# Data processing for the sandwiched Rényi divergence: a condition for equality

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## Abstract

The \(\alpha \)-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for \(\alpha \ge 1/2\). In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the \(\alpha \)-sandwiched Rényi divergence for all \(\alpha \ge 1/2\). For the range \(\alpha \in [1/2,1)\), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

## Keywords

Relative entropies Rényi entropies Data processing inequality Equality condition Conditional entropy Entanglement of formation## Mathematics Subject Classification

47N50 81P40 81P45 94A17 94A40## 1 Introduction

^{1}

*z*-Rényi relative entropies [2]. For \(\alpha \in (0,\infty ){\setminus } \lbrace 1\rbrace \) and positive semidefinite operators \(\rho \) and \(\sigma \), the \(\alpha \)-RRE \(D_\alpha (\rho \Vert \sigma )\) [39] is defined as

*f*-divergences). We discuss this result in Sect. 5.

## 2 Main result

Our main result in this paper, Theorem 1 below, is a necessary and sufficient condition for equality in the DPI (3). In order to state it properly, we first introduce some necessary notation and terminology.

Throughout this paper we only consider finite-dimensional Hilbert spaces. All logarithms are taken to base 2. For a Hilbert space \(\mathcal {H}\) we write \(\mathcal {B}(\mathcal {H})\) for the algebra of linear operators on \(\mathcal {H}\), and we denote by \(\mathcal {P}(\mathcal {H}):= \lbrace \rho \in \mathcal {B}(\mathcal {H}):\rho \ge 0\rbrace \) and \(\mathcal {D}(\mathcal {H}):= \lbrace \rho \in \mathcal {P}(\mathcal {H}):\mathrm{Tr\,}\rho =1\rbrace \) the sets of positive semidefinite operators and density matrices (or quantum states), respectively. We denote by \(\mathrm{rk}\,\ A\) the rank of an operator *A*, and by \(\mathrm{\,supp\,}A\) the support of *A*, i.e. the orthogonal complement of the kernel of *A*. We write \(A \not \perp B\) if \(\mathrm{\,supp\,}A \cap \mathrm{\,supp\,}B\) contains at least one non-zero vector. For a Hermitian operator *A*, we denote by \(\mathrm{spec}\,A\subseteq \mathbb {R}\) the set of eigenvalues of *A*. For a pure state \(|\psi \rangle \in \mathcal {H}\) we write \(\psi =|\psi \rangle \langle \psi |\in \mathcal {D}(\mathcal {H})\) for the corresponding rank-1 density matrix. Given a linear map \(\mathcal {L}:\mathcal {B}(\mathcal {H})\rightarrow \mathcal {B}(\mathcal {K})\) between Hilbert spaces \(\mathcal {H}\) and \(\mathcal {K}\), the adjoint map \(\mathcal {L}^\dagger :\mathcal {B}(\mathcal {K})\rightarrow \mathcal {B}(\mathcal {H})\) is the unique map satisfying \(\langle \mathcal {L}^\dagger (Y),X\rangle = \langle Y,\mathcal {L}(X)\rangle \) for all \(X\in \mathcal {B}(\mathcal {H})\) and \(Y\in \mathcal {B}(\mathcal {K})\), where \(\langle A,B\rangle := \mathrm{Tr\,}(A^\dagger B)\) is the Hilbert-Schmidt inner product. A linear map \(\Phi :\mathcal {B}(\mathcal {H})\rightarrow \mathcal {B}(\mathcal {K})\) between Hilbert spaces \(\mathcal {H}\) and \(\mathcal {K}\) is called *n*-positive if \(\mathrm{id}_n\otimes \Phi :\mathcal {B}(\mathbb {C}^n)\otimes \mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathbb {C}^n)\otimes \mathcal {B}(\mathcal {K})\) is positive, where \(\mathrm{id}_n\) denotes the identity map on \(\mathcal {B}(\mathbb {C}^n)\). A map is completely positive if it is *n*-positive for all \(n\in \mathbb {N}\). A quantum operation (or quantum channel) \(\Lambda \) between Hilbert spaces \(\mathcal {H}\) and \(\mathcal {K}\) is a linear, completely positive, and trace-preserving map \(\Lambda :\mathcal {B}(\mathcal {H})\rightarrow \mathcal {B}(\mathcal {K})\).

