Distributed Quantum Faithful Simulation and Function Computation Using Algebraic Structured Measurements
11 Distributed Quantum Faithful Simulation and FunctionComputation Using Algebraic Structured Measurements
Touheed Anwar Atif and S. Sandeep PradhanDepartment of Electrical Engineering and Computer Science,University of Michigan, Ann Arbor, MI 48109, USA.Email: [email protected], [email protected]
Abstract
In this work, we consider the task of faithfully simulating a distributed quantum measurement and functioncomputation, and demonstrate a new achievable information-theoretic rate-region. For this, we develop the techniqueof randomly generating structured POVMs using algebraic codes. To overcome the challenges caused by algebraicconstruction, we develop a Pruning Trace inequality which is a tighter version of the known operator Markovinequality. In addition, we develop a covering lemma which is independent of the operator Chernoff inequality soas to be applicable for pairwise-independent codewords. We demonstrate rate gains for this problem over traditionalcoding schemes. Combining these techniques, we provide a multi-party distributed faithful simulation and functioncomputation protocol. C ONTENTS
I Introduction II Preliminaries III Novel Lemmas IV Measurement Compression using Structured Random Codes V Simulation of Distributed POVMs using Structured Random Codes VI Proof of Theorem 2 VII Conclusion Appendix A: Proof of Lemmas Appendix B: Proof of Propositions References This work was supported by NSF grant CCF-2007878. a r X i v : . [ c s . I T ] J a n . I NTRODUCTION M Easurements provide an interface to the intricate quantum world by associating classical attributes toquantum states. However, quantum phenomena, such as superposition, entanglement and non-commutativitycontribute to uncertainty in measurement outcomes. As a result, these quantum measurements can be treated asspecial channels, with their input as quantum states and output as classical bits. A key concern from an information-theoretic viewpoint, is to characterize the amount of “relevant information” conveyed by a measurement abouta quantum state. This is accomplished, in its fullest potential, not just by compressing the classical outputsobtained from these measurements, but also by eliminating the inherent redundancy present within these quantummeasurements.One of the seminal works in this regard was provided by Winter [1], where he performed a novel informationtheoretic analysis to compress measurements in an asymptotic sense. The major contribution of this work (aselaborated in [2]) was in specifying an optimal rate region in terms of classical communication and commonrandomness needed to faithfully simulate the action of repeated independent measurements performed on manyindependent copies of the given quantum state. Consider an agent (Alice) who performs a measurement M on aquantum state ρ , and sends a set of classical bits to a receiver (Bob). Bob intends to faithfully recover the outcomesof Alice’s measurements without having access to ρ . The measurement compression theorem [1] states that inorder to obtain a faithful simulation one needs at least quantum mutual information ( I p X ; R q ) amount of classicalinformation and conditional entropy ( S p X | R q ) amount of common shared randomness, where R denotes a referenceof the quantum state, and X denotes the auxiliary register corresponding to the random measurement outcome.Wilde et al. [2] extended the measurement compression problem by considering additional resources available toeach of the participating parties. One such formulation allows Bob to process the information received from Aliceusing local private randomness. The authors here also combined the ideas from [1] and [3] to simulate a measurementin presence of quantum side information. In the above problem formulations, authors have derived the results usingthe prevalent random coding techniques analogous to Shannon’s unstructured random codes [4] involving mutuallyindependent codewords. The point-to-point setup [1], [2] requires randomly generating approximating POVMs andanalyzing the error associated with these approximating POVMs, also termed as “covering error”. The key analyticaltool that facilitates this is the operator Chernoff bound [5], which crucially exploits the mutual independence ofcodewords, yielding the quantum covering lemma [6].Furthermore, the authors in [7], [8] considered the task of quantifying “relevant information” for the quantummeasurements performed in a distributed fashion on bipartite entangled states involving three agents. In this multi-terminal setting, a composite bipartite quantum system AB is made available to two agents, Alice and Bob, wherethey have access to the sub-systems A and B , respectively. Two separate measurements, one for each sub-system,are performed in a distributed fashion with no communication taking place between Alice and Bob. A third party,Eve, is connected to Alice and Bob via two separate classical links. The objective of the three parties is to simulatethe action of repeated independent measurements performed on many independent copies of the given compositestate. Further, common randomness at rate C is also shared amidst the three parties. This is achieved using randomunstructured code ensembles while still using the operator Chernoff bound. A similar result is proved for theclassical setting in [9], [10].An ubiquitous application of distributed systems in current quantum settings arises due to the inherent vulnerabilityof the large-scale quantum computation systems to noise. The state-of-art systems exhibits technical difficulties in2ncreasing the number of low-noise qubits in a single quantum device. A solution to this is cooperative processing ofinformation separately on spatially segregated units. This necessitates the need for distributed compression protocolsto compress efficiently and recover the data. In addition, when one is interested in solely reconstructing functions ofthe distributively stored quantum data, the rate of communication may be further reduced by employing structuredcoding techniques. For this, we need to impose further structure on these POVMs. This is to ensure that thejoint decoder (Eve) is able to reconstruct a lower dimensional quantum state with minimal use of the classicalcommunication resource. Hence, structure of the POVM is desired to match with the structure of the function beingcomputed.The traditional random coding techniques using unstructured code ensembles may not always achieve optimalityfor distributed multi-terminal settings. For instance, the work by