Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and $f$ -divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.

arXiv:2507.07065Jul 9, 20250 citations

Hockey stick $f$ -divergences

Fumio Hiai, Milán Mosonyi, Marco Tomamichel

In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$ -divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick $f$ -divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick $f$ -divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$ -divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$ . Moreover, we give some partial results on the characterization of when different notions of quantum $f$ -divergences give the same value on a pair of quantum states.In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick $f$-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$-divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$. Moreover, we give some partial results on the characterization of when different notions of quantum $f$-divergences give the same value on a pair of quantum states.

quant-pharXiv:2607.08760Jul 10, 20260 citations

Some properties and applications of the new quantum $f$ -divergencesSome properties and applications of the new quantum $f$-divergences

Salman Beigi, Christoph Hirche, Marco Tomamichel

Recently, a new definition for quantum $f$ -divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum $f$ -divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum $f$ -divergences. We leverage these expressions to prove new properties of these $f$ -divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new $f$ -divergences, and prove inequalities between these divergences for various functions.Recently, a new definition for quantum $f$-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum $f$-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum $f$-divergences. We leverage these expressions to prove new properties of these $f$-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new $f$-divergences, and prove inequalities between these divergences for various functions.

arXiv:2501.03799Jan 7, 20250 citations

Investigating properties of a family of quantum Rényi divergences

Mingyan Simon Lin, Marco Tomamichel

arXiv:1408.6897Quantum Information ProcessingFeb 2, 201533 citations

Integral representations of quantum Rényi divergences on von Neumann algebras

Tanay Kibe, Pratik Roy

We prove an integral representation of the $α$ - $z$ Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray $z=cα$ , the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders $α$ above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup $L^p$ scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional $α$ - $z$ entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.We prove an integral representation of the $α$-$z$ Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray $z=cα$, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders $α$ above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup $L^p$ scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional $α$-$z$ entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.

quant-pharXiv:2608.21214Aug 24, 20260 citations

f

-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.

In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum

f

-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick

f

-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick

f

-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi

α

-divergences coincide with the Petz-type Rényi divergences for

α\in(0,1)

and with the sandwiched Rényi divergences for

α>1

. Moreover, we give some partial results on the characterization of when different notions of quantum

f

-divergences give the same value on a pair of quantum states.

Some properties and applications of the new quantum

f

-divergences

Recently, a new definition for quantum

f

-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum

f

-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum

f

-divergences. We leverage these expressions to prove new properties of these

f

-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new

f

-divergences, and prove inequalities between these divergences for various functions.

We prove an integral representation of the

α

z

Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray

z=cα

, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders

α

above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup

L^p

scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional

α

z

entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.