Our main result is given by the following theorem:

### Theorem 1

For \(\alpha > 1\) and positive trace-preserving maps, Theorem 1 was also proved using the framework of non-commutative \(L_p\)-spaces [13]. The case of equality in the DPI for the \(\alpha \)-SRD was also discussed in two papers by Hiai and Mosonyi [22] and Jenčová [25], both of which focused on the aspect of sufficiency. The connections between Theorem 1 and these results are discussed in Sect. 5. The rest of this paper is organized as follows: In Sect. 3, we analyze the proof of the DPI (3) for the \(\alpha \)-SRD as given in [17], extracting a necessary and sufficient condition for equality in (3) and thus proving Theorem 1. We present applications of Theorem 1 to entanglement and distance measures in Sect. 4. Finally, in Sect. 5 we compare our result to the recoverability/sufficiency results mentioned above and state some open questions.

## 3 Proof of the main result

*U*and any state \(\tau \), we have

*AB*and

*A*indicate the Hilbert spaces \(\mathcal {H}_{AB}=\mathcal {H}_A\otimes \mathcal {H}_B\) and \(\mathcal {H}_A\) on which the density matrices act, and the partial trace is taken over the

*B*system. Since the logarithm is monotonically increasing, the monotonicity of \(\widetilde{D}_\alpha (\cdot \Vert \cdot )\) under partial trace (6) is in turn equivalent to the following monotonicity properties of \(\widetilde{Q}_\alpha (\cdot \Vert \cdot )\):

*B*. The crucial ingredient in proving (7) is then the joint concavity/convexity of the trace functional \(\widetilde{Q}_\alpha (\cdot \Vert \cdot )\):

### Proposition 2

([17]) The functional \((\rho ,\sigma )\mapsto \widetilde{Q}_\alpha (\rho \Vert \sigma )\) is jointly concave for \(\alpha \in [1/2,1)\) and jointly convex for \(\alpha \in (1,\infty )\).

### Remark 3

The joint convexity/concavity of the trace functional \(\widetilde{Q}_\alpha (\cdot \Vert \cdot )\) is a special case of the joint convexity/concavity of a more general trace functional underlying the \(\alpha \)-z-Rényi relative entropies mentioned in Sect. 1, which was proved by Hiai [21] using the theory of Pick functions. A more accessible proof can be found in the arXiv version of [2].

*H*and the respective ranges of \(\alpha \). Moreover, in the course of proving the validity of (10), Frank and Lieb [17] show that for fixed \(\rho ,\sigma \) a critical point of \(f_\alpha (H,\rho ,\sigma )\) satisfying \(\partial f_\alpha (H,\rho ,\sigma )/\partial H = 0\) is given by

### Theorem 4

Let *A* be a Hermitian matrix with \(\mathrm{spec} \ A\subseteq \mathcal {D}\subseteq \mathbb {R}\), and let \(g:\mathcal {D}\rightarrow \mathbb {R}\) be a continuous, (strictly) convex function. Then the function \(A\mapsto \mathrm{Tr\,}g(A)\) is (strictly) convex.

Let us first consider \(\alpha >1\). The function \(H\mapsto \mathrm{Tr\,}[( \sigma ^{-\gamma } H \sigma ^{-\gamma })^{\alpha /(\alpha -1)}]\) is the composition of the linear function \(X\mapsto \sigma ^{-\gamma }X\sigma ^{-\gamma }\) and the functional \(A\mapsto \mathrm{Tr\,}A^{\alpha /(\alpha -1)}\), the latter being strictly convex by Theorem 4 upon choosing \(g:\mathbb {R}_+\rightarrow \mathbb {R}_+,\, g(x)= x^{\alpha /(\alpha -1)}\). As \(\alpha >1\), the function \(H\mapsto -(\alpha -1)\mathrm{Tr\,}[( \sigma ^{-\gamma } H \sigma ^{-\gamma })^{\alpha /(\alpha -1)}]\) is, therefore, strictly concave, and hence, \(f_\alpha (H,\rho ,\sigma )\) is strictly concave, since it is the sum of a linear function and a strictly concave function. In the case \(\alpha \in [1/2,1)\), a similar argument shows that \(f_\alpha (H,\rho ,\sigma )\) is strictly convex.