Korner-Marton [11] demonstrated this sub-optimality for a classical distributed lossless compression problem with symmetric binary sources using ran-dom linear codes. Traditionally, algebraic-structured codes are used in classical information coding problemstoward achieving computationally efficient (polynomial-time) encoding and decoding algorithms. However, in multi-terminal communication problems, even if computational complexity is a non-issue, random algebraic structuredcodes outperform random unstructured codes in terms of achieving improved asymptotic rate regions, in most cases.In fact, algebraic structured codes may be necessary to achieve optimality in multi-terminal setups [12]–[15].Motivated by this, we consider the quantum distributed faithful measurement simulation problem and present anew achievable rate-region using structured coding techniques. However, there are two main challenges in usingthese algebraic structured codes toward an asymptotic analysis in quantum information theory. The first challengeis to be able to induce arbitrary empirical single-letter distributions. For example, if we were to send codewordsfrom a linear code with uniform probability, then the induced empirical distribution of codeword symbols (single-letter distribution on the symbols of the codewords) is uniform. To address this challenge, we use a collectionof cosets of a linear code called Unionized Coset Codes (UCCs) [16]. The second challenge is that unlike therandom unstructured codes, the codewords generated from a random linear code are only pairwise-independent[17]. This renders the above technique of operator Chernoff bound, or even the covering lemma, unusable. Sinceour approach relies on the use of UCCs for generating the approximating POVMs, the binning of these POVMelements is performed in a correlated fashion as governed by these structured codes. This is in contrast to thecommon technique of independent binning. Due to the correlated binning, the pairwise-independence issue getsexacerbated.We address these challenges using four main ideas summarized as follows: ‚ Random structured generation of pruned POVMs - We generate a collection of structured approximatingPOVMs randomly using the above described UCC technique, and then prune them. This pruning ensures thatthese POVMs form a positive resolution of identity, and thus eliminates any need for the operator Chernoffinequality. However, such pruning comes at the cost of additional approximating error. ‚ Pruning Trace Inequality - To bound the approximating error caused by pruning the POVMs, we develop anew Operator Inequality which provides a handle to convert the pruning error in the form of covering errorexpression (dealt within the next idea). We later prove that this new inequality is indeed a tightening of theexisting Operator Markov Inequality [18]. ‚ Covering Lemma for Pairwise-Independent Ensemble - Since the traditional covering lemma is based onthe Chernoff inequality, we develop an alternative proof for the aforementioned covering lemma [18]. This3lternative proof is based on the second-order analysis using the operator trace inequalities and hence requiresthe operators to be only pairwise-independent. ‚ Multi-partite Packing Lemma - We develop a binning technique for performing computation on the fly so asto achieve a low dimensional reconstruction of a function at the location of Eve. In an effort towards analysingthis binning technique, we develop a multi-partite packing Lemma for the structured POVMs.Combining these techniques, we provide a multi-party distributed faithful simulation and function computationprotocol in a quantum information theoretic setting. We provide a characterization of the asymptotic performancelimit of this protocol in terms of a computable single-letter achievable rate-region, which is the main result of thepaper (see Theorem 2).The organization of the paper is as follows. In Section II, we set the notation, state requisite definitions and alsoprovide related results. In Section III we provide the two crucial lemmas mentioned above i.e., (i) The New OperatorInequality and (ii) The Soft Covering Lemma. In Section IV, we consider the point-to-point setup and provide atheorem characterizing the rate-region. In Section V we state our main result on the distributed measurementcompression and provide the theorem (Theorem 2) characterizing the rate-region. We proof this theorem in SectionVI. Finally, we conclude the paper in Section VII.II. P
RELIMINARIES
Notation:
Given any natural number M , let the finite set t , , ¨ ¨ ¨ , M u be denoted by r , M s . Let B p H q denote thealgebra of all bounded linear operators acting on a finite dimensional Hilbert space H . Further, let D p H q denote theset of all unit trace positive operators acting on H . Let I denote the identity operator. The trace distance betweentwo operators A and B is defined as } A ´ B } “ ∆ Tr | A ´ B | , where for any operator Λ we define | Λ | “ ∆ ? Λ : Λ .The von Neumann entropy of a density operator ρ P D p H q is denoted by S p ρ q . The quantum mutual informationfor a bipartite density operator ρ AB P D p H A b H B q is defined as I p A ; B q ρ “ ∆ S p ρ A q ` S p ρ B q ´ S p ρ AB q . A positive-operator valued measure (POVM) acting on a Hilbert space H is a collection M “ ∆ t Λ x u x P X of positiveoperators in B p H q that form a resolution of the identity: Λ x ě , @ x P X , ÿ x P X Λ x “ I, where X is a finite set. If instead of the equality above, the inequality ř x Λ x ď I holds, then the collection is saidto be a sub-POVM. A sub-POVM M can be completed to form a POVM, denoted by r M s , by adding the operator Λ “ ∆ p I ´ ř x Λ x q to the collection. Let Ψ ρRA denote a purification of a density operator ρ P D p H A q . Given aPOVM M “ ∆ t Λ Ax u x P X acting on ρ , the post-measurement state of the reference together with the classical outputsis represented by p id b M qp Ψ ρRA q “ ∆ ÿ x P X | x yx x | b Tr A tp I R b Λ Ax q Ψ ρRA u . (1)Consider two POVMs M A “ t Λ Ax u x P X and M B “ t Λ By u y P Y acting on H A and H B , respectively. Define M A b M B “ ∆ t Λ Ax b Λ By u x P X ,y P Y With this definition, M A b M B is a POVM acting on H A b H B . By M b n denote the n -foldtensor product of the POVM M with itself. 4 efinition 1 (Faithful simulation [2]) . Given a POVM M “ ∆ t Λ x u x P X acting on a Hilbert space H and a densityoperator ρ P D p H q , a sub-POVM ˜ M “ ∆ t ˜Λ x u x P X acting on H is said to be (cid:15) -faithful to M with respect to ρ , for (cid:15) ą , if the following holds: ÿ x P X ››› ? ρ p Λ x ´ ˜Λ x q? ρ ››› ` Tr p I ´ ÿ x ˜Λ x q ρ + ď (cid:15). (2) Lemma 1.
