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SECTION I

INTRODUCTION

MUTUAL information has been an important concept from the beginning of information theory. In classical information theory, the Shannon mutual information, Formula TeX Source $$I({\rm A}:{\rm B})=H({\rm A})-H({\rm A}\vert{\rm B}),\eqno{\hbox{(1)}}$$ serves as a measure for the capacity of communication channels [1]. In quantum information theory, its analogue is given by the von Neumann mutual information which is defined in terms of von Neumann entropy in the same way as in (1). It generally presents a measure of correlation between the subsystems A and B of a composite quantum system. The operational drawback of these quantities from a practical point of view is that they only characterize processes under the assumption that they can be repeated an arbitrary number of times and that these repetitions are completely uncorrelated. In other words, the assumption states that the available resources are independent and identically distributed or i.i.d. However, this assumption is not justified in more realistic settings. Channels for instance need not be memoryless and the outputs for consecutive inputs may therefore be correlated. Also, assuming an i.i.d. structure in cryptographic protocols may compromise their security since an adversary may perform an attack that is not i.i.d. A great amount of research has consequently been devoted to scenarios where the resources are not i.i.d., commonly called the one-shot setting. This scenario is not only closer to realistic communication settings, but can also be regarded as strictly more general. The i.i.d. case is a limiting case and can thus be reproduced from one-shot results. Hence, one-shot information theory also serves as a method for proving i.i.d. statements.

In order to characterize processes in the one-shot scenario, the smooth min- and max-entropies Formula${H^{\varepsilon}_{\min}}$ and Formula${H^{\varepsilon}_{\max}}$ have been introduced [2], [3] and studied extensively both operationally and formally (see for example [4], [5], [6], [7], [8], [9]). They satisfy properties like data processing inequalities [3], [10] and a set of chain rules [8], [11]. Operationally, min- and max-entropies can be used to characterize various information theoretic tasks, including randomness extraction and state merging [4]. When the i.i.d.-limit is taken, i.e., if we evaluate them on average over Formula$n$ for states of the form Formula$\rho^{\otimes n}=\rho\otimes\rho\otimes\cdots\otimes\rho$ with asymptotically large Formula$n$, they indeed reproduce the von Neumann entropy [5], [10] (this is called the quantum asymptotic equipartition property, or QAEP). Furthermore, smooth entropies have been shown to be asymptotically equivalent to an independent approach to non-asymptotic information theory [7], [12], [13], namely the information spectrum method as introduced by Han and Verdú in classical information theory [14], [15] and later generalized to the quantum setting by Nagaoka, Hayashi, Bowen, and Datta [16], [17], [18]. In light of the success of the smooth entropy formalism, the question arises of how it can be extended to mutual information in a meaningful way.

Recent research has produced a whole variety of expressions that appear to be useful one-shot generalizations for mutual information. Motivated by (1), generalized mutual information quantities can be defined as Formula TeX Source $$\eqalignno{& I^{\varepsilon}_{\rm gen}({\rm A}:{\rm B})\mathrel{\mathop:}=H_{\min}^{\varepsilon}({\rm A})-H_{\min}^{\varepsilon}({\rm A}\vert{\rm B})\cr&\qquad~~{\rm or}~H_{\min}^{\varepsilon}({\rm A})\qquad~\,- H_{\max}^{\varepsilon}({\rm A}\vert{\rm B})\cr&\qquad~~{\rm or}~H_{\max}^{\varepsilon}({\rm A})\qquad~-H_{\max}^{\varepsilon}({\rm A}\vert{\rm B})\cr&\qquad~~{\rm or}~H_{\max}^{\varepsilon}({\rm A})\qquad~- H_{\min}^{\varepsilon}({\rm A}\vert{\rm B}).&{\hbox{(2)}}}$$ Several of these expressions have been found to have useful applications as bounds on one-shot capacities [19], [20]or in the study of area laws in quantum statistical physics [21].

On the other hand, it is well known that the von Neumann entropy and mutual information can be defined as special cases of the quantum relative entropy Formula TeX Source $${D\left({\rho}\Vert{\sigma}\right)}={\mathop{\rm tr}}\left(\rho (\log\rho-\log\sigma)\right)\!,\eqno{\hbox{(3)}}$$ where tr denotes the trace and log is the logarithm with base 2 throughout the paper. Therefore, it appears natural to define generalized information theoretic quantities in terms of generalized relative entropies. Min- and max-entropies for example are derived from the max- and min-relative entropy [7], Formula TeX Source $${D_{\max}\left({\rho}\Vert{\sigma}\right)}=\min\{\lambda\vert 2^{\lambda}\sigma\geq \rho\}\eqno{\hbox{(4)}}$$ and Formula TeX Source $${D_{\min}\left({\rho}\Vert{\sigma}\right)}=-\log\left\Vert{\sqrt\rho\sqrt\sigma}\right\Vert_{1}^{2},\eqno{\hbox{(5)}}$$ respectively. In this paper, we focus on the mutual information quantity corresponding to the max-relative entropy, called the max-information. Recent work has established the max- information as a relevant quantity in different information theoretic tasks. It has been identified by Berta et al. as a measure for the quantum communication cost of state splitting and state merging protocols [22], [23]. In addition, Datta et al. found the smooth max-information to characterize the minimal one-shot qubit compression size for a quantum rate distortion code [24]. Apart from its information theoretic applications, it also appears to give a good characterization for the amount of correlation in spin systems [25]. However, there is again a priori no unique way in which such a quantity should be defined. Von Neumann mutual information can be defined in multiple ways in terms of the quantum relative entropy Formula${D\left({\rho}\Vert{\sigma}\right)}$ [26], since Formula TeX Source $$\eqalignno{I({\rm A}:{\rm B})_{\rho}=&\,{D\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\rho_{\rm B}}\right)}&{\hbox{(6)}}\cr=&\,\min_{\sigma_{\rm B}}{D\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}&{\hbox{(7)}}\cr=&\,\min_{\sigma_{\rm A},\sigma_{\rm B}}{D\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)},&{\hbox{(8)}}}$$ where the minimizations run over all density operators Formula$\sigma_{\rm A}$ and Formula$\sigma_{\rm B}$ on Formula${\cal H}{\rm A}$ and Formula${\cal H}{\rm B}$, respectively. For other relative entropy measures these equalities do not hold in general. In fact, if we replace the quantum relative entropy with the max-relative entropy Formula${D_{\max}\left({\rho}\Vert{\sigma}\right)}$, the values of the three expressions above can lie arbitrarily far apart [22]: while the expressions of the form (7) and (8) have a general upper bound given by Formula$2\cdot\log\min\{\vert A\vert,\vert B\vert\}$, the expression of the form (6) is unbounded. Furthermore, the expression of the form (7) is not symmetric in A and B, unlike the von Neumann mutual information.

In order to consolidate and possibly unify these various approaches, it is of great interest to understand more about the relations among all these different quantities. In this paper, we show that smoothed versions of the max-information can be related to each other and regarded as approximately equivalent up to terms that depend only on the smoothing parameter and not on the specific quantum state or Hilbert space. These results can be employed to obtain chain rules in which we relate the max-information to differences of entropies as in (2). When evaluated for i.i.d.-states, these chain rules reproduce the well known relation (1) and thus imply the QAEP for the max-information. Since max-information and min-entropy are formally related via their definitions in terms of the max-relative entropy, we can adapt proof techniques from earlier work on min-entropy.

The organization of the paper is as follows. In Section II we present the mathematical terminology and formal definitions necessary for the formulation of our results. Our results concerning the comparability of the different definitions and chain rules are summarized in Sections III and IV. Longer proofs, along with useful technical results, can be found in the appendices.

SECTION II

MATHEMATICAL PRELIMINARIES

A. Basic Notations and Definitions

In this paper we deal exclusively with finite dimensional Hilbert spaces Formula${\cal H}{\rm A}$, Formula${\cal H}{\rm B}$ corresponding to physical systems A, B. To extend our results to infinite dimensional Hilbert spaces, the techniques of [27] could be used. For tensor products of Hilbert spaces, we use the short notation Formula${\cal H}{\rm AB}={\cal H}{\rm A}\otimes{\cal H}{\rm B}$. Let Formula${\mathop{\rm Herm}}({\cal H})$ be the space of Hermitian operators that act on Formula${\cal H}$ and Formula${\cal P}({\cal H})\subseteq{\mathop{\rm Herm}}({\cal H})$ the set of positive semi-definite operators on Formula${\cal H}$. For Formula$A,B\in{\mathop{\rm Herm}}({\cal H})$ we write Formula$A\geq B$ iff Formula$A-B\in{\cal P}({\cal H})$. In this sense, we will sometimes write Formula$A\geq 0$ in order to state that Formula$A\in{\cal P}({\cal H})$. The sets of normalized and subnormalized density operators on Formula${\cal H}$ are defined as Formula TeX Source $$S_{=}\left({\cal H}\right)\mathrel{\mathop:}=\{\rho\in{\cal P}({\cal H}):{\mathop{\rm tr}}\rho=1\}\eqno{\hbox{(9)}}$$ and Formula TeX Source $$S_{\leq}\left({\cal H}\right)\mathrel{\mathop:}=\{\rho\in{\cal P}({\cal H}):0<{\mathop{\rm tr}}\rho\leq 1\},\eqno{\hbox{(10)}}$$ respectively. Operators are usually written with a subscript that specifies on which system they act, e.g., Formula$\rho_{\rm AB}\in{\mathop{\rm Herm}}({\cal H}{\rm AB})$. Given an operator Formula$O_{\rm AB}$ on a composite Hilbert space Formula${\cal H}{\rm AB}$, we obtain the reduced operator Formula$O_{\rm A}$ on Formula${\cal H}{\rm A}$ by taking the partial trace over the subsystem Formula${\cal H}{\rm B}$: Formula$O_{\rm A}={\mathop{\rm tr}}_{\rm B}O_{\rm AB}$. The identity operator on Formula${\cal H}{\rm A}$ is denoted by Formula${\BBI}_{\rm A}$.

