Strong Converse Exponents of Quantum Soft Covering and Privacy Amplification

Abstract: We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders $α\in[\frac{1}{2},\infty)$. For $α\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $α\in[1,\infty)$, it is characterized by the order-$α$ sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for $α\in(2,\infty)$, expressed in terms of the corresponding order-$α$ sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $α\in[\frac{1}{2},1)$.

• The paper derives exact soft-covering exponents: for α≥1, failure is governed by the positive part of Rényi mutual information minus the coding rate, while α<1 requires a new two-parameter club-sandwiched quantity.
• The paper completes the privacy-amplification exponent for α>2 as the positive part of the output rate minus the order-α conditional Rényi entropy, complementing prior results across α≥1/2.
• The paper introduces an exact exponential-rate analysis of Peetre’s K-functional in noncommutative Lp spaces, combining interpolation, reverse Rosenthal inequalities, and tensorization to prove achievability and optimality.

This paper determines the exact strong converse exponents of two fundamental randomization tasks—quantum soft covering and quantum privacy amplification against quantum side information—when the approximation error is measured by the sandwiched Rényi divergence. For soft covering with i.i.d. random codebooks, the exponent is characterized for every order $\alpha \in [\tfrac{1}{2},\infty)$: for $\alpha \in [1,\infty)$ it is the positive part of the order-$\alpha$ sandwiched Rényi mutual information minus the rate, while for $\alpha \in [\tfrac{1}{2},1)$ it involves a new two-parameter quantity, the club-sandwiched mutual information. For privacy amplification, the paper closes the remaining regime $\alpha > 2$ with an exponent given by the positive part of the rate minus the order-$\alpha$ sandwiched Rényi conditional entropy. The central technical innovation is an exact exponential-rate analysis of Peetre's $K$-functional in noncommutative $L_p$ spaces.

Problem setting

The starting point is a classical–quantum (C–Q) state $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$. In quantum soft covering, a random codebook $\mathcal{C} = \{X_1,\dots,X_M\}$ of i.i.d. codewords drawn from $\alpha \in [1,\infty)$0 induces the state $\alpha \in [1,\infty)$1, which should approximate the average output $\alpha \in [1,\infty)$2. In privacy amplification, a hash function $\alpha \in [1,\infty)$3 applied to $\alpha \in [1,\infty)$4 should render the key $\alpha \in [1,\infty)$5 nearly uniform and independent of the adversary's system $\alpha \in [1,\infty)$6, i.e., the hashed C–Q state should approximate $\alpha \in [1,\infty)$7.

Both tasks are analyzed under the order-$\alpha \in [1,\infty)$8 sandwiched Rényi divergence $\alpha \in [1,\infty)$9, restricted to $\alpha$0 because only this range satisfies the data-processing inequality—a structural requirement for an operational distinguishability measure. The strong converse exponents are defined as asymptotic normalized limits:

$\alpha$1

and analogously for privacy amplification with a minimization over hash functions $\alpha$2.

Main results

For soft covering at orders $\alpha$3, the exact strong converse exponent is

$\alpha$4

where $\alpha$5. This single-letter formula exhibits the expected threshold behavior: covering fails exponentially precisely when the coding rate falls below the Rényi mutual information.

For $\alpha$6, no single Rényi order suffices; the exponent is instead

$\alpha$7

where the newly introduced two-parameter club-sandwiched mutual information is defined via an optimization over auxiliary states $\alpha$8 of a mixed trace functional interpolating between $\alpha$9 and $\alpha \in [\tfrac{1}{2},1)$0 weights. At the boundary values $\alpha \in [\tfrac{1}{2},1)$1 and $\alpha \in [\tfrac{1}{2},1)$2 this quantity reduces continuously to $\alpha \in [\tfrac{1}{2},1)$3 and to the von Neumann mutual information respectively, so the supremum over the closed interval $\alpha \in [\tfrac{1}{2},1)$4 is well defined. To the authors' knowledge, this is the first exact strong converse exponent for quantum soft covering and the first precise operational interpretation of the club-sandwiched mutual information in the quantum setting.

For privacy amplification at $\alpha \in [\tfrac{1}{2},1)$5, the result is

$\alpha \in [\tfrac{1}{2},1)$6

with $\alpha \in [\tfrac{1}{2},1)$7. Combined with Li–Yao–Hayashi's result for $\alpha \in [\tfrac{1}{2},1)$8 and Rubboli–Tomamichel's composable fidelity exponent at $\alpha \in [\tfrac{1}{2},1)$9 (which extends to all $\alpha > 2$0), this completes the characterization of privacy amplification exponents across the entire data-processing range $\alpha > 2$1.

The $\alpha > 2$2-functional method

The proof for $\alpha > 2$3 rests on a noncommutative harmonic analysis toolkit. Working in a finite von Neumann algebra $\alpha > 2$4 with subalgebra $\alpha > 2$5 and conditional expectation $\alpha > 2$6, the authors consider the noncommutative $\alpha > 2$7 space $\alpha > 2$8 together with the conditional column space $\alpha > 2$9, both multiplicative under tensor products and reflexive. For $\alpha$0 decomposed as $\alpha$1, Peetre's $\alpha$2-functional is

$\alpha$3

and its exponential rate $\alpha$4 is shown, via Fekete's lemma and supermultiplicativity, to be finite, nondecreasing, 1-Lipschitz, and concave in $\alpha$5.