Given a density operator ρ AB P D p H AB q , a sub-POVM M Y “ ∆ (cid:32) Λ By : y P Y ( acting on H B , for someset Y , and any Hermitian operator Γ A acting on H A , we have ÿ y P Y ›› ? ρ AB ` Γ A b Λ By ˘ ? ρ AB ›› ď ›› ? ρ A Γ A ? ρ A ›› , (3) with equality if ÿ y P Y Λ By “ I , where ρ A “ Tr B t ρ AB u .Proof. The proof is provided in Lemma 3 of [8].III. N
OVEL L EMMAS
Definition 2 (Pruning Operators) . Consider an operator A ě acting on Hilbert space H A . We say that a projector P prunes A with respect to Identity I A on H A , if P is a projector on to the non-negative eigenspace of I A ´ A . A. Pruning Trace Inequality
Lemma 2.
Consider a random operator X ě acting on a Hilbert space H A . Let P be a pruning operator for X with respect to I A , as in Definition 2. Then we have E r Tr t I A ´ P us ď E r Tr t X us . Proof.
The proof follows by noting that Tr t I A ´ P u ď Tr t X u . Remark 1.
To demonstrate the significance of this inequality, we compare it with the popular Operator MarkovInequality [18]. We know from Operator Markov inequality P p X ę I A q ď E r Tr t X us . One can observe that t X ę I A u ď Tr t I A ´ P u . Taking expectation, we obtain P p X ę I A q ď E r Tr t I A ´ P us . Moreover, one can also note that Tr t I A ´ P u ď Tr t X u , and expectation gives E r Tr t I A ´ P us ď E r Tr t X us . Hence we conclude that the new inequality is indeed tighter than the operator Markov inequality.
Lemma 3. (Pruning Trace Inequality) Consider the above random operator X ě acting on a Hilbert space H A . Further, suppose E r X s ď p ´ η q I A for η P p , q . Let P be a pruning operator for X with respect to I A , as inDefinition 2. Then, we have E r Tr t I A ´ P us ď η E r} X ´ E r X s} s . (4) Proof.
The proof is provided in Appendix A-A 5 . Covering Lemma with Change of Measure for Pairwise-Independent Ensemble
Lemma 4.
Let t λ x , σ x u x P X be an ensemble, with σ x P D p H q for all x P X , X being a finite set, and σ “ ř x P X λ x σ x . Further, suppose we are given a total subspace projector Π and a collection of codeword subspaceprojectors t Π x u x P X which satisfy the following hypotheses Tr t Π σ x u ě ´ (cid:15), (5a) Tr t Π x σ x u ě ´ (cid:15), (5b) } Π ? σ } ď D, (5c) Π x σ x Π x ď d Π x , and (5d) Π x σ x Π x ď σ x . (5e) for some (cid:15) P p , q and d ă D . Let M be a finite non-negative integer. Additionally, assume that there exists someset ¯ X containing X , with σ x “ ∆ (null operator) and λ x “ ∆ for x P ¯ X z X . Suppose t µ ¯ x u ¯ x P ¯ X be any distributionon the set ¯ X such that the distribution is t λ x u x P X is absolutely continuous with respect to the distribution t µ ¯ x u ¯ x P ¯ X .Further, assume that λ x { µ x ď κ for all x P X . Let a random covering code C “ ∆ t C m u m Pr ,M s be defined as acollection of codewords C m that are chosen pairwise independently according to the distribution t µ ¯ x u ¯ x P ¯ X . Thenwe have E C «››› ÿ x P ¯ X λ x σ x ´ M M ÿ m “ λ C m µ C m σ C m ››› ff ď c κDM d ` δ p (cid:15) q , (6) where δ p (cid:15) q “ ? (cid:15) . Futhermore, for ˜ σ x defined as ˜ σ x “ ∆ ΠΠ x σ x Π x Π , we have E C «››› ÿ x P ¯ X λ x ˜ σ x ´ M M ÿ m “ λ C m µ C m ˜ σ C m ››› ff ď c κDM d . (7) Proof.
The proof is provided in Appendix A-BIV. M
EASUREMENT C OMPRESSION USING S TRUCTURED R ANDOM C ODES
We begin by considering a point-to-point setup of the measurement compression problem and achieve theoptimal rate-region using structured random codes. Since the algebraic structured codes can only induce a uniformdistribution, we consider a collection of cosets of a random linear code for this task. The problem setup is describedas follows. An agent (Alice) performs a measurement M on a quantum state ρ , and sends a set of classical bits toa receiver (Bob). Bob has access to additional private randomness, and he is allowed to use this additional resourceto perform any stochastic mapping of the received classical bits. The overall effect on the quantum state can beassumed to be a measurement which is a concatenation of the POVM Alice performs and the stochastic map Bobimplements. Formally, the problem is stated as follows. A. Problem Formulation
Definition 3.
For a given finite set Z , and a Hilbert space H , a measurement simulation protocol with stochasticprocessing with parameters p n, Θ , N q is characterized by a collections of Alice’s sub-POVMs ˜ M p µ q , µ P r , N s each acting on H b n and with outcomes in a subset L satisfying | L | ď Θ . 6 ) a Bob’s classical stochastic map P p µ q p z n | l q for all l P L , z n P Z n and µ P r , N s .The overall sub-POVM of this distributed protocol, given by ˜ M , is characterized by the following operators: ˜Λ z n “ ∆ N ÿ µ,l P p µ q p z n | l q Λ p µ q l , @ z n P Z n , (8)where Λ p µ q l are the operators corresponding to the sub-POVMs ˜ M p µ q .In the above definition, Θ characterizes the amount of classical bits communicated from Alice to Bob, and theamount of common randomness is determined by N , with µ being the common randomness bits distributed amongthe parties. The classical stochastic mappings induced by P p µ q represents the action of Bob on the received classicalbits. Definition 4.