Quantum operations are represented by completely positive and trace preserving (CPTP) maps, i.e., linear maps Formula${\cal E}: S_{\leq}\left({{\cal H}}\right)\mapsto S_{\leq}\left({{\cal H}^{\prime}}\right)$ with the properties Formula TeX Source $$\rho\geq 0\Rightarrow{\cal E}(\rho)\geq 0,$$ and Formula TeX Source $${\mathop{\rm tr}}\rho={\mathop{\rm tr}}{\cal E}(\rho),$$ for all Formula$\rho\in S_{\leq}\left({{\cal H}}\right)$. Note that the (partial) trace is a CPTP map.

Given any operator Formula$O$, its operator norm Formula$\left\Vert{O}\right\Vert_{\infty}$ is given by its maximal singular value. Its trace norm is defined as Formula$\left\Vert{O}\right\Vert_{1}\mathrel{\mathop:}={\mathop{\rm tr}}\sqrt{O^{\dagger}O}$, where Formula$O^{\dagger}$ is the adjoint of Formula$O$. We will also require a notion of distance between density operators. For this purpose, we make use of the generalized fidelity, which is defined as Formula TeX Source $$F(\rho,\sigma)\mathrel{\mathop:}=\left\Vert{\sqrt\rho\sqrt\sigma}\right\Vert_{1}+\sqrt{(1-{\mathop{\rm tr}}\rho)(1-{\mathop{\rm tr}}\sigma)},\eqno{\hbox{(11)}}$$ for any Formula$\rho$, Formula$\sigma\in S_{\leq}\left({{\cal H}}\right)$. Note that when at least one of the states Formula$\rho$ and Formula$\sigma$ is normalized, Formula TeX Source $$F(\rho,\sigma)=\left\Vert{\sqrt{\rho}\sqrt{\sigma}}\right\Vert_{1},\eqno{\hbox{(12)}}$$ which corresponds to the standard definition for fidelity. We use the fidelity to define a distance measure on Formula$S_{\leq}\left({{\cal H}}\right)$ as Formula TeX Source $$P(\rho,\sigma)\mathrel{\mathop:}=\sqrt{1-F^{2}(\rho,\sigma)}\eqno{\hbox{(13)}}$$ which is a metric (Lemma 5 in [6]). Formula$P(\rho,\sigma)$ is called the purified distance between Formula$\rho$ and Formula$\sigma$. We say that two states Formula$\rho$ and Formula$\sigma$ are Formula$\varepsilon$- close and write Formula$\rho~{\approx_{\varepsilon}}\sigma$ iff Formula$P(\rho,\sigma)\leq\varepsilon$.

Using the purified distance as a distance measure has many technical advantages. We summarize its essential properties, along with important properties of the fidelity in Appendix A.

For any given Formula$\rho\in S_{\leq}\left({{\cal H}}\right)$, we can now define the ball of Formula$\varepsilon$-close states around Formula$\rho$ as Formula TeX Source $${\cal B}^{\varepsilon}\left(\rho\right)\mathrel{\mathop:}=\{\rho^{\prime}\in S_{\leq}\left({\cal H}\right):P(\rho,\rho^{\prime})\leq\varepsilon\},\eqno{\hbox{(14)}}$$ where Formula$\varepsilon$ is called the smoothing parameter and satisfies Formula$0\leq\varepsilon<\sqrt{{\mathop{\rm tr}}\rho}$, since we want to exclude the zero operator from the ball. In all of our statements, we make the implicit assumption that the involved smoothing parameters are small enough in this sense.

B. Generalized Entropy Measures

Let us now give the definitions for two types of generalized relative entropy, the max- and the min-relative entropy [7].

Definition 1

For Formula$\rho,\sigma\in{\cal P}({\cal H})$, the max-relative entropy is defined as Formula TeX Source $${D_{\max}\left({\rho}\Vert{\sigma}\right)}\mathrel{\mathop:}=\min\{\lambda\vert 2^{\lambda}\sigma\geq \rho\}.\eqno{\hbox{(15)}}$$ Note that Formula${D_{\max}\left({\rho}\Vert{\sigma}\right)}$ to be well defined requires Formula$\mathop{\rm supp}\rho\subseteq\mathop{\rm supp}\sigma$, where supp Formula$O$ denotes the support of the operator Formula$O$, i.e., the space orthogonal to the kernel of Formula$O$. If this is satisfied, there is an alternative way to express the max-relative entropy that we use frequently [22]: Formula TeX Source $${D_{\max}\left({\rho}\Vert{\sigma}\right)}=\log\left\Vert{\sigma^{-{{1}\over{2}}}\rho\sigma^{-{{1}\over{2}}}}\right\Vert_{\infty}.\eqno{\hbox{(16)}}$$ The inverses here are generalized inverses: given Formula$\sigma\in{\cal P}({\cal H})$, its generalized inverse Formula$\sigma^{-1}$ is the unique minimum rank operator such that Formula$\sigma^{0}\mathrel{\mathop:}=\sigma\sigma^{-1}=\sigma^{-1}\sigma$ is the projector onto Formula$\mathop{\rm supp}\sigma$.

Definition 2

For Formula$\rho$, Formula$\sigma\in{\cal P}({\cal H})$, the min-relative entropy of Formula$\rho$ with respect to Formula$\sigma$ is Formula TeX Source $${D_{\min}\left({\rho}\Vert{\sigma}\right)}\mathrel{\mathop:}=-\log\left\Vert{\sqrt\rho\sqrt\sigma}\right\Vert_{1}^{2}.\eqno{\hbox{(17)}}$$

Given any Formula$\rho_{\rm AB}\in S_{\leq}\left({{\cal H}{\rm AB}}\right)$, we can now define the (conditional) min- and max-entropies as Formula TeX Source $${H_{\min}}({\rm A}\vert{\rm B})_{\rho}\mathrel{\mathop:}=-\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{{\BBI}_{\rm A}\otimes\sigma_{\rm B}}\right)}\eqno{\hbox{(18)}}$$ and Formula TeX Source $${H_{\max}}({\rm A}\vert{\rm B})_{\rho}\mathrel{\mathop:}=-\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\min}\left({\rho_{\rm AB}}\Vert{{\BBI}_{\rm A}\otimes\sigma_{\rm B}}\right)},\eqno{\hbox{(19)}}$$ along with their smoothed versions: Formula TeX Source $$H_{\min}^{\varepsilon}\left({\rm A}\vert{\rm B}\right)_{\rho}\mathrel{\mathop:}=\max_{\rho^{\prime}\in{\cal B}^{\varepsilon}\left(\rho\right)}H_{\min}({\rm A}\vert{\rm B})_{\rho^{\prime}},\eqno{\hbox{(20)}}$$ and Formula TeX Source $$H_{\max}^{\varepsilon}\left({\rm A}\vert{\rm B}\right)_{\rho}\mathrel{\mathop:}=\min_{\rho^{\prime}\in{\cal B}^{\varepsilon}\left(\rho\right)}H_{\max}({\rm A}\vert{\rm B})_{\rho^{\prime}}.\eqno{\hbox{(21)}}$$ Min- and max-entropy are duals of each other in the sense that for pure Formula$\rho_{\rm ABC}$ [6] Formula TeX Source $$H^{\varepsilon}_{\min}({\rm A}\vert{\rm B})_{\rho}=-H^{\varepsilon}_{\max}({\rm A}\vert{\rm C})_{\rho}.\eqno{\hbox{(22)}}$$ If the system B is trivial, we obtain the definitions for the non-conditional entropies: Formula TeX Source $${H_{\min}}({\rm A})_{\rho}=-\log\lambda_{\max}(\rho_{\rm A}),\eqno{\hbox{(23)}}$$ where Formula$\lambda_{\max}(\rho)$ is the largest eigenvalue of Formula$\rho$, while Formula TeX Source $${H_{\max}}({\rm A})_{\rho}=\log\left\Vert{\sqrt\rho_{\rm A}}\right\Vert_{1}^{2}\eqno{\hbox{(24)}}$$ presents a measure for the fidelity between Formula$\rho_{\rm A}$ and the completely mixed state on Formula${\cal H}{{\rm A}}$.

C. (Smooth) Max-Information

As argued before, there is no unique way in which generalized mutual information measures should be obtained from the introduced relative entropies. Based on (6)(8), we define three different versions of max-information: Formula TeX Source $$\eqalignno{^{1}\!I_{\max}({\rm A}:{\rm B})_{\rho}\mathrel{\mathop:}=&\,{D_{\max}\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\rho_{\rm B}}\right)},\cr^{2}\!I_{\max}({\rm A}:{\rm B})_{\rho}\mathrel{\mathop:}=&\,\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)},\cr^{3}\!I_{\max}({\rm A}:{\rm B})_{\rho}\mathrel{\mathop:}=&\,\min_{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}{\rm A}\right),}\atop\scriptstyle\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}.&{\hbox{(25)}}}$$ For Formula$\rho\in S_{\leq}\left({{\cal H}{\rm AB}}\right)$ and Formula$\varepsilon\geq 0$, we obtain smooth max-information from Formula$^{i}I_{\max}\left({\rm A}:{\rm B}\right)_{\rho}$ as Formula TeX Source $$^{i}I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}\mathrel{\mathop:}=\min_{{\rho^{\prime}}\in{\cal B}^{\varepsilon}\left(\rho\right)}{^{i}I_{\max}\left({\rm A}:{\rm B}\right)_{\rho^{\prime}}}.\eqno{\hbox{(26)}}$$ It should be pointed out that earlier literature making use of smooth max-information usually refers to Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$. In particular, a chain rule, a data processing inequality and the QAEP have been proven for Formula$^{2}\!I_{\max}$ in [22]. The proof of the data processing inequality can straightforwardly be extended to all smooth definitions.