### Proposition 5

We are now in a position to prove our main result, Theorem 1:

### Proof of Theorem 1

*U*acting on \(\mathcal {H}\otimes \mathcal {H}'\otimes \mathcal {K}\) such that for every \(\rho \in \mathcal {B}(\mathcal {H})\) we have

*f*and unitary

*U*, this is equivalent to

## 4 Applications

In this section we discuss applications of Theorem 1. Our goal is to generalize a set of results by Carlen and Lieb [9] about the Araki–Lieb inequality and entanglement of formation by proving the corresponding results for Rényi quantities. In Sect. 4.1 we state a Rényi version of the Araki–Lieb inequality (Lemma 8) and analyze the case of equality (Theorem 9). In Sect. 4.2 we first prove a general lower bound on the Rényi entanglement of formation (analogous to the corresponding bound on the entanglement of formation in [9]) and then use the results from Sect. 4.1 to show that this lower bound is achieved by states saturating the Rényi version of the Araki–Lieb inequality. These results are presented in Theorem 13. Finally, in Sect. 4.3 we discuss the case of equality in a well-known upper bound on the entanglement fidelity in terms of the usual fidelity, which we state in Proposition 15.

### Proposition 6

### 4.1 Rényi version of Araki–Lieb inequality and the case of equality

### Theorem 7

- (i)
\(r_B = r_A r_{AB}\)

- (ii)The state \(\rho _{AB}\) has a spectral decomposition of the formwhere the vectors \(\lbrace |i\rangle _{AB}\rbrace _{i=1}^{r_{AB}}\) are such that \(\mathrm{Tr\,}_B|i\rangle \langle j|_{AB} = \delta _{ij}\rho _A\) for \(i,j=1,\ldots ,r_{AB}\).$$\begin{aligned} \rho _{AB} = \sum _{i=1}^{r_{AB}} \lambda _i |i\rangle \langle i|_{AB}, \end{aligned}$$

### Lemma 8

### Theorem 9

- (i)
\(r_B = r_A r_{AB}.\)

- (ii)The state \(\rho _{AB}\) has a spectral decomposition of the formwhere the vectors \(\lbrace |i\rangle _{AB}\rbrace _{i=1}^{r_{AB}}\) are such that \(\mathrm{Tr\,}_B|i\rangle \langle j|_{AB} = \delta _{ij}\rho _A\) for \(i,j=1,\ldots ,r_{AB}\).

### Remark 10

For the upper bound \(\widetilde{S}_\alpha (A|B)_\rho \le S_\alpha (A)_\rho \) in Lemma 8, we have equality if and only if \(\widetilde{D}_\alpha (\rho _{AB}\Vert \mathbbm {1}_A\otimes \tilde{\sigma }_B) = \widetilde{D}_\alpha (\rho _A\Vert \mathbbm {1}_A)\), where \(\tilde{\sigma }_B\) is a state optimizing \(\widetilde{S}_\alpha (A|B)_\rho \). Similar to above, we obtain from Proposition 5 that this is the case if and only if \(\rho _{AB} = \rho _A\otimes \tilde{\sigma }_B\).

### 4.2 Rényi entanglement of formation

### Theorem 11

### Remark 12

If \(S(A|B)_\rho = -S(A)_\rho \), then \(E_F(\rho _{AB}) = -S(A|B)_\rho \ge -S(B|A)_\rho \) by (23).

### Theorem 13

### Proof

*P*and

*Q*on an alphabet \(\mathcal {X}\) as \(D(P\Vert Q) = \sum _{x\in \mathcal {X}} P(x)\log P(x)/Q(x)\), provided that \(P(x)=0\) whenever \(Q(x)=0\). Note that the latter is satisfied as \(\mathrm{\,supp\,}\rho _C\subseteq \mathrm{\,supp\,}\tilde{\sigma }_C\) holds for the optimizing state \(\tilde{\sigma }_C\) of \(\widetilde{S}_\alpha (A|C)_\rho \) [36]. The bound \(E_{F,\alpha }(\rho _{AB})\ge -\widetilde{S}_\beta (B|A)_\rho \) follows in an analogous way, yielding (24).

### Remark 14

- (i)The proof method of the lower bound (24) for \(E_{F,\alpha }(\rho _{AB})\) in Theorem 13 can be specialized to the quantum relative entropy \(D(\cdot \Vert \cdot )\), providing a new proof of (23) in Theorem 11:The bound \(S(B|A)_\rho \ge - E_F(\rho _{AB})\) can be proved in an analogous way.$$\begin{aligned} S(A|B)_\rho&= - S(A|C)_\rho \\&= D(\rho _{AC}\Vert \mathbbm {1}_A\otimes \rho _C)\\&= D\!\left( \sum _{i,j} \sqrt{q_i q_j} \mathrm{Tr\,}_B|\phi _i\rangle \langle \phi _j|_{AB}\otimes |i\rangle \langle j|_{C} \,\Vert \,\mathbbm {1}_A\otimes \rho _C\right) \\&\ge D\!\left( \sum _i q_i \mathrm{Tr\,}_B\phi _i \otimes |i\rangle \langle i|_C \,\Vert \,\mathbbm {1}_A \otimes \sum _i \lambda _i |i\rangle \langle i|_C\right) \\&= D(\lbrace q_i\rbrace \Vert \lbrace \lambda _i\rbrace ) + \sum _i q_i D\!\left( \mathrm{Tr\,}_B\phi _i\Vert \mathbbm {1}_A\right) \\&= D(\lbrace q_i\rbrace \Vert \lbrace \lambda _i\rbrace ) - \sum _i q_i S(\mathrm{Tr\,}_B\phi _i)\\&\ge - E_F(\rho _{AB}). \end{aligned}$$
- (ii)
If a state \(\rho _{AB}\) satisfies \(\widetilde{S}_\beta (A|B)_\rho = -S_\alpha (A)_\rho \) for \(a>1\) and \(\beta = \alpha /(2\alpha -1)\), then \(E_{F,\alpha }(\rho _{AB}) = -\widetilde{S}_\beta (A|B)_\rho \ge -\widetilde{S}_\beta (B|A)_\rho \) by (24) in Theorem 13.

### 4.3 Entanglement fidelity

^{2}

^{3}This can be used to prove the following upper bound on the entanglement fidelity, where we write \(\mathcal {N}(\psi ^\rho )\equiv (\mathcal {N}\otimes \mathrm{id}_{\mathcal {H}'})(\psi ^\rho )\):

### Proposition 15

### Proof

## 5 Conclusion and open questions

Taking a closer look at the condition (30) for equality in the DPI for the \(\alpha \)-SRD, it is easy to see that choosing \(\alpha =2\) in (30) yields precisely the statement \(\mathcal {R}_{\sigma ,\Lambda }(\Lambda (\rho )) = \rho \). Hence, in the case \(\alpha =2\) we have equality in the DPI for the 2-SRD if and only if the recovery map \(\mathcal {R}_{\sigma ,\Lambda }\) defined in (32) satisfies (31). This was already observed in [13] for positive trace-preserving maps.