Given a POVM M acting on H , and a density operator ρ P D p H q , a pair p R, C q is said to beachievable, if for all (cid:15) ą and for all sufficiently large n , there exists a measurement simulation protocol withstochastic processing with parameters p n, Θ , N q such that its overall sub-POVM ˜ M is (cid:15) -faithful to M b n withrespect to ρ b n (see Definition 1), and n log Θ ď R ` (cid:15), n log N ď C ` (cid:15). The set of all achievable pairs is called the achievable rate region.The following theorem characterizes the achievable rate region.
Theorem 1.
For any density operator ρ P D p H q and any POVM M “ ∆ t Λ z u z P Z acting on the Hilbert space H ,a pair p R, C q is achievable if and only if there exist a POVM ¯ M “ ∆ t ¯Λ w u w P W , with W being a finite set, and astochastic map P Z | W : W Ñ Z such that R ě I p R ; W q σ and R ` C ě I p R, Z ; W q σ , Λ z “ ÿ w P W P Z | W p z | w q ¯Λ w , @ z P Z . (9) where σ RW Z “ ∆ ř w,z ? ρ ¯Λ w ? ρ b P Z | W p z | w q | w yx w | b | z yx z | , for some orthogonal sets t | w yu w P W and t | z yu z P Z .B. Proof of Theorem 1 Using UCC Code Ensemble We provide a proof of achievability using the Unionized Coset Codes. Similar proofs have been discussed in [2],[8] using unstructured random codes, however, we proof here (the achievability result) using algebraic structuredcodes. The main objective of proving this theorem using structured codes is to build a platform for the main theoremof the paper (Theorem 2). In doing so, we observe that the structured POVMs constructed below are only pairwiseindependent. Since the results in [8] are based on the assumption that approximating POVMs are all mutuallyindependent, the proofs below become significantly different from [8].Suppose there exist a POVM ¯ M “ ∆ t ¯Λ w u w P W and a stochastic map P Z | W : W Ñ Z , such that M “ ∆ t Λ z u z P Z can be decomposed as Λ z “ ÿ w P W P Z | W p z | w q ¯Λ w , @ z P Z . (10)We generate the canonical ensemble corresponding to ¯ M as λ w “ ∆ Tr t ¯Λ w ρ u , ˆ ρ w “ ∆ λ w ? ρ ¯Λ w ? ρ. (11)7et T p n q δ p W q denote a δ -typical set corresponding to the probability distribution induced by t λ w u w P W , correspondingto a random variable W . Let Π ρ and Π w n denote the δ -typical and the conditional typical projectors (as in [19])for marginal density operator ρ “ ř w P W λ w ˆ ρ w and ˆ ρ w n for w n P T p n q δ p W q , respectively. For each w n P T p n q δ p W q ,define ˜ ρ w n “ ∆ Π ρ Π w n ˆ ρ w n Π w n Π ρ , and ˜ ρ w n “ , for w n R T p n q δ p W q , with ˆ ρ w n “ ∆ Â i ˆ ρ w i .
1) Construction of Structured POVMs:
We now construct the random structured POVM elements. Fix a blocklength n ą , a positive integer N, and a finite field F p with p ě | W | . Without loss of generality, we assume W “ ∆ t , , ¨ ¨ ¨ , | W | ´ u . Furthermore, we assume λ w “ for all | W | ´ ă w ă p . From now on, we assumethat W takes values in F p with this distribution. Let µ P r , N s denote the common randomness shared betweenthe encoder and decoder. In building the code, we use the Unionized Coset Code (UCC) [16] defined as below.These codes involve two layers of codes (i) a coarse code and (ii) a fine code. The coarse code is a coset of thelinear code and the fine code is the union of several cosets of the linear code.For a fixed k ˆ n matrix G P F k ˆ np with k ď n , and a ˆ n vector B P F np , define the coset code as C p G, B q “ ∆ t x n : x n “ a k G ` B, for some a k P F kp u . In other words, C p G, B q is a shift of the row space of the matrix G . The row space of G is a linear code. If therank of G is k , then there are p k codewords in the coset code. Definition 5. An p n, k, l, p q UCC is a pair p G, h q consisting of a k ˆ n matrix G P F k ˆ np , and a mapping h : F lp Ñ F np .In the context of UCC, define the composite code as C “ Ť i P F lp C p G, h p i qq .For every µ P r , N s , consider a UCC p G, h p µ q q with parameters p n, k, l, p q . For each µ , the generator matrix G along with the function h µ generates p k ` l codewords. Each of these codewords are characterized by a triple p a, i, µ q , where a P F kp and i P F lp correspond to the coarse code and the coset indices, respectively. Let W n, p µ q p a, i q denote the codewords associated with the encoder (Alice), generated using the above procedure, where W n, p µ q p a, i q “ aG ` h p µ q p i q . (12)Now, construct the operators ¯ A p µ q w n “ ∆ α w n ˆa ρ b n ´ ˜ ρ w n a ρ b n ´ ˙ and α w n “ ∆ p ` η q p n λ w n nS , (13)with η P p , q being a parameter to be determined, and S “ ∆ k ` ln log p . Note that, following the definition of ˜ ρ w n ,we have ¯ A p µ q w n “ for w n R T p n q δ p W q . Having constructed the operators ¯ A p µ q w n , we normalize these operators, sothat they constitute a valid sub-POVM. To do so, we define Σ p µ q “ ÿ w n γ p µ q w n ¯ A p µ q w n and γ p µ q w n “ ∆ |tp a, i q : W n, p µ q p a, i q “ w n u| . Now, we define Π µ as the pruning operator for Σ p µ q with respect to Π ρ using Definition 2. Note that, the pruningoperator Π µ depends on the pair p G, h p µ q q . For ease of analysis, the subspace of Π µ is restricted to Π ρ and hence Π µ is a projector onto a subspace of Π ρ . Using these pruning operators, for each µ P r , N s , construct the sub-POVM ˜ M p n,µ q as ˜ M p n,µ q “ ∆ t γ p µ q w n A p µ q w n u w n P W n , (14)8here A p µ q w n “ ∆ Π µ ¯ A p µ q w n Π µ . Further, using Π µ we have ř w n γ p µ q w n A p µ q w n “ Π µ Σ p µ q Π µ ď Π ρ ď I, and thus ˜ M p n,µ q is a valid sub-POVM for all µ P r , N s . Moreover, the collection ˜ M p n,µ q is completed using the operators I ´ ř w n P W n γ p µ q w n A p µ q w n .