Lemma 1

Let Formula$\rho_{\rm AB}\in S_{\leq}\left({{\cal H}{\rm AB}}\right)$, Formula$\varepsilon\geq 0$ and let Formula${\cal E}$ be a CPTP map of the form Formula${\cal E}={\cal E}_{\rm A}\otimes{\cal E}_{\rm B}$. Then Formula TeX Source $$^{i}I_{\max}^{\varepsilon}({\rm A}:{\rm B})_{{\cal E}(\rho)}\leq{^{i}I_{\max}^{\varepsilon}}({\rm A}:{\rm B})_{\rho},\eqno{\hbox{(27)}}$$ for any Formula$i\in\{1,2,3\}$.

Proof

We provide the proof for Formula$i=2$, the other cases being similar. Let Formula$\rho^{\prime}_{\rm AB}\in{\cal B}^{\varepsilon}\left(\rho_{\rm AB}\right)$ be a state that optimizes Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$, i.e., Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}={^{2}\!I_{\max}}({\rm A}:{\rm B})_{\rho^{\prime}}$. Then there exists Formula${\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}$ such that Formula TeX Source $$\eqalignno{{^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}=&\,{D_{\max}\left({\rho^{\prime}_{\rm AB}}\Vert{\rho^{\prime}_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr\geq &\,{D_{\max}\left({{\cal E}(\rho^{\prime}_{\rm AB})}\Vert{{\cal E}_{\rm A}(\rho^{\prime}_{\rm A})\otimes{\cal E}_{\rm B}(\sigma_{\rm B})}\right)}\cr\geq &\,\min_{\omega_{\rm B}\in S_{=}\left({{\cal H}{\rm B}}\right)}{D_{\max}\left({{\cal E}(\rho^{\prime}_{\rm AB})}\Vert{{\cal E}_{\rm A}(\rho^{\prime}_{\rm A})\otimes\omega_{\rm B}}\right)}\cr\geq &\,\min_{{\scriptstyle\bar\rho\in{\cal B}^{\varepsilon}\left({\cal E}(\rho)\right),}\atop\scriptstyle\omega_{\rm B}\in S_{=}\left({{\cal H}_{\rm B}}\right)}{D_{\max}\left({\bar\rho_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\omega_{\rm B}}\right)},}$$ where the first inequality follows from the data processing inequality for the max-relative entropy (cf. Lemma 20) and the last inequality is a consequence of the monotonicity of the purified distance under trace non-increasing CPMs (cf. Lemma 12). Formula$\blackboxfill$

SECTION III

APPROXIMATE EQUIVALENCE RELATIONS FOR Formula$^{i}I_{\max}^{\varepsilon}$

Let us now turn to our main problem of relating alternative expressions for smooth max-information to each other. Our key results are given by the following two theorems. For convenience of notation, we introduce the two functions Formula TeX Source $$f(\varepsilon,{\varepsilon^{\prime}})\mathrel{\mathop:}=\log\left({{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+{{1}\over{1-{\varepsilon^{\prime}}}}\right)\eqno{\hbox{(28)}}$$ and Formula TeX Source $$g(\varepsilon)\mathrel{\mathop:}=\log\left({{2(1-\varepsilon)+3}\over{(1-\varepsilon)(1-\sqrt{1-\varepsilon^{2}})}}\right)\!.\eqno{\hbox{(29)}}$$ Note that both functions grow logarithmically in Formula${{1}\over{\varepsilon}}$ when Formula$\varepsilon$, Formula${\varepsilon^{\prime}}\rightarrow 0$.

Theorem 2

Let Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$, Formula${\varepsilon^{\prime}}\geq 0$. Then Formula TeX Source $$\eqalignno{{^{3}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq &\,{^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\cr\leq &\,{^{3}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+f(\varepsilon,{\varepsilon^{\prime}}).&{\hbox{(30)}}}$$

Theorem 3

Let Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$, Formula${\varepsilon^{\prime}}\geq 0$. Then, Formula TeX Source $$\eqalignno{{^{2}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq &\,{^{1}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\cr\leq &\,{^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+g(\varepsilon).&{\hbox{(31)}}}$$

We provide the proofs of these theorems in the appendix and turn immediately to the corollaries. First we complete our set of approximate equivalence relations. In order to compare Formula$^{1}\!I_{\max}^{\varepsilon}$ and Formula$^{3}\!I_{\max}^{\varepsilon}$, we only need to combine Theorems 2 and 3.

Corollary 4

Let Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon$, Formula${\varepsilon^{\prime}}>0$, Formula$\varepsilon^{\prime\prime}\geq 0$. Then Formula TeX Source $$\eqalignno{{^{3}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}+\varepsilon^{\prime\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq &\,{^{1}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}+\varepsilon^{\prime\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}\cr\leq &\,{^{3}\!I^{\varepsilon^{\prime\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}\cr&+f(\varepsilon^{\prime},\varepsilon^{\prime\prime})+g(\varepsilon).&{\hbox{(32)}}}$$

We thus conclude that all three definitions for Formula$I_{\max}^{\varepsilon}$ are pairwise approximately equivalent, meaning that since the differences between them are independent of the given state or Hilbert space, they must carry the same qualitative content.

These relations further imply an estimate on the approximate symmetry of Formula$^{2}\!I^{\varepsilon}_{\max}$.

Corollary 5

Let Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$, Formula${\varepsilon^{\prime}}\geq 0$. Then Formula TeX Source $$\eqalignno{\indent{^{2}\!I^{2\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq &\,{^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm B}:{\rm A}\right)_{\rho}}+f(\varepsilon,\varepsilon+{\varepsilon^{\prime}})\cr\leq&\,{^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+f(\varepsilon,\varepsilon+{\varepsilon^{\prime}})+f(\varepsilon,{\varepsilon^{\prime}}).\cr& &{\hbox{(33)}}}$$

Proof

Note that Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm B}:{\rm A}\right)_{\rho}}\geq {^{3}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$, which follows directly from the definitions. Then, using Theorem 2 and the apparent symmetry of Formula$^{3}\!I_{\max}^{\varepsilon}$, we find that Formula TeX Source $${^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm B}:{\rm A}\right)_{\rho}}\leq{^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+f(\varepsilon,{\varepsilon^{\prime}}),$$ as well as Formula TeX Source $${^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq{^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm B}:{\rm A}\right)_{\rho}}+f(\varepsilon,{\varepsilon^{\prime}}),$$ and the claim follows. Formula$\blackboxfill$

SECTION IV

CHAIN RULES FOR Formula$^{i}I^{\varepsilon}_{\max}$

In this section we prove chain rules for smooth max-information of the form Formula TeX Source $$\eqalignno{H^{\varepsilon}_{\min}({\rm A})_{\rho}-H^{\varepsilon}_{\min}({\rm A}\vert{\rm B})_{\rho}\lesssim&\, I_{\max}^{\varepsilon}({\rm A}:{\rm B})_{\rho}\cr\lesssim & \,H_{\max}^{\varepsilon}({\rm A})_{\rho}-H_{\min}^{\varepsilon}({\rm A}\vert{\rm B})_{\rho}.}$$ An upper bound chain rule for Formula$^{2}\!I_{\max}^{\varepsilon}$ is already known from Lemma B.15 in [22]: for Formula$\rho\in S_{=}\left({{\cal H}}\right)$ and Formula$\varepsilon>0$, Formula TeX Source $${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq H_{\max}^{\varepsilon^{2}/48}({\rm A})_{\rho}-H_{\min}^{\varepsilon^{2}/48}({\rm A}\vert{\rm B})_{\rho}-l(\varepsilon),\eqno{\hbox{(34)}}$$ where Formula$l(\varepsilon)\mathrel{\mathop:}=2\cdot\log{{\varepsilon^{2}}\over{24}}$. Let us first derive a lower bound chain rule for Formula$^{2}\!I_{\max}^{\varepsilon}$. Having both bounds for one of the definitions will allow us to write down chain rules for all Formula$^{i}I_{\max}^{\varepsilon}~{\rm by}$ exploiting the approximate equivalence relations from the previous section.

Lemma 6

Let Formula$\rho\in S_{=}\left({{\cal H}{\rm AB}}\right)$ and Formula$\varepsilon\geq 0$. Then Formula TeX Source $${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\geq H_{\min}^{\varepsilon}({\rm A})_{\rho}-H_{\min}^{4\sqrt{2\varepsilon}}({\rm A}\vert{\rm B})_{\rho}.\eqno{\hbox{(35)}}$$