The key structural theorem identifies this rate exactly through complex interpolation:

$\alpha$6

The upper bound follows from complex interpolation of the embedding maps into the sum space; the reverse inequality uses the real-to-complex interpolation embedding plus a sharp tail-splitting argument on the integral representation, exploiting the piecewise-linear majorization of $\alpha$7. Tensorization of the interpolation norm is established via Calderón duality and reflexivity of the endpoint spaces.

Achievability and optimality for $\alpha$8

Achievability proceeds by fixing the optimizer $\alpha$9 of the club-sandwiched mutual information and bounding $K$0 through a chain of Hölder's inequality (with indices satisfying $K$1), Jensen's inequality, and McCarthy's inequality, yielding

$K$2

which tensorizes directly since the relevant quantities are additive over product states.

Optimality is more delicate. A one-shot lower bound on $K$3 is obtained by encoding the quantity of interest as the Schatten $K$4-norm ($K$5) of a block matrix built from $K$6, then applying the reverse matrix Rosenthal inequality of Junge–Xu after Rademacher symmetrization; the row term is absorbed into the diagonal term using concavity of $K$7 for $K$8. The resulting infimum over decompositions $K$9 is exactly the $L_p$0-functional, whose exponential rate is computed by identifying the interpolation space $L_p$1 with an amalgamated factorization space via Junge–Parcet's amalgamated interpolation theorem. This identification yields

$L_p$2

so that Fenchel–Moreau inversion of the concave rate function produces precisely the club-sandwiched formula. The endpoint $L_p$3 is handled separately by a limiting argument $L_p$4, requiring joint upper semicontinuity of the objective in $L_p$5, established through an $L_p$6-regularization of the trace functional and compactness of the state space.

Orders $L_p$7

For $L_p$8, the one-shot achievability bound combines Hayashi's pinching inequality with operator concavity of $L_p$9 and data processing, giving

$$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$0

where the pinching factor $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$1 grows only polynomially in $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$2 and hence vanishes under the exponential normalization. For $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$3, the authors leverage their earlier mixed-order one-shot bounds involving $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$4, combined with the ordering $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$5, to obtain a uniform one-shot bound implying the same exponent. The lower bound for all $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$6 follows from McCarthy's inequality applied termwise over the codebook. The case $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$7 is obtained by a chain-rule argument on the mutual information $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$8 with a uniformly chosen index $$\rho_{XE} = \sum_x P_X(x)|x\rangle\langle x| \otimes \rho_E^x$$9, showing $\mathcal{C} = \{X_1,\dots,X_M\}$0, together with monotonicity of $\mathcal{C} = \{X_1,\dots,X_M\}$1 in $\mathcal{C} = \{X_1,\dots,X_M\}$2 to pass from $\mathcal{C} = \{X_1,\dots,X_M\}$3.

Privacy amplification for $\mathcal{C} = \{X_1,\dots,X_M\}$4

The achievability direction uses random binning functions and the authors' prior one-shot bound expressed through the mixed-order conditional entropy $\mathcal{C} = \{X_1,\dots,X_M\}$5, again combined with the ordering $\mathcal{C} = \{X_1,\dots,X_M\}$6 to produce a bound of the form $\mathcal{C} = \{X_1,\dots,X_M\}$7. Additivity of $\mathcal{C} = \{X_1,\dots,X_M\}$8 under tensor products then yields the exponent. The matching converse was already known from Li–Yao–Hayashi. An implication worth noting is that the same club-sandwich methodology developed here for soft covering can be adapted to give the low-order ($\mathcal{C} = \{X_1,\dots,X_M\}$9) privacy amplification exponents directly, providing an alternative route to the Rubboli–Tomamichel result.

Limitations and open questions

Several restrictions are inherent to the analysis. The restriction to $\alpha \in [1,\infty)$00 is structural—the sandwiched Rényi divergence fails data processing below this order—but leaves the sub-$\alpha \in [1,\infty)$01 regime outside the framework entirely. The soft-covering result is stated for i.i.d. random codebooks; constant-composition ensembles are not treated. Under the trace distance criterion, exact strong converse exponents remain open for both tasks: Cheng–Gao established only a lower bound for soft covering, and Shen–Gao–Cheng and Salzmann–Datta only lower bounds for privacy amplification. Finally, whether the two-parameter club-sandwiched mutual information admits operational interpretations beyond these two tasks is left unresolved.

Conclusion

The paper delivers exact strong converse exponents for quantum soft covering across the full data-processing range of the sandwiched Rényi divergence, revealing a phase transition at order one: above it, a single-order Rényi mutual information suffices; below it, a genuine two-parameter optimization over club-sandwiched quantities is necessary. The complementary structure between the two tasks—soft covering failure governed by mutual information exceeding the rate, privacy amplification failure governed by the rate exceeding the conditional entropy—is made precise. Methodologically, the exact exponential-rate theorem for the noncommutative $\alpha \in [1,\infty)$02-functional, combined with amalgamated interpolation and reverse Rosenthal inequalities, constitutes a transferable analytic machinery likely applicable to other exponential analyses in quantum information theory.