Shortly after completion of the present paper, the connection between sufficiency and equality in the DPI for the \(\alpha \)-SRD was presented by Jenčová [25] and Hiai and Mosonyi [22]. The main result of [25] is that a 2-positive trace-preserving map \(\Lambda \) is sufficient with respect to \(\lbrace \rho ,\sigma \rbrace \) if and only if \(\widetilde{D}_\alpha (\Lambda (\rho )\Vert \Lambda (\sigma )) = \widetilde{D}_\alpha (\rho \Vert \sigma )\) holds for some \(\alpha >1\). By the theorem of Petz [40] mentioned above, we, therefore, have equality in the DPI for the \(\alpha \)-SRD for any \(\alpha >1\) if and only if the map \(\mathcal {R}_{\sigma ,\Lambda }\) defined in (32) satisfies (31). Furthermore, a similar argument as in (33) for \(\widetilde{D}_\alpha (\cdot \Vert \cdot )\) shows that equality holds in the DPI for the \(\alpha \)-SRD for all \(\alpha >1\) if it holds for some \(\alpha >1\). This result settles the sufficiency question for the \(\alpha \)-SRD for the range \(\alpha >1\) and 2-positive trace-preserving maps (which include all quantum operations). In [22] sufficiency is analyzed for 2-positive bistochastic maps \(\Lambda \), that is, both \(\Lambda \) and \(\Lambda ^\dagger \) are 2-positive and trace-preserving. The main theorem of [22] regarding the \(\alpha \)-SRD states conditions for sufficiency of \(\Lambda \) for certain ranges of \(\alpha \) (including the range \(\alpha \in [1/2,1)\)) under the additional assumption that one of the two states \(\rho \) and \(\sigma \) is a fixed point of \(\Lambda \). In fact, this result is obtained as a corollary of a more general theorem analyzing sufficiency for the \(\alpha \)-*z*-Rényi relative entropies under similar assumptions.

It the light of our main result (Theorem 1) and the results of [22] and [25], it remains an open question whether equality in the DPI for the \(\alpha \)-SRD in the range \(\alpha \in (1/2,1)\) is equivalent to sufficiency of \(\Lambda \) in our setting, in which \(\Lambda \) is an arbitrary quantum operation and \(\rho \) and \(\sigma \) are states with \(\rho \not \perp \sigma \). Note that for \(\alpha =1/2\) it is known that such a general sufficiency result cannot hold [32].^{4}

Regarding our results in Sect. 4 about entanglement measures and distances, it would be interesting to see whether the entropic bounds proved therein can be used to characterize information-theoretic tasks.

## Footnotes

- 1.
See Sect. 2 for definitions and notation.

- 2.
For an arbitrary operator

*A*, the trace norm is defined as \(\Vert A\Vert _1 = \mathrm{Tr\,}\sqrt{A^\dagger A}\). - 3.
Note that this can also be proved directly, e.g. via Uhlmann’s Theorem [48].

- 4.
This can be seen as follows: The 1 / 2-SRD can be expressed in terms of the fidelity as \(\widetilde{D}_{1/2}(\rho \Vert \sigma ) = -2\log F(\rho ,\sigma )\). It is well-known that for given \(\rho \) and \(\sigma \) there exists a measurement \(M=\lbrace M_x\rbrace _{x\in \mathcal {X}}\) for some finite set \(\mathcal {X}\) such that the fidelity \(F(\rho ,\sigma )\) is equal to the classical fidelity of the measurement outcomes \(\lbrace \mathrm{Tr\,}(M_x\rho )\rbrace _{x\in \mathcal {X}}\) and \(\lbrace \mathrm{Tr\,}(M_x\sigma )\rbrace _{x\in \mathcal {X}}\) obtained from measuring \(\rho \) and \(\sigma \) (see e.g. [51]). Hence, for any two states \(\rho \) and \(\sigma \) we have equality in the DPI for the 1 / 2-SRD with respect to the quantum operation \(\mathcal {M}(\omega ) = \sum _{x\in \mathcal {X}} \mathrm{Tr\,}(\omega M_x) |x\rangle \langle x|\), and it is impossible to recover the states \(\rho \) and \(\sigma \) from the measurement outcomes alone. This proves that a general sufficiency result as stated above cannot hold for \(\alpha =1/2\).

## Notes

### Acknowledgements

ND is grateful to Anna Jenčová and Mark M. Wilde for earlier discussions on the issue of equality in the DPI for the \(\alpha \)-SRD in the \(L_p\)-space framework during the workshop ‘Beyond IID in Information Theory’ (5–10 July 2016) in Banff, Canada. The authors would also like to thank Will Matthews and Michał Studziński for interesting discussions.

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