2) Binning of POVMs:
The next step using the UCC is to bin the above constructed sub-POVMs. We proceedas follows. Since, UCC is a union of several cosets, we associate a bin to each coset, and hence place all thecodewords of a coset in the same bin. For each i P F lp , let B p µ q p i q “ ∆ C p G, h p µ q p i qq denote the i th bin. Further, forall i P F lp , we define Γ A, p µ q i “ ∆ ÿ w n P W n ÿ a P F kp A p µ q w n t aG ` h p µ q p i q“ w n u . Using these operators, we form the following collection: M p n,µ q “ ∆ t Γ A, p µ q i u i P F lp . Note that if the collection ˜ M p n,µ q is a sub-POVM for each µ P r , N s , then so is the collection M p n,µ q , whichis due to the relation ř i P F lp Γ A, p µ q i “ ř w n P W n γ p µ q w n A p µ q w n ď I. To complete M p n,µ q , we define Γ A, p µ q as Γ A, p µ q “ I ´ ř i Γ A, p µ q i . Now, we intend to use the completions r M p n,µ q s as the POVM for the encoder.
3) Decoder mapping :
We create a decoder which, on receiving the classical bits from the encoder, generates asequence W n P F np as follows. The decoder first creates a set D p µ q i and a function F p µ q defined as D p µ q i “ ∆ (cid:32) ˜ a P F kp :˜ aG ` h p µ q p i q P T p n q δ p W q , ( and F p µ q p i q “ ∆ ˜ aG ` h p µ q p i q if D p µ q i ” t ˜ a u w n otherwise , (15)where w n is an additional sequence added to F np . Further, F p µ q p i q “ w n for i “ . Given this and the stochasticprocessing P Z | W , we obtain the approximating sub-POVM ˆ M p n q with the following operators. ˆΛ z n “ ∆ N N ÿ µ “ ÿ w n P F np Ť t w n u ÿ i : F p µ q p i q“ w n Γ A, p µ q i P nZ | W p z n | w n q , @ z n P Z n . Code Ensemble:
The generator matrix G and the function h p µ q are chosen randomly uniformly and independently.
4) Trace Distance:
In what follows, we show that ˆ M p n q is (cid:15) -faithful to M b n with respect to ρ b n (accordingto Definition 1), where (cid:15) ą can be made arbitrarily small. More precisely, using (10), we show that, E r K s ď (cid:15), where K “ ∆ ÿ z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ÿ w n a ρ b n ¯Λ w n a ρ b n P nZ | W p z n | w n q ´ a ρ b n ˆΛ z n a ρ b n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (16)where the expectation is with respect to the codebook generation. Step 1: Isolating the effect of error induced by not covering
Consider the second term within K , which can be written as a ρ b n ˆΛ z n a ρ b n “ N ÿ µ ÿ i a ρ b n Γ A, p µ q i a ρ b n P nZ | W p z n | F p µ q p i qq ÿ w n t F p µ q p i q“ w n u loooooooomoooooooon “ “ T ` r T , Note that Γ A, p µ q “ I ´ ř i Γ A, p µ q i “ I ´ ř w n P T p n q δ p W q γ p µ q w n A p µ q w n . T “ ∆ N ÿ µ ÿ i ą a ρ b n Γ A, p µ q i a ρ b n P nZ | W p z n | F p µ q p i qq , r T “ ∆ N ÿ µ a ρ b n Γ A, p µ q a ρ b n P nZ | W p z n | w n q . Hence, we have K ď S ` r S, (17)where S “ ∆ ÿ z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ÿ w n a ρ b n ¯Λ w n a ρ b n P nZ | W p z n | w n q ´ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (18)and r S “ ∆ ř z n } r T } . Note that r S captures the error induced by not covering the state ρ b n . We further bound r S as r S ď N ÿ µ ÿ z n P nZ | W p z n | w n q ›››a ρ b n Γ A, p µ q a ρ b n ››› ď N ÿ µ ›››››a ρ b n p I ´ ÿ w n γ p µ q w n A p µ q w n q a ρ b n ››››› ď N ÿ µ ›››››ÿ w n λ w n ˆ ρ w n ´ ÿ w n a ρ b n γ p µ q w n ¯ A p µ q w n a ρ b n ››››› ` N ÿ µ ›››››ÿ w n a ρ b n γ p µ q w n ´ ¯ A p µ q w n ´ A p µ q w n ¯ a ρ b n ››››› ď r S ` r S , where r S “ ∆ N ÿ µ ›››››ÿ w n λ w n ˆ ρ w n ´ ÿ w n a ρ b n γ p µ q w n ¯ A p µ q w n a ρ b n ››››› , r S “ ∆ N ÿ µ ÿ w n ›››a ρ b n γ p µ q w n ´ ¯ A p µ q w n ´ A p µ q w n ¯ a ρ b n ››› . To provide a bound for the term r S , we (i) develop a n-letter version of Lemma 4 and (ii) provide a propositionusing this n-letter lemma. Lemma 5.
Let t λ w , θ w u w P W be an ensemble, with θ w P D p H q for all w P W , W Ď F p for some finite prime p .Then, for any δ ą , there exist functions δ c p δ q and (cid:15) c p δ q , such that for all sufficiently large n , E »–›››››ÿ w n λ w n θ w n ´ p n nS N N ÿ µ “ ÿ w n ÿ a,m λ w n p ` η q θ w n t W n, p µ q p a,m q“ w n u ››››› fifl ď (cid:15) c , (19) holds if S ` n log N ě I p W ; R q σ θ ´ S p W q σ θ ` log p ` δ c , where θ w n “ ∆  ni “ θ w i and λ w n “ ∆ λ w λ w ¨ ¨ ¨ λ w n , σ RWθ “ ∆ ř w P W λ w θ w b | w yx w | , for some orthogonal set t | w yu w P W , and t W n, p µ q p a, m q : a P F kp , m P F lp , µ P r nC su are as defined in (12) , with G and h p µ q generated randomly uniformly and independently, S “ ∆ k ` ln log p , and δ c ,(cid:15) c Œ as δ Œ .Proof. The proof of the lemma is provided in Appendix A-CNow we provide the following proposition. 10 roposition 1.
There exist functions (cid:15) r S p δ q , and δ r S p δ q , such that for all sufficiently small δ and sufficiently large n , we have E r r S s ď (cid:15) r S p δ q , if S ą I p W ; R q σ ´ S p W q σ ` log p ` δ r S , where σ is the auxiliary state defined in thetheorem and (cid:15) r S Œ , δ r S Œ as δ Œ .Proof. The proof is provided in Appendix B-A.Now we provide the bound for r S . For that, we first develop another n-letter lemma as follows.
Lemma 6.
For γ p µ q w n , ¯ A p µ q w n , and A p µ q w n as defined above, we have ÿ w n γ p µ q w n ›››a ρ b n ´ ¯ A p µ q w n ´ A p µ q w n ¯ a ρ b n ››› ď nδ ρ ˜ H ` a p ´ ε qp ` η q a H ` H ` H ¸ , where H “ ∆ ˇˇˇ ∆ p µ q ´ E r ∆ p µ q s ˇˇˇ , H “ ∆ Tr p Π ρ ´ Π µ q ÿ w n λ w n ˜ ρ w n + ,H “ ∆ ›››››ÿ w n λ w n ˜ ρ w n ´ p ´ ε q ÿ w n α w n γ p µ q w n E r ∆ p µ q s ˜ ρ w n ››››› , H “ ∆ p ´ ε q ›››››ÿ w n α w n γ p µ q w n ∆ p µ q ˜ ρ w n ´ ÿ w n α w n γ p µ q w n E r ∆ p µ q s ˜ ρ w n ››››› , (20) ∆ p µ q “ ř w n P T p n q δ p W q α w n γ p µ q w n , ε “ ∆ ř w n R T p n q δ p W q λ w n and δ ρ p δ q Œ as δ Œ .Proof. The proof is provided in Appendix A-DUsing the above lemma on r S gives r S ď N N ÿ µ “ nδ ρ ˜ H ` a p ´ ε qp ` η q a H ` H ` H ¸ . Let us first consider H . By observing ř w n λ w n ˜ ρ w n ď Π ρ ρ b n Π ρ ď ´ n p S p ρ q´ δ ρ q Π ρ , we bound H as H ď ´ n p S p ρ q´ δ ρ q Tr tp Π ρ ´ Π µ qu . Note that E r Σ p µ q s “ E «ÿ w n α w n γ p µ q w n a ρ b n ´ ˜ ρ w n a ρ b n ´ ff “ p ` η q ÿ w n λ w n a ρ b n ´ ˜ ρ w n a ρ b n ´ ď p ` η q Π ρ . Now, we use the Pruning Trace Inequality developed in Lemma 3 on Σ p µ q , with η P p , q to obtain E r H s ď ´ n p S p ρ q´ δ ρ q p ` η q η E ” } Σ p µ q ´ E r Σ p µ q s} ı ď ´ n p S p ρ q´ δ ρ q p ` η q η ››› Π ρ a ρ b n ´ ››› E «›››››ÿ w n α w n γ p µ q w n ˜ ρ w n ´ E “ ÿ w n α w n γ p µ q w n ˜ ρ w n ‰››››› ff ››› Π ρ a ρ b n ´ ››› ď nδ ρ p ` η q η E »–›››››ÿ w n λ w n ˜ ρ w n p ` η q ´ p ` η q p n nS ÿ w n ÿ a,i λ w n ˜ ρ w n t W n, p µ q p a,i q“ w n u ››››› fifl “ nδ ρ p ´ ε q η E r r H s , (21)where the second inequality follows from H´olders inequality and the equality follows by defining r H as r H “ ∆ ›››››ÿ w n λ w n p ´ ε q ˜ ρ w n ´ p n nS ÿ w n ÿ a,i λ w n p ´ ε q ˜ ρ w n t W n, p µ q p a,i q“ w n u ››››› . (22)11imilarly, using E r ∆ p µ q s “ p ´ ε qp ` η q , H can be simplified as H “ ›››››ÿ w n λ w n ˜ ρ w n ´ p n nS ÿ w n ÿ a,i λ w n ˜ ρ w n t W n, p µ q p a,i q“ w n u ››››› “ p ´ ε q ˜ H, (23)where the inequality above is obtained using the triangle inequality. Now we consider H and convert it into asimilar expression as H H ď p ´ ε q ÿ w n P T p n q δ p W q α w n γ p µ q w n ˇˇˇˇ p µ q ´ E r ∆ p µ q s ˇˇˇˇ ď p ` η q ˇˇˇ ∆ p µ q ´ E r ∆ p µ q s ˇˇˇ “ p ` η q H . (24)Using the above simplification and the concavity of square-root function gives E r r S s ď N nδ ρ N ÿ µ “ ˜ E r H s ` a p ´ ε qp ` η q d p ´ ε q ˆ nδ ρ η ` ˙ E r r H s ` p ` η q E r H s ¸ ď N nδ ρ N ÿ µ “ ˜ E r H s ` p ´ ε qp ` η q dˆ nδ ρ η ` ˙ E r r H s ` d p ´ ε qp ` η q a E r H s ¸ . The following proposition provides a bound on the above term.
Proposition 2.
There exists (cid:15) r S p δ q , δ r S p δ q such that for all sufficiently small δ and sufficiently large n , we have E ” r S ı ď (cid:15) r S if S ě I p W ; R q σ ` log p ´ S p W q σ ` δ r S , where σ is the auxiliary state defined in the theorem and (cid:15) r S , δ r S Œ as δ Œ .Proof. The proof is provided in Appendix B-B
Remark 2.
The term corresponding to the operators that complete the sub-POVMs M p n,µ q , i.e., I ´ ř w n P T p n q δ p W q γ p µ q w n A p µ q w n is taken care in r T . The expression T excludes these completing operators. Step 2: Isolating the effect of error induced by binning
For this, we simplify T as T “ N ÿ µ ÿ w n ÿ i ą ÿ a P F kp a ρ b n A p µ q w n a ρ b n P nZ | W p z n | F p µ q p i qq t aG ` h p µ q p i q“ w n u . We substitute the above expression into S defined in (18), and isolate the effect of binning by adding and subtractingan appropriate term within S and applying triangle inequality to obtain S ď S ` S , where S “ ∆ ÿ z n ›››››ÿ w n a ρ b n ˜ ¯Λ w n ´ N ÿ µ γ p µ q w n A p µ q w n ¸ a ρ b n P nZ | W p z n | w n q ››››› ,S “ ∆ ÿ z n ››››› N ÿ µ ÿ a,i ą ÿ w n a ρ b n A p µ q w n a ρ b n t aG ` h p µ q p i q“ w n u ´ P nZ | W p z n | w n q ´ P nZ | W ´ z n | F p µ q p i q ¯¯››››› , where F p µ q p¨q is as defined in (15). Note that the term S characterizes the error introduced by approximation ofthe original POVM with the collection of approximating sub-POVM ˜ M p n,µ q , and the term S characterizes the errorcaused by binning this approximating sub-POVM. In this step, we analyze S and prove the following proposition. Proposition 3.
There exist (cid:15) S p δ q , such that for all sufficiently small δ and sufficiently large n , we have E r S s ď (cid:15) S p δ q , if S ´ R ă log p ´ S p W q σ , where σ is the auxiliary state defined in the theorem and (cid:15) S Œ as δ Œ .Proof. The proof is provided in Appendix B-C 12 tep 3: Isolating the effect of approximating measurement
In this step, we finally analyze the error induced from employing the approximating measurement, given by theterm S . We add and subtract appropriate terms within S and use triangle inequality obtain S ď S ` S ` S ,where S “ ∆ ÿ z n ›››››ÿ w n a ρ b n ˜ ¯Λ w n ´ N N ÿ µ “ α w n γ p µ q w n λ w n ¯Λ w n ¸ a ρ b n P nZ | W p z n | w n q ››››› ,S “ ∆ ÿ z n ››››› N N ÿ µ “ ÿ w n a ρ b n ˜ α w n γ p µ q w n λ w n ¯Λ w n ´ γ p µ q w n ¯ A p µ q w n ¸ a ρ b n P nZ | W p z n | w n q ››››› ,S “ ∆ ÿ z n ››››› N N ÿ µ ÿ w n a ρ b n ´ γ p µ q w n ¯ A p µ q w n ´ γ p µ q w n A p µ q w n ¯ a ρ b n P nZ | W p z n | w n q ››››› . Now with the intention of employing Lemma 5, we express S as S “ ›››››ÿ w n λ w n ˆ ρ w n b φ w n ´ N p ` η q p n nS ÿ µ ÿ w n ÿ a,i ‰ t W n, p µ q p a,i q“ w n u ˆ ρ w n b φ w n ››››› , where the equality above is obtained by defining φ w n “ ř z n P nZ | W p z n | w n q b | z n yx z n | and using the definitions of α w n , γ p µ q w n and ˆ ρ w n , followed by using the triangle inequality for the block diagonal operators, Note that the triangleinequality becomes an equality for such block diagonal operators. By identifying θ w with ˆ ρ w b φ w in Lemma 5 weobtain, E r S s ď (cid:15) S p δ q , if S ` n log N ě I p W ; R, Z q σ ` log p ´ S p W q σ ` δ S , where σ is the auxiliary statedefined in the theorem and (cid:15) S , δ S Œ as δ Œ .Now, we consider the term corresponding to S and prove that its expectation is small. Recalling S , we get S ď N N ÿ µ “ ÿ w n ÿ z n P nZ | W p z n | w n q ›››››a ρ b n ˜ α w n γ p µ q w n λ w n ¯Λ w n ´ γ p µ q w n ¯ A p µ q w n ¸ a ρ b n ››››› , “ N N ÿ µ “ ÿ w n α w n γ p µ q w n ››››a ρ b n ˆ λ w n ¯Λ w n ´ a ρ b n ´ ˜ ρ w n a ρ b n ´ ˙ a ρ b n ›››› , where the inequality above is obtained by using triangle inequality and the next equality follows from the fact that ř z n P nZ | W p z n | w n q “ for w n P W n , and using the definition of ¯ A p µ q w n . Applying expectation, we get E r S s ď p ` η q ÿ w n λ w n ››››a ρ b n ˆ λ w n ¯Λ w n ´ a ρ b n ´ ˜ ρ w n a ρ b n ´ ˙ a ρ b n ›››› , ď p ` η q ÿ w n P T p n q δ p W q λ w n }p ˆ ρ w n ´ ˜ ρ w n q} ` p ` η q ÿ w n R T p n q δ p W q λ w n } ˆ ρ w n } ď p ? ε ` ? ε q ` ε p ` η q “ (cid:15) S , where we have used the fact that E r α w n γ p µ q w n s “ λ wn p ` η q and the last inequality is obtained by the repeated usage ofthe Average Gentle Measurement Lemma [6] and (cid:15) S Œ as δ Œ (see (35) in [2] for details). Now, we moveon to bounding the last term within S , i.e., S . We start by applying triangle inequality to obtain S ď ÿ z n ÿ w n P nZ | W p z n | w n q ››››› N N ÿ µ “ a ρ b n ´ γ p µ q w n ¯ A p µ q w n ´ γ p µ q w n A p µ q w n ¯ a ρ b n ››››› ď N N ÿ µ “ ÿ w n γ p µ q w n ›››a ρ b n ´ ¯ A p µ q w n ´ A p µ q w n ¯ a ρ b n ››› “ r S , (25)13ince the above term is exactly same as r S , we obtain the same rate constraints as in r S to bound S , i.e., E r S s ď (cid:15) r S if S ě I p W ; R q σ ` log p ´ S p W q σ ` δ r S .Since S ď S ` S ` S , S can be made arbitrarily small for sufficiently large n, if S ` n log N ą I p W ; RZ q σ ´ S p W q σ ` log p ` δ S and S ě I p W ; R q σ ´ S p W q σ ` log p ` δ r S .
5) Rate Constraints:
To sum-up, we showed E r K s ď (cid:15) holds for sufficiently large n if the following boundshold: S ě I p W ; R q σ ´ S p W q σ ` log p, (26a) S ` C ě I p W ; RZ q σ ´ S p W q σ ` log p, (26b) S ´ R ď log p ´ S p W q σ , (26c) S ě R ě , C ě , (26d)where C “ ∆ 1 n log N and (cid:15) Œ as δ Œ . Therefore, there exists a distributed protocol with parameters p n, nR , nC q such that its overall POVM ˆ M is (cid:15) -faithful to M b n with respect to ρ b n . Lastly, we complete the proof ofthe theorem using the following lemma. Lemma 7.
Let R denote the set of all p R, C q for which there exists S such that the triple p S, R, C q satisfies theinequalities in (26) . Let, R denote the set of all tuples p R, C q that satisfies the inequalities in (9) given in thestatement of the theorem. Then, R “ R .Proof. This follows from Fourier-Motzkin elimination [20].V. S
IMULATION OF D ISTRIBUTED
POVM
S USING S TRUCTURED R ANDOM C ODES
Let ρ AB be a density operator acting on a composite Hilbert Space H A b H B . Consider two measurements M A and M B on sub-systems A and B , respectively. Imagine again that we have three parties, named Alice, Bob andCharlie, that are trying to collectively simulate the action of a given measurement M AB performed on the state ρ AB , as shown in Fig. 1. Charlie additionally has access to unlimited private randomness. The problem is definedin the following. Fig. 1. The diagram depicting the distributed POVM simulation problem with stochastic processing. In this setting, Charlie additionally hasaccess to unlimited private randomness.
Definition 6.
For a given finite set Z , and a Hilbert space H A b H B , a distributed protocol with stochastic processingwith parameters p n, Θ , Θ , N q is characterized by a collections of Alice’s sub-POVMs ˜ M p µ q A , µ P r , N s each acting on H b nA and with outcomes in a subset L satisfying | L | ď Θ . 14 ) a collections of Bob’s sub-POVMs ˜ M p µ q B , µ P r , N s each acting on H b nB and with outcomes in a subset L ,satisfying | L | ď Θ . Charlie’s classical stochastic map P p µ q p z n | l , l q for all l P L , l P L , z n P Z n and µ P r , N s .The overall sub-POVM of this distributed protocol, given by ˜ M AB , is characterized by the following operators: ˜Λ z n “ ∆ N ÿ µ,l ,l P p µ q p z n | l , l q Λ A, p µ q l b Λ B, p µ q l , @ z n P Z n , where Λ A, p µ q l and Λ B, p µ q l are the operators corresponding to the sub-POVMs ˜ M p µ q A and ˜ M p µ q B , respectively.In the above definition, p Θ , Θ q determines the amount of classical bits communicated from Alice and Bob toCharlie. The amount of common randomness is determined by N . The classical stochastic maps P p µ q p z n | l , l q represent the action of Charlie on the received classical bits. Definition 7.
Given a POVM M AB acting on H A b H B , and a density operator ρ AB P D p H A b H B q , a triple p R , R , C q is said to be achievable, if for all (cid:15) ą and for all sufficiently large n , there exists a distributed protocolwith stochastic processing with parameters p n, Θ , Θ , N q such that its overall sub-POVM ˜ M AB is (cid:15) -faithful to M b nAB with respect to ρ b nAB (see Definition 1), and n log Θ i ď R i ` (cid:15), i “ , , and n log N ď C ` (cid:15). The set of all achievable triples p R , R , C q is called the achievable rate region. Definition 8 (Joint Measurements) . A POVM M AB “ t Λ ABz u z P Z , acting on a joint state ρ AB P D p H A b H B q , issaid to have a separable decomposition with stochastic modulo-sum integration with respect to a prime finite field F p if there exist POVMs ¯ M A “ t ¯Λ Au u u P U and ¯ M B “ t ¯Λ Bv u v P V and a stochastic mapping P Z | W : W Ñ Z such that Λ ABz “ ