We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

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Consider a unital CP map Φ:AB. Using the Stinespring dilation theorem, the map decomposes as Φ(a) = WaW, where W is an isometry since Φ is unital. The action of the map on the GNS Hilbert space is given by
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Note that, by definition, ‖T = ‖T(2→2).

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Note that the algebra itself is a linear vector space.

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Since |aa⟩ is in the natural cone, it follows from (36) that the vector that saturates the Holder inequality is also in the natural cone. Therefore, in the definition of the q-norm in (39) for |aa⟩, we can restrict the supremum to the vectors |ψ1/2⟩ in the natural cone.

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We can define an alternate (p, ω, *)-norm to beap,ω,*ap,ω=ω1/pap=Δω|e1/p|ap. As opposed to the p-norm, the (p, ω)-norm is not invariant under auav with u and v unitaries. Instead, we have
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u|Ψp,ω=|Ψp,ω,u|Ψp,ω,*=|Ψp,ω,*.
In other words, for unitaries uA and uA, we haveMore generally, one can define the Kosaki (p, σ, ω)-norms ‖ap,σ,ω = ‖σ1−1/p1/pp.80 
27.
We can also define the alternate (p, ω, *)-norm of a vector
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ua|ω1/2p,ω,*uap,ω,*=a|ω1/2p,ω,*.
The (2, ω, *) is the Hilbert space norm of a|ω1/2⟩. The (p, ω, *)-norm of a vector has the advantage that it is independent of unitary rotations uA:Therefore, it only depends on the reduced state on A that is aa, and not a particular purification choice u′|a⟩.
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Note that, in finite dimensions, we can take θ > 1 as well. However, in this work, we restrict to the range because it generalizes to infinite dimensions.

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We remind the reader that a contraction is defined with respect to the infinity norm, and not any other norms we discuss in this work.

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See Theorem 2.1 of Zhang9 for a proof of the data processing inequality in the extended range of (θ, r) for matrix algebras.

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In general, the range of a CP map is a *-closed subspace of observables inside B(HA), otherwise known as an operator system.

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See Petz45 and Witten33 for a review of its proof using the Tomita–Takesaki modular theory.

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Consider the CP map that sends all states to the same ωB. After the channel, the measure is zero. Since it has not increased, it was non-negative before applying the channel.

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Since the measure does not depend on ⃗α, we suppress it in the notation.

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