Proof

The proof is similar to the one of (34) in [22]. We rearrange Lemma B.13 from [22] as Formula TeX Source $$H_{\min}({\rm A}\vert{\rm B})_{\rho}\geq H_{\min}({\rm A})_{\rho}-{^{2}\!I_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}.$$ Thus, Formula TeX Source $$\eqalignno{& H_{\min}^{\varepsilon}({\rm A}\vert{\rm B})_{\rho}\cr&\quad\geq \max_{\rho^{\prime}\in{\cal B}^{\varepsilon}\left(\rho\right)}\big [H_{\min}({\rm A})_{\rho^{\prime}}-{^{2}\!I}_{\max}({\rm A}:{\rm B})_{\rho^{\prime}}\big]\cr&\quad\geq \max_{\omega\in{\cal B}^{\varepsilon^{2}/32}\left(\rho\right)}\max_{\Pi_{\rm A}}\big[H_{\min}({\rm A})_{\Pi\omega\Pi}-{^{2}\!I}_{\max}({\rm A}:{\rm B})_{\Pi\omega\Pi}\big],}$$ where the maximization runs over all Formula$0\leq\Pi_{\rm A}\leq{\BBI}_{\rm A}$ with Formula$\Pi_{\rm A}\omega\Pi_{\rm A}~{\approx_{\varepsilon/2}}\omega$. Next, choose Formula$\omega^{\prime}\in{\cal B}^{\varepsilon^{2}/32}\left(\rho\right)$ such that Formula$^{2}\!I_{\max}({\rm A}:{\rm B})_{\omega^{\prime}}={^{2}\!I^{\varepsilon^{2}/32}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$. This gives us Formula TeX Source $$\eqalignno{H_{\min}^{\varepsilon}({\rm A}\vert{\rm B})_{\rho}\geq &\,\max_{\Pi_{\rm A}}\big[H_{\min}({\rm A})_{\Pi\omega^{\prime}\Pi}-{^{2}\!I}_{\max}({\rm A}:{\rm B})_{\Pi\omega^{\prime}\Pi}\big]\cr\geq &\,\max_{\Pi_{\rm A}}H_{\min}({\rm A})_{\Pi\omega^{\prime}\Pi}-{^{2}\!I}_{\max}({\rm A}:{\rm B})_{\omega^{\prime}},}$$ with the maximization running over all Formula$0\leq\Pi_{\rm A}\leq{\BBI}_{\rm A}$ with Formula$\Pi_{\rm A}\omega^{\prime}\Pi_{\rm A}~{\approx_{\varepsilon/2}}\omega^{\prime}$. The second inequality is a consequence of Remark 1 (cf. Appendix B). According to Lemma 19, we can choose a Formula$\Pi_{\rm A}$ such that Formula$H_{\min}^{\varepsilon^{2}/16}({\rm A})_{\omega^{\prime}}\leq H_{\min}({\rm A})_{\Pi\omega^{\prime}\Pi}$. Doing so yields Formula TeX Source $$\eqalignno{H_{\min}^{\varepsilon}({\rm A}\vert{\rm B})_{\rho}\geq &\, H_{\min}^{\varepsilon^{2}/16}({\rm A})_{\omega^{\prime}}-{^{2}\!I^{\varepsilon^{2}/32}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\cr\geq &\, H_{\min}^{\varepsilon^{2}/32}({\rm A})_{\rho}-{^{2}\!I^{\varepsilon^{2}/32}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}.}$$ Relabelling Formula$\varepsilon^{2}/32\rightarrow\varepsilon$ concludes the proof. Formula$\blackboxfill$

We can now obtain chain rules for alternative definitions of Formula$I_{\max}^{\varepsilon}$ as well.

Corollary 7

Let Formula$\rho\in S_{=}\left({{\cal H}{\rm AB}}\right)$ and Formula$\varepsilon$, Formula${\varepsilon^{\prime}}>0$. Then Formula TeX Source $$\eqalignno{&{^{1}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq H_{\max}^{{\varepsilon^{\prime}}^{2}/48}({\rm A})_{\rho}-H_{\min}^{{\varepsilon^{\prime}}^{2}/48}({\rm A}\vert{\rm B})_{\rho}\cr&\hskip9.3em+g(\varepsilon)-l({\varepsilon^{\prime}}).&{\hbox{(36)}}}$$

Proof

With (34) and using Theorem 3 to estimate Formula${^{1}\!I^{\varepsilon+2\sqrt\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$ in terms of Formula${^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}$, the claim follows immediately. Formula$\blackboxfill$

Similarly, we obtain a lower bound chain rule for Formula${^{3}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$:

Corollary 8

For Formula$\rho\in S_{=}\left({{\cal H}{\rm AB}}\right)$ and Formula$\varepsilon>0$, Formula${\varepsilon^{\prime}}\geq 0$, Formula TeX Source $$\eqalignno{{^{3}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}\geq &\, H_{\min}^{\varepsilon+{\varepsilon^{\prime}}}({\rm A})_{\rho}-H_{\min}^{4\sqrt{2\varepsilon+2{\varepsilon^{\prime}}}}({\rm A}\vert{\rm B})_{\rho}\cr&-f(\varepsilon,{\varepsilon^{\prime}}).&{\hbox{(37)}}}$$

Proof

The claim is a direct consequence of Theorem 2 and Lemma 6. Formula$\blackboxfill$

SECTION V

CONCLUSION

We have investigated properties of smooth max-information defined as a special case of the max-relative entropy. In earlier work, it has been shown to be an operational quantity in one-shot state splitting and state merging [22], [23]. It is also found to be a useful quantity in quantum rate distortion theory [24] and the statistical physics of many body systems [25]. We have shown that it exhibits some properties that we would expect from previous results on smooth entropies. Alternative definitions of max-information turn out to be essentially equivalent. Furthermore, they satisfy upper and lower bound chain rules in terms of min- and max-entropies. Chain rules are generally an important technical tool in information theory. In this case, they also relate max-information to alternative definitions for one-shot mutual information, made up from differences of entropies as used in [19], [20], [21].

The primary goal of further research on these quantities is to gain a better understanding of their operational relevance. We hope that the formal tools provided in this paper will be useful for this purpose.

APPENDIX A

PROPERTIES OF THE FIDELITY AND PURIFIED DISTANCE

Here we summarize the essential properties of the purified distance. For a more extensive discussion, we refer the reader to [8]. The main reasons the purified distance is preferred over the trace distance are the following two lemmas, which state that for given Formula$\rho$, Formula$\sigma$, we can always find purifications or extensions Formula$\bar{\rho},\bar{\sigma}$ such that Formula$P(\rho,\sigma)=P(\bar\rho,\bar\sigma)$. This is due to Uhlmann's theorem [28]: for any states Formula$\rho_{\rm A},\sigma_{\rm A}\in S_{=}\left({{\cal H}{\rm A}}\right)$ Formula TeX Source $$\left\Vert{\sqrt{\rho_{\rm A}}\sqrt{\sigma_{\rm A}}}\right\Vert_{1}=\max_{\rho_{\rm AB},\sigma_{\rm AB}}\left\Vert{\sqrt{\rho_{\rm AB}}\sqrt{\sigma_{\rm AB}}}\right\Vert_{1}\!,\eqno{\hbox{(A.38)}}$$ where the maximization runs over all purifications Formula$\rho_{\rm AB}$ and Formula$\sigma_{\rm AB}$ of Formula$\rho_{\rm A}$ and Formula$\sigma_{\rm A}$.

Lemma 9(Lemma 8 in [6])

Let Formula$\rho,\sigma\in S_{\leq}\left({{\cal H}}\right)$, Formula${\cal H}^{\prime}\cong{\cal H}$ and Formula$\varphi\in{\cal H}\otimes{\cal H}^{\prime}$ be a purification of Formula$\rho$. Then there exists a purification Formula$\vartheta\in{\cal H}\otimes{\cal H}^{\prime}$ of Formula$\sigma$ with Formula$P(\rho,\sigma)=P(\varphi,\vartheta)$.

Lemma 10 (Corollary 9 in [6])

Let Formula$\rho,\sigma\in S_{\leq}\left({{\cal H}}\right)$ and Formula$\bar\rho\in S_{\leq}\left({{\cal H}\otimes{\cal H}^{\prime}}\right)$ be an extension of Formula$\rho$. Then there exists an extension Formula$\bar\sigma\in S_{\leq}\left({{\cal H}\otimes{\cal H}^{\prime}}\right)$ of Formula$\sigma$ with Formula$P(\rho,\sigma)=P(\bar\rho,\bar\sigma)$.

Still, the purified distance is equivalent to the generalized trace distance given by Formula TeX Source $$D(\rho,\sigma)={{1}\over{2}}\left\Vert{\rho-\sigma}\right\Vert_{1}+{{1}\over{2}}\vert{\mathop{\rm tr}}\rho-{\mathop{\rm tr}}\sigma\vert.\eqno{\hbox{(A.39)}}$$ Therefore it retains an operational interpretation as a measure for the maximum guessing probability [29]: the maximal probability Formula$p_{\rm dist}(\rho,\sigma)$ for correctly distinguishing between two quantum states Formula$\rho$, Formula$\sigma$ satisfies Formula TeX Source $$p_{\rm dist}(\rho,\sigma)\leq{{1}\over{2}}(1+D(\rho,\sigma)).\eqno{\hbox{(A.40)}}$$

Lemma 11 (Lemma 7 in [6])

Let Formula$\rho,\sigma\in S_{\leq}\left({{\cal H}}\right)$. Then Formula TeX Source $$D(\rho,\sigma)\leq P(\rho,\sigma)\leq\sqrt{2D(\rho,\sigma)}.\eqno{\hbox{(A.41)}}$$

Another useful property of the purified distance is that it cannot increase under trace non-increasing CPMs.

Lemma 12 (Lemma 7 in [6])

Let Formula$\rho,\sigma\in S_{\leq}\left({{\cal H}}\right)$ and let Formula${\cal E}$ be a trace non-increasing CPM. Then Formula$P(\rho,\sigma)\geq P({\cal E}(\rho),{\cal E}(\sigma))$.

We make use of the following properties of the standard fidelity.

Lemma 13 ([18])

Let Formula$\rho,\sigma\in{\cal P}({\cal H})$.

  • For any Formula$\omega\geq \rho$, Formula TeX Source $$\left\Vert{\sqrt\omega\sqrt\sigma}\right\Vert_{1}\geq \left\Vert{\sqrt\rho\sqrt\sigma}\right\Vert_{1}\!.\eqno{\hbox{(A.42)}}$$
  • For any projector Formula$\Pi\in{\cal P}({\cal H})$, Formula TeX Source $$\eqalignno{\left\Vert{\sqrt{\Pi\rho\Pi}\sqrt{\sigma}}\right\Vert_{1}=&\,\left\Vert{\sqrt{\rho}\sqrt{\Pi\sigma\Pi}}\right\Vert_{1}\cr=&\,\left\Vert{\sqrt{\Pi\rho\Pi}\sqrt{\Pi\sigma\Pi}}\right\Vert_{1}.&{\hbox{(A.43)}}}$$

We conclude this section by stating a few useful technical facts.

Lemma 14 (Lemma 17 in [9])

Let Formula$\rho\in S_{\leq}\left({{\cal H}}\right)$ and Formula$\Pi$ a projector on Formula${\cal H}$, then Formula TeX Source $$P(\rho,\Pi\rho\Pi)\leq\sqrt{2\cdot{\mathop{\rm tr}}(\Pi^{\bot}\rho)-\left({\mathop{\rm tr}}(\Pi^{\bot}\rho)\right)^{2}},\eqno{\hbox{(A.44)}}$$ where Formula$\Pi^{\bot}={\BBI}-\Pi$.

Lemma 15 (Lemma A.7 in [30])

Let Formula$\rho\in S_{\leq}\left({{\cal H}}\right)$ and Formula$\Pi\in{\cal P}({{\cal H}})$ such that Formula$\Pi\leq{\BBI}$. Then Formula TeX Source $$P(\rho,\Pi\rho\Pi)\leq{{1}\over{\sqrt{{\mathop{\rm tr}}\rho}}}\sqrt{\left({\mathop{\rm tr}}(\rho)\right)^{2}-\left({\mathop{\rm tr}}(\Pi^{2}\rho)\right)^{2}}.\eqno{\hbox{(A.45)}}$$

Corollary 16

Let Formula$\rho\in S_{\leq}\left({{\cal H}}\right)$ and Formula$0<k\leq 1$. Then Formula TeX Source $$P(\rho,k\cdot\rho)\leq\sqrt{1-k^{2}}.\eqno{\hbox{(A.46)}}$$

Proof

Apply Lemma 15 to Formula$\Pi=\sqrt k\cdot{\BBI}$ and use Formula$\sqrt{{\mathop{\rm tr}}\rho}\leq 1$. Formula$\blackboxfill$

APPENDIX B

TECHNICAL LEMMAS

The following lemma introduces a notion of duality between projectors on subsystems of a multi-partite quantum system with respect to a given pure state. It is essential in the proofs of Theorems 2 and 3.

Lemma 17 (Corollary 16 in [9])

Let Formula$\rho_{\rm AB}={\vert{\varphi}\rangle\langle{\varphi}\vert}_{\rm AB}\in{\cal P}({\cal H}{\rm AB})$ be pure, Formula$\rho_{\rm A}={\mathop{\rm tr}}_{\rm B}\rho_{\rm AB}$, Formula$\rho_{\rm B}={\mathop{\rm tr}}_{\rm A}\rho_{\rm AB}$ and let Formula$\Pi_{\rm A}\in{\cal P}({\cal H}{\rm A})$ be a projector in Formula$\mathop{\rm supp}\rho_{\rm A}$. Then, there exists a dual projector Formula$\Pi_{\rm B}$ on Formula${\cal H}{\rm B}$ such that Formula TeX Source $$(\Pi_{\rm A}\otimes\rho_{\rm B}^{-1/2}){\vert{\varphi}\rangle}_{\rm AB}=(\rho_{\rm A}^{-1/2}\otimes\Pi_{\rm B}){\vert{\varphi}\rangle}_{\rm AB}.\eqno{\hbox{(B.47)}}$$

The Proof of Theorem 3 further requires the following inequality for the operator norm.

Lemma 18

Let Formula$A$, Formula$B$, Formula$C\in{\cal P}({\cal H})$ be such that Formula$\mathop{\rm supp}A\subseteq\mathop{\rm supp}B$ and Formula$B\leq C$. Then Formula TeX Source $$\left\Vert{C^{-1/2}AC^{-1/2}}\right\Vert_{\infty}\leq\left\Vert{B^{-1/2}AB^{-1/2}}\right\Vert_{\infty}\!.\eqno{\hbox{(B.48)}}$$

Proof

We know from (16) that Formula$\lambda=\left\Vert{B^{-1/2}AB^{-1/2}}\right\Vert_{\infty}$ is the smallest number such that Formula$A\leq\lambda B$. Then Formula$B\leq C$ implies Formula$A\leq\lambda C$ and the claim follows. Formula$\blackboxfill$

In proving the chain rules for Formula$I_{\max}$, we have used the following facts on different entropic quantities.

Lemma 19 (lemma 5 in [31])

For any Formula$\rho\in S_{\leq}\left({{\cal H}{\rm A}}\right)$ and Formula$\varepsilon\geq 0$, there exists an operator Formula$0\leq\Pi\leq{\BBI}_{\rm A}$ such that Formula$\rho~{\approx_{\varepsilon/2}}\Pi\rho\Pi$ and Formula TeX Source $$H_{\min}^{\varepsilon^{2}/16}({\rm A})_{\rho}\leq H_{\min}({\rm A})_{\Pi\rho\Pi}.\eqno{\hbox{(B.49)}}$$

Lemma 20 (lemma 7 in [7])

Let Formula$\rho,\sigma\in{\cal P}({\cal H})$ and Formula${\cal E}$ be a CPTP map on Formula${\cal H}$. Then Formula TeX Source $${D_{\max}\left({\rho}\Vert{\sigma}\right)}\geq {D_{\max}\left({{\cal E}(\rho)}\Vert{{\cal E}(\sigma)}\right)}.\eqno{\hbox{(B.50)}}$$

Remark 1

This actually holds more generally even if the CPM Formula${\cal E}$ is not trace preserving. In particular, for any Formula$\Pi\in{\cal P}({\cal H})$, Formula TeX Source $${D_{\max}\left({\rho}\Vert{\sigma}\right)}\geq {D_{\max}\left({\Pi\rho\Pi}\Vert{\Pi\sigma\Pi}\right)}.\eqno{\hbox{(B.51)}}$$

APPENDIX C

PROOFS OF THEOREMS 2 AND 3

Auxiliary Lemmas

Before turning to the main proofs, we want to make a few observations on the normalization of optimal operators for Formula$^{i}I_{\max}^{\varepsilon}$. Lemma 22 will prove especially useful in the proof of Thereom 3.

The proofs of these lemmas rely on the following fact.

Lemma 21

Let Formula$\rho_{\rm AB}\in S_{=}\left({{\cal H}{\rm AB}}\right)$, Formula$\varepsilon\geq 0$ and Formula$\rho^{\prime}\in{\cal B}^{\varepsilon}\left(\rho_{\rm AB}\right)$. Then Formula${{\rho^{\prime}}\over{{\mathop{\rm tr}}\rho^{\prime}}}\in{\cal B}^{\varepsilon}\left(\rho_{\rm AB}\right)$ as well.

Proof

Remember that the generalized fidelity Formula$F(\sigma,\tau)$ is equal to Formula$\left\Vert{\sqrt{\sigma}\sqrt{\tau}}\right\Vert_{1}~{\rm if}$ at least one of the arguments is normalized. Note also that every subnormalized operator Formula$\omega^{\prime}$ can be written as Formula$\omega^{\prime}={\mathop{\rm tr}}\omega^{\prime}\cdot\omega$ with a normalized operator Formula$\omega$. Let Formula$\omega_{\rm AB}={{\rho^{\prime}_{\rm AB}}\over{{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}}}$. Then Formula TeX Source $$\eqalignno{F(\rho^{\prime}_{\rm AB},\rho_{\rm AB})=&\,\left\Vert{\sqrt{\rho^{\prime}_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}\cr=&\,\left\Vert{\sqrt{{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}\cdot\omega_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}\cr\leq &\,\left\Vert{\sqrt{\omega_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}=F(\omega_{\rm AB},\rho_{\rm AB}).}$$ Therefore, the purified distance is Formula TeX Source $$\eqalignno{P(\rho^{\prime}_{\rm AB},\rho_{\rm AB})=&\,\sqrt{1-F^{2}(\rho^{\prime}_{\rm AB},\rho_{\rm AB})}\cr\geq &\,\sqrt{1-F^{2}(\omega_{\rm AB},\rho_{\rm AB})}\cr=&\,P(\omega_{\rm AB},\rho_{\rm AB}),}$$ which concludes the proof. Formula$\blackboxfill$

With this lemma, we can show that there always exists an optimal operator for Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$ that is normalized.

Lemma 22

Let Formula$\rho\in S_{=}\left({{\cal H}{\rm AB}}\right)$ and Formula$\varepsilon\geq 0$. Then there exists a normalized state Formula$\rho^{\prime}\in{\cal B}^{\varepsilon}\left(\rho\right)$ with Formula$^{2}\!I_{\max}({\rm A}:{\rm B})_{\rho^{\prime}}={^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$.

Proof

Let Formula$\bar\rho_{\rm AB}\in{\cal B}^{\varepsilon}\left(\rho\right)$ be any operator satisfying Formula$^{2}\!I_{\max}({\rm A}:{\rm B})_{\bar\rho}={^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$ and let Formula${\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}$ be such that Formula TeX Source $$\eqalignno{&\quad~~2^{^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\bar\rho_{\rm A}\otimes\sigma_{\rm B}\geq \bar\rho_{\rm AB}\cr&\Rightarrow 2^{^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}{{\bar\rho_{\rm A}}\over{{\mathop{\rm tr}}\bar\rho_{\rm AB}}}\otimes\sigma_{\rm B}\geq {{\bar\rho_{\rm AB}}\over{{\mathop{\rm tr}}\bar\rho_{\rm AB}}}.}$$ Hence, Formula TeX Source $${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\geq {D_{\max}\left({{\bar\rho_{\rm AB}}\over{{\mathop{\rm tr}}\bar\rho_{\rm AB}}}\Vert{{{\bar\rho_{\rm A}}\over{{\mathop{\rm tr}}\bar\rho_{\rm AB}}}\otimes\sigma_{\rm B}}\right)},$$ but because of Lemma 21 we find that actually equality holds. We thus conclude that if Formula$\bar\rho$ optimizes Formula${^{2}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$, then so does Formula$\rho^{\prime}={{\bar\rho}\over{{\mathop{\rm tr}}\bar\rho}}$. Formula$\blackboxfill$

We can prove an analogous and in fact stricter statement about Formula${^{1}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$. We give it here for the sake of completeness.

Lemma 23

Let Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$, Formula$\varepsilon\geq 0$ and let Formula$\rho^{\prime}_{\rm AB}\in{\cal B}^{\varepsilon}\left(\rho_{\rm AB}\right)$ optimize Formula${^{1}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$. Then Formula${\mathop{\rm tr}}\rho^{\prime}_{\rm AB}=1$.

Proof

Let Formula$\omega_{\rm AB}={{\rho^{\prime}_{\rm AB}}\over{{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}}}$. It holds that for Formula$k=2^{^{1}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$ Formula TeX Source $$\eqalignno{&\quad~~k\cdot\rho^{\prime}_{\rm A}\otimes\rho^{\prime}_{\rm B}\geq \rho^{\prime}_{\rm AB}\cr&\Rightarrow k\cdot{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}\cdot\omega_{\rm A}\otimes\omega_{\rm B}\geq {{\rho^{\prime}_{\rm AB}}\over{{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}}}=\omega_{\rm AB}\cr&\Rightarrow{^{1}\!I^{\varepsilon}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}+\log{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}\geq {^{1}\!I_{\max}({\rm A}:{\rm B})}_{\omega},}$$ where Formula$\log{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}\leq 0$. If however this inequality is strict, i.e., Formula$\log{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}<0$, this would be a contradiction to the optimality of Formula$\rho^{\prime}_{\rm AB}$ according to Lemma 21 and therefore Formula${\mathop{\rm tr}}\rho^{\prime}_{\rm AB}=1$. Formula$\blackboxfill$

Proof of Theorem 2

The proof is analogous to the reasoning in Lemma 20 in [9]. We divide it into three steps. Claim 1 is a crucial step in the proof of Claim 2, from which in turn the result follows.

Claim 1

Let Formula$\rho_{\rm ABC}$ be a purification of Formula$\rho_{\rm AB}\in S_{\leq}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$. Then there exists a projector Formula$\Pi_{\rm BC}$ on Formula${\cal {H}}_{\rm BC}$ such that Formula${\mathtilde{\rho}}_{\rm ABC}\mathrel{\mathop:}=\Pi_{\rm BC}\rho_{\rm ABC}\Pi_{\rm BC}\in{\cal B}^{\varepsilon}\left(\rho_{\rm ABC}\right)$ and Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq\min_{{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}{\rm A}\right)},}\atop\scriptstyle{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}+\log{{1}\over{1-\sqrt{1-\varepsilon^{2}}}}.\cr&&{\hbox{(C.52)}}}$$

Proof

The strategy of the proof is to define Formula$\Pi_{\rm BC}$ as the dual projector with respect to Formula$\rho_{\rm ABC}$ (in the sense of Lemma 17) of a conveniently chosen Formula$\Pi_{\rm A}$ with Formula$\mathop{\rm supp}\Pi_{\rm A}\subseteq\mathop{\rm supp}\rho_{\rm A}$. Fix Formula$\lambda$, Formula$\bar{\sigma}_{\rm A}$, and Formula$\bar{\sigma}_{\rm B}$ such that Formula TeX Source $$\eqalignno{&\quad~\min_{{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}{\rm A}\right)},}\atop\scriptstyle{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&={D_{\max}\left({\rho_{\rm AB}}\Vert{\bar{\sigma}_{\rm A}\otimes\bar{\sigma}_{\rm B}}\right)}=\log\lambda.}$$ Note that by construction we have Formula${\mathtilde{\rho}}_{\rm A}\leq\rho_{\rm A}$ and Formula$\mathop{\rm supp}{\mathtilde{\rho}}_{\rm B}\subseteq\mathop{\rm supp}\rho_{\rm B}$, so that we find Formula TeX Source $$\eqalignno{\mathop{\rm supp}{\mathtilde\rho}_{\rm AB}\subseteq &\,\mathop{\rm supp}({\mathtilde\rho}_{\rm A}\otimes{\mathtilde\rho}_{\rm B})\cr\subseteq &\,\mathop{\rm supp}(\rho_{\rm A}\otimes\rho_{\rm B})\cr\subseteq &\,\mathop{\rm supp}(\rho_{\rm A}\otimes\bar\sigma_{\rm B}).}$$ Therefore Formula${D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\bar\sigma_{\rm B}}\right)}$ is well defined and we can write Formula TeX Source $$\eqalignno{&\quad~\,\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\bar{\sigma}_{\rm B}}\right)}\cr&=\log\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}{\mathtilde{\rho}}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}.}$$ Defining Formula$\Pi_{\rm A}$ as the dual projector of Formula$\Pi_{\rm BC}$ and using the inequality Formula$\lambda\bar{\sigma}_{\rm A}\otimes\bar{\sigma}_{\rm B}\geq \rho_{\rm AB}$ we obtain Formula TeX Source $$\eqalignno{&\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}{\mathtilde{\rho}}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\quad~=\left\Vert{\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}{\mathop{\rm tr}}_{\rm C}\left(\rho_{\rm A}^{-{{1}\over{2}}}\otimes\Pi_{\rm BC}{\rho}_{\rm ABC}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\Pi_{\rm BC}\right)\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\quad~=\left\Vert{\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}\Pi_{\rm A}\rho_{\rm A}^{-{{1}\over{2}}}{\rho}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\Pi_{\rm A}\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\quad~\leq\lambda\left\Vert{\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}\Pi_{\rm A}\rho_{\rm A}^{-{{1}\over{2}}}\bar{\sigma}_{\rm A}\otimes\bar{\sigma}_{\rm B}\rho_{\rm A}^{-{{1}\over{2}}}\Pi_{\rm A}\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\quad~=\lambda\left\Vert{\Pi_{\rm A}\rho_{\rm A}^{-{{1}\over{2}}}\bar{\sigma}_{\rm A}\rho_{\rm A}^{-{{1}\over{2}}}\Pi_{\rm A}\otimes\bar{\sigma}_{\rm B}^{0}}\right\Vert_{\infty}\cr&\quad~=\lambda\left\Vert{\Pi_{\rm A}\Gamma_{\rm A}\Pi_{\rm A}}\right\Vert_{\infty},}$$ where we have introduced Formula$\Gamma_{\rm A}\mathrel{\mathop:}=\rho_{\rm A}^{-{{1}\over{2}}}\bar{\sigma}_{\rm A}\rho_{\rm A}^{-{{1}\over{2}}}$. Thus, we find Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq\min_{{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}_{\rm A}\right)},}\atop\scriptstyle{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}+\log\left\Vert{\Pi_{\rm A}\Gamma_{\rm A}\Pi_{\rm A}}\right\Vert_{\infty}.}$$ By Lemma 14 it holds that Formula TeX Source $$\eqalignno{P(\rho_{\rm ABC},{\mathtilde{\rho}}_{\rm ABC})\leq &\,\sqrt{2\cdot{\mathop{\rm tr}}(\Pi_{\rm BC}^{\bot}\rho_{\rm ABC})-\left({\mathop{\rm tr}}(\Pi_{\rm BC}^{\bot}\rho_{\rm ABC})\right)^{2}}\cr=&\,\sqrt{2\cdot{\mathop{\rm tr}}(\Pi_{\rm A}^{\bot}\rho_{\rm A})-\left({\mathop{\rm tr}}(\Pi_{\rm A}^{\bot}\rho_{\rm A})\right)^{2}},\cr&&{\hbox{(C.53)}}}$$ where Formula$\Pi^{\bot}={\BBI}-\Pi$. Now we define Formula$\Pi_{\rm A}$ to be the minimum rank projector on the smallest eigenvalues of Formula$\Gamma_{\rm A}$ such that Formula${\mathop{\rm tr}}(\Pi_{\rm A}^{\bot}\rho_{\rm A})\leq{1-\sqrt{1-\varepsilon^{2}}}$. With (C.53) this implies Formula$P(\rho_{\rm ABC},{\mathtilde{\rho}}_{\rm ABC})\leq\varepsilon$ since Formula$t\mapsto\sqrt{2t-t^{2}}$ is monotonically increasing in the interval Formula$[{0,1}]$. It remains to show that with our choice of Formula$\Pi_{\rm A}$ Formula TeX Source $$\left\Vert{\Pi_{\rm A}\Gamma_{\rm A}\Pi_{\rm A}}\right\Vert_{\infty}\leq{{1}\over{1-\sqrt{1-\varepsilon^{2}}}}$$ holds. This, however, can be shown in an identical manner as it is done in the Proof of Lemma 21 in [9]. The only difference is that we have chosen Formula$\Pi_{\rm A}$ such that Formula${\mathop{\rm tr}}(\Pi_{\rm A}^{\bot}\rho_{\rm A})\leq{1-\sqrt{1-\varepsilon^{2}}}$, instead of Formula${\mathop{\rm tr}}(\Pi_{\rm A}^{\bot}\rho_{\rm A})\leq{{\varepsilon^{2}}\over{2}}$, which eventually leads to slightly tighter correction terms. Formula$\blackboxfill$

Claim 2

For any Formula$\rho_{\rm AB}\in S_{\leq}\left({\cal H}{\rm AB}\right)$ there exists a state Formula$\bar\rho_{\rm AB}\in{\cal B}^{\varepsilon}\left(\rho_{\rm AB}\right)$ that satisfies Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({\bar{\rho}_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq\min_{{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}{\rm A}\right)},}\atop\scriptstyle{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}+c(\varepsilon,\rho_{\rm AB}),&{\hbox{(C.54)}}}$$ where Formula$c(\varepsilon,\rho_{\rm AB})\mathrel{\mathop:}=\log\left({{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+{{1}\over{{\mathop{\rm tr}}\rho_{\rm AB}}}\right)$.

Proof

Let Formula$\lambda$, Formula$\bar{\sigma}_{\rm A}$, Formula$\bar{\sigma}_{\rm B}$ be such that Formula TeX Source $$\eqalignno{&\quad~\min_{{\scriptstyle{\sigma_{\rm A}\in S_{=}\left({\cal H}{\rm A}\right)},}\atop\scriptstyle{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}}{D_{\max}\left({\rho_{\rm AB}}\Vert{\sigma_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&={D_{\max}\left({\rho_{\rm AB}}\Vert{\bar{\sigma}_{\rm A}\otimes\bar{\sigma}_{\rm B}}\right)}=\log\lambda.}$$ Let us also define the positive semi-definite operator Formula$\Delta_{\rm A}\mathrel{\mathop:}=\rho_{\rm A}-{\mathtilde\rho}_{\rm A}$ and set Formula$\bar\rho_{\rm AB}={\mathtilde\rho}_{\rm AB}+\Delta_{\rm A}\otimes\bar\sigma_{\rm B}$. It holds that Formula$\bar\rho_{\rm A}=\rho_{\rm A}$ and Formula$\bar\rho_{\rm AB}~{\approx_{\varepsilon}}\rho_{\rm AB}$, which can be seen as follows: since Formula${\mathtilde\rho}_{\rm AB}\leq\bar\rho_{\rm AB}$, it also holds that Formula$\left\Vert{\sqrt{{\mathtilde\rho}_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}\leq\left\Vert{\sqrt{\bar\rho_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}$. Hence, Formula TeX Source $$\eqalignno{F(\bar\rho_{\rm AB},\rho_{\rm AB})\geq &\,\left\Vert{\sqrt{{\mathtilde\rho}_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}+1-{\mathop{\rm tr}}\rho_{\rm AB}\cr\geq &\,\left\Vert{\sqrt{{\mathtilde\rho}_{\rm ABC}}\sqrt{\rho_{\rm ABC}}}\right\Vert_{1}+1-{\mathop{\rm tr}}\rho_{\rm AB}\cr=&\, 1-{\mathop{\rm tr}}(\Pi_{\rm BC}^{\bot}\rho_{\rm BC})\cr\geq &\,{\sqrt{1-\varepsilon^{2}}},}$$ and thus Formula$P(\bar\rho_{\rm AB},\rho_{\rm AB})\leq\varepsilon$. The equality in the penultimate line is a consequence of (A.43) in Lemma 13.

Finally, we use Formula$\bar\rho_{\rm A}=\rho_{\rm A}$ and Formula$\bar\rho_{\rm AB}\leq{\mathtilde\rho}_{\rm AB}+\rho_{\rm A}\otimes\bar\sigma_{\rm B}$ to find Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({\bar{\rho}_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq\log\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}\bar{\rho}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\leq\log\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}({\mathtilde\rho}_{\rm AB}+\rho_{\rm A}\otimes\bar\sigma_{\rm B})\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&=\log\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}^{-{{1}\over{2}}}_{\rm B}{\mathtilde\rho}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\bar{\sigma}_{\rm B}^{-{{1}\over{2}}}+\rho_{\rm A}^{0}\otimes\bar\sigma^{0}_{\rm B}}\right\Vert_{\infty}\cr&\leq\log\left(\lambda{{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+1\right)\!.}$$ The first inequality is justified, as Formula TeX Source $$\mathop{\rm supp}\bar\rho_{\rm B}=\mathop{\rm supp}\left({\mathtilde\rho}_{\rm B}+{\mathop{\rm tr}}(\Delta_{\rm A})\cdot\bar\sigma_{\rm B}\right)\subseteq\mathop{\rm supp}\bar\sigma_{\rm B}.$$ Since Formula$\lambda\geq {\mathop{\rm tr}}\rho_{\rm AB}$, we conclude Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({\bar{\rho}_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\leq\log\lambda+\log\left({{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+{{1}\over{{\mathop{\rm tr}}\rho_{\rm AB}}}\right)\!,}$$ thus completing the proof of Claim 2. Formula$\blackboxfill$

It is now straightforward to prove the upper bound in the theorem, the lower bound given by Formula TeX Source $${^{3}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq{^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$$ being clear from the definitions. Let Formula$\rho^{\prime}_{\rm AB}\in{\cal B}^{\varepsilon^{\prime}}\left(\rho_{\rm AB}\right)$ be the operator that optimizes Formula${^{3}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}$. Then, by Claim 2, there exists an operator Formula$\bar\rho_{\rm AB}\in{\cal B}^{\varepsilon+{\varepsilon^{\prime}}}\left(\rho_{\rm AB}\right)$ such that Formula TeX Source $$\eqalignno{&\quad~\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}_{\rm B}\right)}{D_{\max}\left({\bar{\rho}_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\!\leq{^{3}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+\log\left({{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+{{1}\over{{\mathop{\rm tr}}\rho^{\prime}_{\rm AB}}}\right)\cr&\!\leq{^{3}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}+\log\left({{1}\over{1-\sqrt{1-\varepsilon^{2}}}}+{{1}\over{1-{\varepsilon^{\prime}}}}\right)\!.}$$ It remains to notice that by definition of Formula$^{2}\!I_{\max}^{\varepsilon}$ Formula TeX Source $${^{2}\!I^{\varepsilon+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\bar{\rho}_{\rm AB}}\Vert{\bar\rho_{\rm A}\otimes\sigma_{\rm B}}\right)},$$ which concludes the Proof of Theorem 2.

Proof of Theorem 3

The derivation of the equivalence between Formula$^{2}\!I_{\max}^{\varepsilon}$ and Formula$^{1}\!I_{\max}^{\varepsilon}$ is very similar to the one of Theorem 2 and is therefore not carried out with all of its details here. Again, we only need to prove the upper bound of the theorem, since Formula TeX Source $${^{2}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}\leq{^{1}\!I^{\varepsilon+2\sqrt{\varepsilon}+{\varepsilon^{\prime}}}_{\max}\left({\rm A}:{\rm B}\right)_{\rho}}$$ follows directly from the definitions of the quantities.

We find the following fact, analogous to Claim 1:

Claim 3

Let Formula$\rho_{\rm ABC}$ be a purification of Formula$\rho_{\rm AB}\in S_{\leq}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$. Then there exists a projector Formula$\Pi_{\rm AC}$ on Formula${\cal {H}}_{\rm AC}$ such that Formula${\mathtilde{\rho}}_{\rm ABC}\mathrel{\mathop:}=\Pi_{\rm AC}\rho_{\rm ABC}\Pi_{\rm AC}\in{\cal B}^{\varepsilon}\left(\rho_{\rm ABC}\right)$ and Formula TeX Source $$\eqalignno{&\quad~{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\rho_{\rm B}}\right)}\cr&\!\leq\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}+\log{{1}\over{1-\sqrt{1-\varepsilon^{2}}}}.\cr&&{\hbox{(C.55)}}}$$

Proof

The proof of this claim can straightforwardly be adapted from the proof of Claim 1. We then end up estimating a term Formula$\left\Vert{\Pi_{\rm B}\Gamma_{\rm B}\Pi_{\rm B}}\right\Vert_{\infty}$ (with Formula$\Gamma_{\rm B}=\rho_{\rm B}^{-{{1}\over{2}}}\sigma_{\rm B}\rho_{\rm B}^{-{{1}\over{2}}}$) on system B instead of Formula${\rm A}$. As before, we can choose Formula$\Pi_{\rm B}$ such that its dual Formula$\Pi_{\rm AC}$ satisfies Formula${\mathop{\rm tr}}(\Pi^{\bot}_{\rm AC}\rho_{\rm ABC})\leq{1-\sqrt{1-\varepsilon^{2}}}$. Formula$\blackboxfill$

In the following it is sufficient for our purposes to assume that Formula$\rho_{\rm AB}$ is normalized, thanks to Lemma 22. Now define Formula$\Delta_{\rm ABC}\mathrel{\mathop:}=\rho_{\rm ABC}-{\mathtilde\rho}_{\rm ABC}$ and Formula TeX Source $$\bar\rho_{\rm AB}\mathrel{\mathop:}=k\cdot\left({\mathtilde\rho}_{\rm AB}+\rho_{\rm A}\otimes\Delta_{\rm B}+\Delta_{\rm A}\otimes\rho_{\rm B}\right),$$ with Formula$k\mathrel{\mathop:}={{1}\over{1+{\mathop{\rm tr}}\Delta_{\rm ABC}}}$. Notice that Formula$\Delta_{\rm B}\geq 0$, but Formula$\Delta_{\rm A}$ and thus Formula$\bar\rho_{\rm AB}$ is not necessarily positive semi-definite. However, Formula${\mathop{\rm tr}}\bar\rho_{\rm AB}=1$ and by construction we have that Formula$\bar\rho_{\rm A}=\rho_{\rm A}$ and Formula$\bar\rho_{\rm B}=\rho_{\rm B}$. We now want to construct from it a positive semi-definite and sub-normalized operator Formula${\mathhat\rho}_{\rm AB}$ such that Formula$^{1}\!I_{\max}({\rm A}:{\rm B})_{\mathhat\rho}$ is a lower bound to Formula${D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\rho_{\rm B}}\right)}$ and Formula$P(\rho_{\rm AB},{\mathhat\rho}_{\rm AB})\leq c(\varepsilon)$ with Formula$c(\varepsilon)$ a function that vanishes as Formula$\varepsilon\rightarrow 0$.

We can write Formula$\Delta_{\rm A}$ as Formula$\Delta_{\rm A}=\Delta_{\rm A}^{+}-\Delta_{\rm A}^{-}$, where Formula$\Delta_{\rm A}^{+}$ and Formula$\Delta_{\rm A}^{-}$ are positive semi-definite operators with mutually orthogonal supports. Now we define Formula TeX Source $$\eqalignno{{\mathhat\rho}_{\rm AB}\mathrel{\mathop:}=&\,n\cdot (\bar\rho_{\rm AB}+k\cdot\Delta_{\rm A}^{-}\otimes\rho_{\rm B})\cr=&\, nk\cdot\left({\mathtilde\rho}_{\rm AB}+\rho_{\rm A}\otimes\Delta_{\rm B}+\Delta_{\rm A}^{+}\otimes\rho_{\rm B}\right)\!,}$$ where Formula$n\mathrel{\mathop:}=(1+k\cdot{\mathop{\rm tr}}\Delta_{\rm A}^{-})^{-1}$ is a normalization constant such that Formula${\mathop{\rm tr}}{\mathhat\rho}_{\rm AB}=1$. Clearly now, Formula${\mathhat\rho}_{\rm AB}$ is positive semi-definite and we want to estimate Formula${D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}$. Notice that Formula${\mathhat\rho}_{\rm A}=n\cdot (\rho_{\rm A}+k\Delta_{\rm A}^{-})$ and Formula${\mathhat\rho}_{\rm B}=\rho_{\rm B}$. Hence, Formula TeX Source $$\eqalignno{&\,{D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\cr=&\log\left\Vert{{\mathhat\rho}_{\rm A}^{-{{1}\over{2}}}\otimes{\mathhat\rho}^{-{{1}\over{2}}}_{\rm B}{\mathhat\rho}_{\rm AB}{\mathhat\rho}_{\rm A}^{-{{1}\over{2}}}\otimes{\mathhat\rho}_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr=&\log\Big\Vert (\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}\bar\rho_{\rm AB}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}\cr&\qquad\quad~~\,+(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}k\Delta_{\rm A}^{-}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\otimes\rho^{0}_{\rm B}\Big\Vert_{\infty},}$$ and, with the triangle inequality, Formula TeX Source $$\eqalignno{&~{D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\cr&\!\leq\log\!\Bigg (\!\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}\bar\rho_{\rm AB}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\!\otimes\!\rho_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\qquad\quad+\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}k\Delta_{\rm A}^{-}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}}\right\Vert_{\infty}\Bigg).}$$ We can decompose the first term in the logarithm even further and with Formula$k\leq 1$, Formula$\Delta_{\rm B}\leq\rho_{\rm B}$ obtain Formula TeX Source $$\eqalignno{&~{D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\cr&\leq\log\!\Bigg (\!\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\!\otimes\!\rho_{\rm B}^{-{{1}\over{2}}}{\mathtilde\rho}_{\rm AB}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\!\otimes\!\rho_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\,\qquad\quad+2\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}\rho_{\rm A}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\,\qquad\quad+\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}{\mathtilde\rho}_{\rm A}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}}\right\Vert_{\infty}\cr&\,\qquad\quad+\left\Vert{(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}k\Delta_{\rm A}^{-}(\rho_{\rm A}+k\Delta_{\rm A}^{-})^{-{{1}\over{2}}}}\right\Vert_{\infty}\Bigg).}$$

Now we apply Lemma 18 to all terms inside the logarithm and replace Formula$(\rho_{\rm A}+k\Delta_{\rm A}^{-})$ with Formula$\rho_{\rm A}$ in the first three terms and with Formula$k\Delta_{\rm A}^{-}$ in the last one, obtaining Formula TeX Source $$\eqalignno{&\quad~{D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\cr&\leq\log\left(2\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}{\mathtilde\rho}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}+3\right)\cr&\leq{D_{\max}\left({{\mathtilde{\rho}}_{\rm AB}}\Vert{\rho_{\rm A}\otimes\rho_{\rm B}}\right)}+\log\left(2+{{3}\over{1-\varepsilon}}\right)\!.}$$ In the last line, we have used that Formula TeX Source $$\left\Vert{\rho_{\rm A}^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}{\mathtilde\rho}_{\rm AB}\rho_{\rm A}^{-{{1}\over{2}}}\otimes\rho_{\rm B}^{-{{1}\over{2}}}}\right\Vert_{\infty}\geq 1-\varepsilon.$$ Thus, Formula TeX Source $$\eqalignno{&\quad~{D_{\max}\left({{\mathhat\rho}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\cr&\!\leq\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&+\log\left({{2(1-\varepsilon)+3}\over{(1-\varepsilon)(1-\sqrt{1-\varepsilon^{2}})}}\right).}$$

Let us finally find an estimate for Formula$P(\rho_{\rm AB},{\mathhat\rho}_{\rm AB})$. Recall that Formula TeX Source $${\mathop{\rm tr}}\Delta_{\rm ABC}={\mathop{\rm tr}}(\Pi^{\bot}_{\rm AC}\rho_{\rm ABC})\leq{1-\sqrt{1-\varepsilon^{2}}}$$ according to our choice of Formula$\Pi_{\rm AC}$ in Claim 3, which implies Formula TeX Source $$k\geq {{1}\over{2-\sqrt{1-\varepsilon^{2}}}}.$$ We further have that Formula${\mathop{\rm tr}}\Delta_{\rm A}^{-}\leq 2\varepsilon$ and therefore Formula$n\geq {{1}\over{1+2\varepsilon}}$. Thus, with Corollary 16, Formula TeX Source $$P({\mathtilde\rho}_{\rm AB},nk\cdot{\mathtilde\rho}_{\rm AB})\leq\sqrt{1-n^{2}k^{2}}\leq2\sqrt\varepsilon,$$ and consequently Formula TeX Source $$\eqalignno{P(\rho_{\rm AB},nk\cdot{\mathtilde\rho}_{\rm AB})\leq &\, P(\rho_{\rm AB},{\mathtilde\rho}_{\rm AB})+P({\mathtilde\rho}_{\rm AB},nk\cdot{\mathtilde\rho}_{\rm AB})\cr\leq &\,\varepsilon+2\sqrt\varepsilon.}$$ As Formula$nk\cdot{\mathtilde\rho}_{\rm AB}\leq{\mathhat\rho}_{\rm AB}$ and therefore with Lemma 13 Formula TeX Source $$\left\Vert{\sqrt{nk\cdot{\mathtilde\rho}_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1}\leq\left\Vert{\sqrt{{\mathhat\rho}_{\rm AB}}\sqrt{\rho_{\rm AB}}}\right\Vert_{1},$$ we conclude that also Formula$P({\mathhat\rho}_{\rm AB},\rho_{\rm AB})\leq\varepsilon+2\sqrt{\varepsilon}$.

In summary, we have just proven the following claim.

Claim 4

For any Formula$\rho_{\rm AB}\in S_{=}\left({\cal H}{\rm AB}\right)$ and Formula$\varepsilon>0$, there exists a state Formula${\mathhat\rho}_{\rm AB}~{\approx_{\varepsilon+2\sqrt{\varepsilon}}}\rho_{\rm AB}$ that satisfies Formula TeX Source $$\eqalignno{&{D_{\max}\left({{\mathhat{\rho}}_{\rm AB}}\Vert{{\mathhat\rho}_{\rm A}\otimes{\mathhat\rho}_{\rm B}}\right)}\leq\min_{\sigma_{\rm B}\in S_{=}\left({\cal H}{\rm B}\right)}{D_{\max}\left({\rho_{\rm AB}}\Vert{\rho_{\rm A}\otimes\sigma_{\rm B}}\right)}\cr&\hskip10.2em+\log\left({{2(1-\varepsilon)+3}\over{(1-\varepsilon)(1-\sqrt{1-\varepsilon^{2}})}}\right).\cr& &{\hbox{(C.56)}}}$$

To conclude the Proof of Theorem 3, let Formula$\rho_{\rm AB}\in S_{=}\left({{\cal H}{\rm AB}}\right)$ and let Formula$\rho^{\prime}_{\rm AB}\in{\cal B}^{\varepsilon^{\prime}}\left(\rho_{\rm AB}\right)$ be a normalized operator such that Formula${^{2}\!I^{\varepsilon^{\prime}}_{\!\!\max}\left({\rm A}:{\rm B}\right)_{\rho}}={^{2}\!I}_{\max}({\rm A}:{\rm B})_{\rho^{\prime}}$. Applying Claim 4 to Formula$\rho^{\prime}_{\rm AB}$ yields the result.

ACKNOWLEDGMENT

NC would like to thank Joseph M. Renes and Frédéric Dupuis for helpful discussions.

Footnotes

This work was supported in part by the Swiss National Science Foundation through the National Centre of Competence in Research Quantum Science and Technology, Grant 200020-135048, and in part by the European Research Council under Grant 258932. This paper was presented at the 2013 Beyond i.i.d. in Information Theory Workshop, Cambridge, UK.

N. Ciganović is with the Department of Bioengineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: n.ciganovic13@imperial.ac.uk).

N. J. Beaudry and R. Renner are with the Institute for Theoretical Physics, ETH Zürich, Zürich 8093, Switzerland (e-mail: nbeaudry@phys.ethz.ch; renner@phys.ethz.ch).

Communicated by A. Holevo, Associate Editor for Quantum Information Theory.

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Nikola Ciganović

Nikola Ciganović was born on October 23, 1987, in Zagreb (Croatia). He studied physics at ETH Zurich (Switzerland) and received his MSc degree in 2012. Since November 2013, he is pursuing postgraduate studies at Imperial College London, where his research focuses on the biophysics of hearing.

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Normand J. Beaudry

Normand J. Beaudry was born on September 19, 1984, in Scarborough (Ontario, Canada). He studied mathematical physics at the University of Waterloo and graduated in 2006. After one year working away from academia, he returned to the University of Waterloo (Canada) to complete a Masters degree in physics at the Institute of Quantum Computing in 2009. Since 2010, he has been working towards his PhD at ETH Zurich (Switzerland). His research interests include information theory and quantum cryptography.

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Renato Renner

Renato Renner was born on December 11, 1974, in Lucerne (Switzerland). He studied physics, first at EPF Lausanne and later at ETH Zurich (Switzerland), where he graduated in theoretical physics in 2000. He then moved to the Computer Science Department of ETH to work on a thesis in the area of quantum cryptography. He received his PhD degree in 2005.

Between 2005 and 2007, he held a HP research fellowship in the Department for Applied Mathematics and Theoretical Physics at the University of Cambridge (UK). Since 2007, he has been with the ETH Zurich Physics Department, first as an Assistant Professor and later as an Associate Professor for Theoretical Physics. His research interests are ranging from quantum information science to foundations of quantum mechanics and thermodynamics.

Prof. Renner is a member of the American Physical Society and of the IEEE. He has been a member of the technical program committee of ISIT 2008.

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