sigma^{\frac{1-\alpha}{2\alpha}})^\alpha]$ ; they agree only when $[\rho,\sigma]=0$ .
Data processing (monotonicity under any CPTP map) holds for Petz on $\alpha\in[0,2]$ and sandwiched on $\alpha\in[\tfrac12,\infty]$ ; both rise with $\alpha$ , and $\widetilde D_\alpha\le D_\alpha$ always.
Both families converge to the Umegaki relative entropy $D(\rho\|\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma)$ as $\alpha\to1$ , so the ordinary theory sits at the centre of the family.
Limiting orders are named operational quantities: $\widetilde D_\infty=D_{\max}$ (one-shot resource cost), $\widetilde D_{1/2}=-2\log F$ (fidelity), $H_\infty=H_{\min}$ (extractable randomness), $H_2$ (privacy amplification).
Quantum hypothesis testing is organised by the two families: $D_\alpha$ with $\alpha<1$ gives the achievable (Hoeffding) exponent, $\widetilde D_\alpha$ with $\alpha>1$ the strong-converse exponent, both squeezing the Stein exponent $D$ .
Key idea: because $\rho$ and $\sigma$ need not commute, the classical $\sum_x p_x^\alpha q_x^{1-\alpha}$ has several non-equivalent quantum lifts; the Petz $D_\alpha$ and the sandwiched $\widetilde D_\alpha$ are the two that matter, and their data-processing windows ( $[0,2]$ and $[\tfrac12,\infty]$ ) together with their common $\alpha\to1$ limit $D(\rho\|\sigma)$ are what make them the correct tools for finite-block and strong-converse information theory.

The core insight

Because $\rho$ and $\sigma$ need not commute, the classical $\sum_x p_x^\alpha q_x^{1-\alpha}$ has several non-equivalent quantum lifts; the Petz $D_\alpha$ and the sandwiched $\widetilde D_\alpha$ are the two that matter, and their domains of data processing ( $[0,2]$ and $[\tfrac12,\infty]$ ) and their $\alpha\to1$ limits to $D(\rho\|\sigma)$ are what make them the correct tools for finite-block and strong-converse information theory.

Where this lives today

The sandwiched divergence is not a curiosity; it is the working tool that made a decade of "impossible-task" theorems single-letter and tight. Its most quoted victory is the strong converse for channel capacities: using sandwiched Rényi mutual information $\widetilde I_\alpha$ with $\alpha>1$ , Wilde, Winter and Yang proved that the entanglement-assisted capacity is a genuine sharp threshold — push the rate a hair above it and the success probability decays to zero exponentially, rather than merely failing to reach one — and the same machinery gives the strong converse to the Holevo capacity. In finite-block-length theory the smooth min- and max-entropies $H_{\min}^\varepsilon,H_{\max}^\varepsilon$ — optimising $\widetilde D_\infty$ and $\widetilde D_{1/2}$ over an $\varepsilon$ -ball of states — govern data compression, state merging, and the second-order corrections that tell an engineer how many extra qubits a length- $n$ code really costs. Most concretely, in quantum key distribution the leftover hashing lemma sets the number of extractable secret bits equal to the conditional min-entropy $H_{\min}(A|E)=-\widetilde D_\infty(\rho_{AE}\|I_A\otimes\rho_E)$ , and the entropy accumulation theorem behind modern device-independent QKD security proofs is built directly on Rényi entropies. Every time a real key is certified against a quantum adversary, a sandwiched Rényi quantity is doing the accounting.

The physical picture

Read $\alpha$ as a dial that chooses which part of a state's eigenvalue spectrum gets to speak, through $\operatorname{Tr}\rho^\alpha=\sum_i\lambda_i^\alpha$ . For $\alpha>1$ the large eigenvalues are amplified, so the sum is dominated by the spectral peaks and the entropy slides toward the min-entropy $-\log\lambda_{\max}$ — the dial hears only the most likely outcome. For $\alpha<1$ the small eigenvalues are boosted, the support matters more than the shape, and the entropy climbs toward the max-entropy $\log\operatorname{rank}\rho$ . The von Neumann value $S(\rho)$ at $\alpha=1$ is the smooth handover. The first figure turns this into one curve, computed for a real qubit.

Figure

Rényi entropy Hα of a qubit with eigenvalues 0.7 and 0.3 as α runs from 0 to ∞: it falls monotonically from the max-entropy H₀ = 1 bit through the von Neumann value S = 0.881 bits and the collision entropy H₂ = 0.786 bits down to the min-entropy H∞ = 0.515 bits

For divergences the same dial sets a distinguishability budget. The quantity $\operatorname{Tr}[\rho^\alpha\sigma^{1-\alpha}]$ (or its sandwiched cousin) weights the overlap of the two states, and $\alpha$ decides whether the regions where $\rho$ is large but $\sigma$ is small — the events that betray $\rho$ when you are trying and failing to mistake it for $\sigma$ — dominate ( $\alpha>1$ ) or wash out ( $\alpha<1$ ). That is why $\alpha>1$ controls the strong converse: it quantifies how unavoidable detection becomes when you demand too much. The second figure shows both lifts for a genuinely non-commuting pair, and is the numerical spine of this article.

Figure

The Petz Dα (amber, upper) and sandwiched divergence (blue, lower) for a pure state ρ against σ with eigenvalues 0.8 and 0.2: both meet the quantum relative entropy D = 1.322 bits at α = 1, the sandwiched curve stays below the Petz curve everywhere else, and the sandwiched landmarks run 1.0 (twice the fidelity) at α = 1/2, 1.492 at α = 2, and 1.644 (Dmax) at α → ∞

The core equations

The starting point is the quantum Rényi entropy: a one-parameter family whose only ingredients are the eigenvalues of $\rho$ , re-weighted by the power $\alpha$ . Everything about a single state's spectral shape that finite-block theory needs is a value of it.

H_\alpha(\rho)=\frac{1}{1-\alpha}\log\operatorname{Tr}\big[\rho^{\alpha}\big],\qquad \alpha\in(0,1)\cup(1,\infty) \qquad (1)

where: $\rho$ — density operator, $\rho\ge0$ , $\operatorname{Tr}\rho=1$ ; $\alpha$ — Rényi order; $\operatorname{Tr}\rho^\alpha=\sum_i\lambda_i^\alpha$ over the eigenvalues $\lambda_i$ of $\rho$ . The prefactor $\frac{1}{1-\alpha}$ flips sign at $\alpha=1$ , and the entropy is non-increasing in $\alpha$ so that $H_0\ge S\ge H_2\ge H_\infty$ . The relative entropy is the more powerful object, since distinguishing two states is the primitive every capacity and converse reduces to. Its historically first quantum lift is the Petz divergence, which keeps $\rho^\alpha$ and $\sigma^{1-\alpha}$ in the fixed left-to-right order of the classical formula.

D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log\operatorname{Tr}\big[\rho^{\alpha}\sigma^{1-\alpha}\big] \qquad (2)

where: $\sigma$ — reference (null-hypothesis) state, with $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$ required for $\alpha>1$ ; $\alpha$ — order, with data processing valid on $\alpha\in[0,2]$ . When $[\rho,\sigma]=0$ this reduces term-by-term to the classical $\frac{1}{\alpha-1}\log\sum_x p_x^\alpha q_x^{1-\alpha}$ . The second lift is the sandwiched divergence. It symmetrically wraps powers of $\sigma$ around $\rho$ before taking the $\alpha$ -th power, and this ordering — not the Petz one — keeps data processing alive all the way to $\alpha=\infty$ .

\widetilde D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log\operatorname{Tr}\Big[\big(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}\big)^{\alpha}\Big] \qquad (3)

where: the exponent $\tfrac{1-\alpha}{2\alpha}$ on $\sigma$ is fixed uniquely by demanding that the bracket collapse to $p_x^\alpha q_x^{1-\alpha}$ in the commuting case; data processing holds on $\alpha\in[\tfrac12,\infty]$ . For non-commuting states $\widetilde D_\alpha\le D_\alpha$ strictly. Outside their monotonicity windows both lifts are useless as distinguishability measures, so the ranges are not fine print — they are the operational content. Data processing is the single most important property either family has.

\widetilde D_\alpha(\rho\|\sigma)\;\ge\;\widetilde D_\alpha\big(\Lambda(\rho)\,\big\|\,\Lambda(\sigma)\big),\qquad \Lambda\ \text{CPTP},\ \tfrac12\le\alpha\le\infty \qquad (4)

where: $\Lambda$ — any completely positive trace-preserving map (a physical channel); the inequality says no processing can increase distinguishability. The Petz version holds instead on $\alpha\in[0,2]$ ; below $\tfrac12$ (sandwiched) or above $2$ (Petz) it can fail outright. The two extreme sandwiched orders are named one-shot quantities. At $\alpha\to\infty$ the divergence becomes the max-relative entropy, the currency of one-shot resource conversion; at $\alpha=\tfrac12$ it is minus-twice the log-fidelity.

\widetilde D_{\infty}(\rho\|\sigma)=D_{\max}(\rho\|\sigma)=\inf\{\lambda:\rho\le 2^{\lambda}\sigma\},\qquad \widetilde D_{1/2}(\rho\|\sigma)=-2\log F(\rho,\sigma) \qquad (5)

where: $D_{\max}$ — smallest $\lambda$ making $2^{\lambda}\sigma$ dominate $\rho$ ; $F(\rho,\sigma)=\lVert\sqrt{\rho}\,\sqrt{\sigma}\rVert_1$ — Uhlmann fidelity, $0\le F\le1$ . These pin the top and bottom of the divergence ladder that runs through $D$ at $\alpha=1$ . Finally, the family lifts to conditional quantities. Choosing $\sigma=I_A\otimes\rho_B$ turns the sandwiched divergence into a Rényi conditional entropy, whose $\alpha\to\infty$ and $\alpha\to\tfrac12$ limits are the smooth min- and max-entropies.

H_\alpha(A|B)_\rho=-\widetilde D_\alpha\big(\rho_{AB}\,\big\|\,I_A\otimes\rho_B\big),\qquad I_\alpha(A\!:\!B)_\rho=\min_{\sigma_B}\widetilde D_\alpha\big(\rho_{AB}\,\big\|\,\rho_A\otimes\sigma_B\big) \qquad (6)

where: $\rho_{AB}$ — bipartite state; $\rho_A,\rho_B$ — its marginals; $\sigma_B$ — an optimised reference. The conditional min-entropy $H_{\min}(A|B)=H_\infty(A|B)$ is what the leftover hashing lemma turns into secret bits.

Derivation ABoth families recover the von Neumann and Umegaki entropy at order one

The goal is to anchor the whole apparatus to the ordinary theory: as $\alpha\to1$ the Rényi entropy $H_\alpha$ becomes the von Neumann entropy $S(\rho)=-\operatorname{Tr}\rho\log\rho$ , and both the Petz $D_\alpha$ and the sandwiched $\widetilde D_\alpha$ become the Umegaki relative entropy $D(\rho\|\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma)$ . Because the prefactor and the logarithm both vanish at $\alpha=1$ , this is a $0/0$ limit that L'Hôpital resolves cleanly, and getting it right is what justifies calling these families extensions of Shannon and von Neumann theory at all.

Step 1. Write $H_\alpha=\dfrac{\log\operatorname{Tr}\rho^\alpha}{1-\alpha}$ and note that at $\alpha=1$ the numerator is $\log\operatorname{Tr}\rho=\log 1=0$ and the denominator is $0$ . Why: this is the indeterminate form that lets us differentiate top and bottom instead of substituting directly.

Step 2. Differentiate using $\partial_\alpha\operatorname{Tr}\rho^\alpha=\operatorname{Tr}[\rho^\alpha\log\rho]$ , so $\partial_\alpha\log\operatorname{Tr}\rho^\alpha=\operatorname{Tr}[\rho^\alpha\log\rho]/\operatorname{Tr}\rho^\alpha$ ; at $\alpha=1$ this is $\operatorname{Tr}[\rho\log\rho]$ , while $\partial_\alpha(1-\alpha)=-1$ . Why: the trace kills the ordering ambiguity of $\rho^\alpha$ and $\log\rho$ , so the matrix derivative follows the scalar rule.

Step 3. Assemble the ratio: $\lim_{\alpha\to1}H_\alpha=\dfrac{\operatorname{Tr}\rho\log\rho}{-1}=-\operatorname{Tr}\rho\log\rho=S(\rho)$ . Why: the L'Hôpital ratio is precisely the von Neumann entropy, so $\alpha=1$ is the smooth crossover marked in the first figure.

Step 4. The same expansion runs on the divergences: $\operatorname{Tr}[\rho^\alpha\sigma^{1-\alpha}]=1+(\alpha-1)\operatorname{Tr}\rho(\log\rho-\log\sigma)+O((\alpha-1)^2)$ , and dividing $\log(1+x)\approx x$ by $(\alpha-1)$ gives $D(\rho\|\sigma)$ . The sandwiched bracket expands identically to first order, so $\lim_{\alpha\to1}\widetilde D_\alpha=\lim_{\alpha\to1}D_\alpha=D$ . Why: the two families differ only at second order in $(\alpha-1)$ , so they share the same tangent — the relative entropy — at the centre. *Verified: standard computation in Wilde Ch. 7; reproduces the Umegaki relative entropy of aq-qci-quantum-relative-entropy.*

Go deeper · Derivation B · 1 minWhy the sandwiched divergence never exceeds the Petz divergence

Derivation BWhy the sandwiched divergence never exceeds the Petz divergence

The goal is to establish the ordering $\widetilde D_\alpha(\rho\|\sigma)\le D_\alpha(\rho\|\sigma)$ for all valid $\alpha$ , with equality exactly when $[\rho,\sigma]=0$ . The route is to first collapse both definitions in the commuting case, then invoke the Araki–Lieb–Thirring trace inequality — the operator statement that reordering powers inside a trace can only move it one way. This ordering is what makes the sandwiched family the tighter, more informative distinguishability measure.

Step 1. Take $[\rho,\sigma]=0$ and diagonalise both in a common eigenbasis with eigenvalues $p_x,q_x$ . Then $\rho^\alpha\sigma^{1-\alpha}$ has diagonal entries $p_x^\alpha q_x^{1-\alpha}$ , and the sandwich $\sigma^{\frac{1-\alpha}{2\alpha}}\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}$ has entries $p_x q_x^{(1-\alpha)/\alpha}$ , whose $\alpha$ -th power is again $p_x^\alpha q_x^{1-\alpha}$ . Why: both traces reduce to the same sum $\sum_x p_x^\alpha q_x^{1-\alpha}$ , so $\widetilde D_\alpha=D_\alpha$ whenever the states commute — the lifts only part company off the commuting locus.

Step 2. For non-commuting states compare $\operatorname{Tr}[(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\,\sigma^{\frac{1-\alpha}{2\alpha}})^\alpha]$ with $\operatorname{Tr}[\rho^\alpha\sigma^{1-\alpha}]$ . The Araki–Lieb–Thirring inequality states that for positive $A,B$ and $\alpha\ge1$ , $\operatorname{Tr}[(B^{1/2}AB^{1/2})^\alpha]\le\operatorname{Tr}[A^\alpha B^\alpha]$ , reversing for $\alpha\le1$ . Why: this is the operator content of "non-commuting powers cannot be freely reordered inside a trace", the single inequality separating the two families.

Step 3. Track the sign of the prefactor $\frac{1}{\alpha-1}$ . For $\alpha>1$ the trace inequality makes the sandwiched trace smaller, and dividing by the positive $\frac{1}{\alpha-1}$ gives $\widetilde D_\alpha\le D_\alpha$ ; for $\alpha<1$ the trace inequality flips but so does the sign of $\frac{1}{\alpha-1}$ , and $\widetilde D_\alpha\le D_\alpha$ survives. Why: the two sign flips cancel, so the ordering holds on the whole common domain.

Step 4. Equality in Araki–Lieb–Thirring requires $[\rho,\sigma]=0$ , so $\widetilde D_\alpha=D_\alpha$ iff the states commute (or trivially at $\alpha=1$ ). The worked example realises the gap numerically: $\widetilde D_2=1.492<D_2=1.644$ bits. Why: the gap is a real number you can compute, not a technicality. *Verified: Araki–Lieb–Thirring ordering and the equality condition in Müller-Lennert et al. (2013); the numerical gap is reproduced in the worked example.*

Go deeper · Derivation C · 1 minQuantum Stein's lemma from the two Rényi families

Derivation CQuantum Stein's lemma from the two Rényi families

The goal is to show how the two families jointly pin the optimal error exponent of asymmetric hypothesis testing — quantum Stein's lemma — from opposite sides, the sandwiched $\alpha>1$ side delivering the strong converse and the Petz $\alpha<1$ side achievability. Set up the test: given $n$ copies, decide between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ with a two-outcome measurement $0\le Q_n\le I$ , and define the Type-I error $\varepsilon_n=\operatorname{Tr}[(I-Q_n)\rho^{\otimes n}]$ and Type-II error $\beta_n=\operatorname{Tr}[Q_n\sigma^{\otimes n}]$ . Stein's lemma says the best Type-II exponent, holding Type-I error below any fixed $\varepsilon\in(0,1)$ , is exactly $D(\rho\|\sigma)$ .

Step 1. Both families are additive on tensor powers: $\widetilde D_\alpha(\rho^{\otimes n}\|\sigma^{\otimes n})=n\,\widetilde D_\alpha(\rho\|\sigma)$ , and likewise for $D_\alpha$ . Why: additivity is what makes a per-copy exponent well defined, so an $n$ -copy bound is just $n$ times a single-letter quantity.

Step 2. Strong converse ( $\alpha>1$ ). Data processing plus additivity gives, for any rate $R$ with $-\tfrac1n\log\beta_n\ge R$ , the bound $\varepsilon_n\ge 1-2^{-n\frac{\alpha-1}{\alpha}(\widetilde D_\alpha(\rho\|\sigma)-R)}$ . For any $R>D$ , monotonicity lets you pick $\alpha>1$ with $\widetilde D_\alpha>R$ , forcing $\varepsilon_n\to1$ . Why: success becomes impossible, not merely hard, above $D$ — the signature of a strong converse — and it needs the sandwiched family because Petz loses data processing before $\alpha$ can be taken large enough.

Step 3. Achievability ( $\alpha<1$ ). A Petz (Hoeffding-type) argument builds tests with $\beta_n\le 2^{-nD_\alpha(\rho\|\sigma)+o(n)}$ while $\varepsilon_n\to0$ ; letting $\alpha\to1$ sends $D_\alpha\to D$ . Why: the $\alpha<1$ family certifies that $D$ is reachable, so the converse is not vacuous.

Step 4. Squeeze. The converse caps the exponent at $D$ and achievability attains it, so $\lim_{n\to\infty}-\tfrac1n\log\beta_n^{*}(\varepsilon)=D(\rho\|\sigma)$ for every $\varepsilon\in(0,1)$ . Why: the two families bound the true exponent from above and below, meeting at $\alpha\to1$ where both equal the relative entropy. *Verified: reproduces Hiai–Petz (1991) achievability and Ogawa–Nagaoka (2000) strong converse; see Wilde Ch. 30 and Tomamichel (2016).*

Limiting cases

The two families share every special value in the limits and at $\alpha=1$ , and differ only in between; the table collects the landmarks that carry names.

Case	Result	Meaning
$\alpha\to1$	$\lim_{\alpha\to1}D_\alpha=\lim_{\alpha\to1}\widetilde D_\alpha=D(\rho\Vert\sigma)$	Both families become the Umegaki relative entropy $\operatorname{Tr}\rho(\log\rho-\log\sigma)$ ; the ordinary theory sits at the centre of the family.
$\alpha\to\infty$ (sandwiched)	$\widetilde D_\infty=\log\big\lVert\sigma^{-1/2}\rho\,\sigma^{-1/2}\big\rVert_\infty=D_{\max}$	Max-relative entropy: smallest $\lambda$ with $\rho\le2^\lambda\sigma$ . Governs one-shot resource cost and smooth-entropy calculus.
$\alpha=\tfrac12$ (sandwiched)	$\widetilde D_{1/2}=-2\log F(\rho,\sigma)$	Tied to the Uhlmann fidelity $F=\lVert\sqrt\rho\sqrt\sigma\rVert_1$ ; the smallest order at which data processing still holds.
Rényi entropy endpoints	$H_0=\log\operatorname{rank}\rho,\ \ H_2=-\log\operatorname{Tr}\rho^2,\ \ H_\infty=-\log\lambda_{\max}(\rho)$	Max-entropy ( $\alpha\to0$ ), collision entropy ( $\alpha=2$ , privacy amplification), min-entropy ( $\alpha\to\infty$ , extractable randomness).
Ordering	$\widetilde D_\alpha\le D_\alpha,\qquad \partial_\alpha D_\alpha\ge0,\ \ \partial_\alpha\widetilde D_\alpha\ge0$	Sandwiched never exceeds Petz; both are non-decreasing in $\alpha$ , so $H_\alpha$ is non-increasing in $\alpha$ .

Common trapThe two divergences are not the same object, and each has its own window

Do not assume the Petz and sandwiched divergences agree: they coincide only when $[\rho,\sigma]=0$ (and trivially at $\alpha=1$ ), and in general $\widetilde D_\alpha\le D_\alpha$ with a strictly positive gap — for the worked pair the gap at $\alpha=2$ is $1.644-1.492=0.152$ bits. The second common error is to use a divergence outside its data-processing window: Petz $D_\alpha$ is monotone under channels only for $\alpha\in[0,2]$ , sandwiched $\widetilde D_\alpha$ only for $\alpha\in[\tfrac12,\infty]$ . Applying $D_\alpha$ at $\alpha>2$ , or $\widetilde D_\alpha$ at $\alpha<\tfrac12$ , as though data processing held silently destroys the one property that makes these quantities meaningful. Finally, $\widetilde D_\alpha$ is the smaller, hence tighter, lift — which is why it, not Petz, closes the strong-converse arguments at $\alpha>1$ .

Worked examples

1 · The Rényi entropy ladder of a qubit (confidence builder)

A qubit is in the state $\rho=\operatorname{diag}(0.7,0.3)$ . Compute the four landmark Rényi entropies $H_0$ , $S=H_1$ , $H_2$ , $H_\infty$ (in bits) and confirm they are ordered.

Step 1. Max-entropy at $\alpha\to0$ counts the support: $H_0=\log_2\operatorname{rank}\rho=\log_2 2=\mathbf{1\ bit}$ . Both eigenvalues are nonzero, so the rank is $2$ and $H_0$ ignores how uneven they are.

Step 2. Von Neumann entropy at $\alpha=1$ : $S=-0.7\log_2 0.7-0.3\log_2 0.3=0.7(0.5146)+0.3(1.7370)=\mathbf{0.881\ bits}$ . This is the Shannon entropy of the eigenvalue distribution and the smooth crossover value.

Step 3. Collision entropy at $\alpha=2$ : $H_2=-\log_2\operatorname{Tr}\rho^2=-\log_2(0.7^2+0.3^2)=-\log_2(0.58)=\mathbf{0.786\ bits}$ . The purity $\operatorname{Tr}\rho^2=0.58$ is what enters two-universal hashing bounds in privacy amplification.

Step 4. Min-entropy at $\alpha\to\infty$ listens only to the largest eigenvalue: $H_\infty=-\log_2\lambda_{\max}=-\log_2 0.7=\mathbf{0.515\ bits}$ . The ordering is confirmed: $H_0=1\ge S=0.881\ge H_2=0.786\ge H_\infty=0.515$ , monotone decreasing in $\alpha$ exactly as the first figure draws. The four landmarks are 1, 0.881, 0.786, 0.515 bits.

2 · The full divergence ladder for a non-commuting pair (fully worked)

Let $\rho=\lvert+\rangle\langle+\rvert$ be the equal-superposition pure state, with matrix elements all equal to $\tfrac12$ , and $\sigma=\operatorname{diag}(0.8,0.2)$ . Note $[\rho,\sigma]\ne0$ . Compute (in bits) the relative entropy $D$ , the sandwiched values $\widetilde D_{1/2}$ , $\widetilde D_2$ , $\widetilde D_\infty$ , and the Petz $D_2$ , and read off the ordering.

Step 1. Relative entropy $\alpha=1$ . Since $\rho$ is pure, $\operatorname{Tr}\rho\log\rho=0$ , so $D=-\operatorname{Tr}\rho\log_2\sigma=-\langle+\rvert\log_2\sigma\lvert+\rangle=-\tfrac12(\log_2 0.8+\log_2 0.2)=-\tfrac12(-0.3219-2.3219)=\mathbf{1.322\ bits}$ .

Step 2. Sandwiched $\alpha=\tfrac12$ is the fidelity. Here $F(\rho,\sigma)=\sqrt{\langle+\rvert\sigma\lvert+\rangle}=\sqrt{\tfrac12(0.8+0.2)}=\sqrt{0.5}$ , so $\widetilde D_{1/2}=-2\log_2\sqrt{0.5}=-2(-0.5)=\mathbf{1.0\ bit}$ . This is the bottom rung of the ladder.

Step 3. Sandwiched $\alpha=2$ . The exponent on $\sigma$ is $\tfrac{1-\alpha}{2\alpha}=-\tfrac14$ , and cyclicity gives $\widetilde Q_2=\operatorname{Tr}[\rho\,\sigma^{-1/2}\rho\,\sigma^{-1/2}]$ . With $\sigma^{-1/2}=\operatorname{diag}(1.1180,2.2361)$ one finds $\widetilde Q_2=2.8125$ , so $\widetilde D_2=\log_2 2.8125=\mathbf{1.492\ bits}$ .

Step 4. Petz $\alpha=2$ . Since $\rho$ is a projector, $\rho^2=\rho$ , and $Q_2=\operatorname{Tr}[\rho\,\sigma^{-1}]=\langle+\rvert\sigma^{-1}\lvert+\rangle=\tfrac12(1.25+5)=3.125$ , so $D_2=\log_2 3.125=\mathbf{1.644\ bits}$ . The sandwiched value is strictly smaller: $\widetilde D_2=1.492<D_2=1.644$ , the gap promised by the ordering derivation.

Step 5. Sandwiched $\alpha\to\infty$ is the max-relative entropy. The operator $\sigma^{-1/2}\rho\,\sigma^{-1/2}$ is rank-one with nonzero eigenvalue $\langle+\rvert\sigma^{-1}\lvert+\rangle=3.125$ , so $\widetilde D_\infty=D_{\max}=\log_2 3.125=\mathbf{1.644\ bits}$ . Reading the sandwiched ladder top to bottom: $\widetilde D_{1/2}=1.0\le D=1.322\le \widetilde D_2=1.492\le \widetilde D_\infty=1.644$ — monotone in $\alpha$ , everything below the Petz curve, exactly the second figure. The ladder is 1.0, 1.322, 1.492, 1.644 bits, with Petz $D_2=1.644$ sitting above the sandwiched $\widetilde D_2$ .

Physics of the portrait

The whole family carries one name, and it is not a physicist's but a mathematician's: Alfréd Rényi, who in 1961 asked which functions of a probability distribution could still count as "information" once you dropped the demand of full additivity, and found a one-parameter answer that quantum information theory is still mining sixty years later.

Origins · 1 minAlfréd Rényi and how this was discovered

Alfréd Rényi

1921–1970 · information theory and quantum divergences

Alfréd Rényi was a Hungarian mathematician whose deepest mark on information theory came from a single question of axiomatics. Shannon's entropy is singled out by a short list of natural postulates, including a strong additivity rule for how information composes. In his 1961 Berkeley Symposium paper "On measures of entropy and information", Rényi asked what survives if that axiom is weakened to additivity over independent events alone. The answer was not unique: an entire one-parameter family $H_\alpha=\frac{1}{1-\alpha}\log\sum_x p_x^\alpha$ , with Shannon's entropy recovered only as $\alpha\to1$ , plus a companion family of divergences. For decades these were treated as analytic conveniences whose $\alpha$ -derivatives produced useful inequalities. Their operational elevation came later: Dénes Petz's 1986 quasi-entropies gave the first quantum lift $D_\alpha$ , and in 2013 Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel — with Wilde, Winter and Yang independently — found the sandwiched form that finally matched the strong-converse exponents of quantum hypothesis testing. Rényi, who died in 1970 and is as well remembered for probabilistic number theory and for the quip that "a mathematician is a machine for turning coffee into theorems", did not live to see his family become the backbone of one-shot quantum information theory — yet every min-entropy in a modern QKD security proof is a value of the function he wrote down.

History

1961

— Rényi introduces the α-family of entropies and divergences

1986

— Petz defines quasi-entropies, giving the traditional quantum Rényi divergence Dα

1991

— Hiai and Petz prove the achievability side of quantum Stein's lemma

2000

— Ogawa and Nagaoka establish the strong converse for quantum hypothesis testing

2013

— Müller-Lennert et al. and Wilde, Winter, Yang introduce the sandwiched Rényi divergence

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information (10th anniv. ed.), Ch. 11 · Wilde, Quantum Information Theory (2nd ed.), Ch. 7 and Ch. 30 (Rényi entropies and the strong converse) · Watrous, The Theory of Quantum Information, Ch. 5 · Tomamichel, Quantum Information Processing with Finite Resources (Springer, 2016) · Preskill, Lecture Notes on Quantum Computation, Ch. 10.

Papers. Müller-Lennert, Dupuis, Szehr, Fehr, Tomamichel, J. Math. Phys. 54, 122203 (2013) · Wilde, Winter, Yang, Comm. Math. Phys. 331, 593 (2014) · Frank & Lieb, J. Math. Phys. 54, 122201 (2013) (data-processing range) · Ogawa & Nagaoka, IEEE Trans. Inf. Theory 46, 2428 (2000) (strong converse).

Provenance. A verifier should (1) confirm both families reduce to $\operatorname{Tr}\rho(\log\rho-\log\sigma)$ as $\alpha\to1$ by L'Hôpital (Wilde Ch. 7); (2) check the windows $[0,2]$ and $[\tfrac12,\infty]$ against Müller-Lennert et al. (2013) and Frank–Lieb (2013); (3) verify $\widetilde D_{1/2}=-2\log F$ and $\widetilde D_\infty=D_{\max}$ from the definitions; (4) reproduce the worked ladder $\widetilde D_{1/2}=1.0$ , $D=1.322$ , $\widetilde D_2=1.492$ , $D_2=1.644$ , $\widetilde D_\infty=1.644$ bits for $\rho=\lvert+\rangle\langle+\rvert$ , $\sigma=\operatorname{diag}(0.8,0.2)$ , and the entropy landmarks $1$ , $0.881$ , $0.786$ , $0.515$ bits for $\rho=\operatorname{diag}(0.7,0.3)$ ; (5) confirm the Stein exponent $D$ and the strong-converse role of $\alpha>1$ (Wilde Ch. 30, Ogawa–Nagaoka 2000).

Prerequisites. Von Neumann Entropy · Quantum Relative Entropy · Data-Processing Inequality · Quantum Channels and CPTP Maps · Shannon Entropy and Classical Information Measures. Leads to. Holevo Bound · Strong Subadditivity of Quantum Entropy · Complementary Channels and Degradability.

Ask the tutor: Ask me to walk you through why the sandwiched form, and not the Petz form, is the one that closes the strong converse — or to recompute the whole divergence ladder for a pair of states you choose.

Why are there two quantum Rényi divergences instead of one?

The classical divergence rests on the product $p_x^\alpha q_x^{1-\alpha}$ , one scalar per outcome. Quantumly you replace $p^\alpha$ by $\rho^\alpha$ and $q^{1-\alpha}$ by $\sigma^{1-\alpha}$ , but because $\rho$ and $\sigma$ generally do not commute, the "product" depends on how you order the factors. The Petz form keeps them side by side, $\operatorname{Tr}[\rho^\alpha\sigma^{1-\alpha}]$ ; the sandwiched form nests $\rho$ inside symmetric powers of $\sigma$ . Both collapse to the same classical expression when the states commute, and both give the relative entropy at $\alpha=1$ , so neither is "wrong" — they are the two orderings with good properties. Which one you reach for is decided by the task, and the wider data-processing window is why strong-converse exponents want the sandwiched form.

What does the order α actually control?

Think of $\alpha$ as a magnifying glass on a spectrum. In $\operatorname{Tr}\rho^\alpha=\sum_i\lambda_i^\alpha$ , raising $\alpha$ above one makes the largest eigenvalues dominate, so the entropy reports the peaks and slides toward the min-entropy; lowering it below one boosts the small eigenvalues, so the entropy reports the support and climbs toward the max-entropy. At $\alpha=1$ the magnification is neutral and you recover the ordinary Shannon or von Neumann average. For divergences the same dial trades between the events that most betray $\rho$ against $\sigma$ (high $\alpha$ ) and a democratic average over all outcomes (low $\alpha$ ) — which is why different operational questions live at different $\alpha$ .

Why does the strong converse specifically need α greater than one?

A strong converse claims that above capacity success does not merely fail — it collapses to zero exponentially. Proving collapse needs a quantity that heavily penalises the rare events an over-ambitious code must rely on: the region where $\rho$ has large weight but $\sigma$ almost none. The order $\alpha>1$ amplifies exactly that region, so $\widetilde D_\alpha$ with $\alpha>1$ gives the exponential lower bound on failure probability a strong converse needs. Orders below one average the region away and can only certify that the exponent $D$ is achievable. The two regimes are complementary halves of the same picture.

When do the Petz and sandwiched divergences give the same number?

Exactly when the states commute, $[\rho,\sigma]=0$ , and trivially at $\alpha=1$ where both equal the Umegaki relative entropy. In the commuting case both traces collapse to the same classical sum $\sum_x p_x^\alpha q_x^{1-\alpha}$ . The moment the states fail to commute the sandwiched value drops strictly below the Petz value, by the Araki–Lieb–Thirring ordering. The worked example makes the gap concrete: at $\alpha=2$ the sandwiched divergence is $1.492$ bits versus $1.644$ bits for Petz, a difference of $0.152$ bits due entirely to non-commutativity.

What are the min-entropy and max-relative entropy good for in practice?

They are the currency of one-shot information theory, where a single use of a resource matters rather than an i.i.d. average. By the leftover hashing lemma the conditional min-entropy $H_{\min}(A|E)=-\widetilde D_\infty(\rho_{AE}\|I_A\otimes\rho_E)$ is the number of nearly perfect secret bits you can hash from a source an adversary partly knows — the heart of every QKD security proof. The max-relative entropy $D_{\max}=\widetilde D_\infty$ lower-bounds the one-shot cost of preparing a state under restricted operations, setting exchange rates in the resource theories of entanglement, coherence and athermality. Both are the $\alpha\to\infty$ extremes of the sandwiched family, which is why it is the analytic glue between one-shot and asymptotic theory.

Does using a divergence outside its α-window just make it less accurate?

No — it can make it meaningless. Data processing (monotonicity under channels) is not a nicety; it is the property that lets a divergence be read as "distinguishability", because it guarantees that no physical processing can make two states easier to tell apart. Outside its window a Rényi divergence can increase under a channel — which would say that throwing information away helps you distinguish states, an operational absurdity. So the Petz form must be confined to $\alpha\in[0,2]$ and the sandwiched form to $\alpha\in[\tfrac12,\infty]$ ; using either beyond its range produces a number that no longer bounds any error probability or conversion rate.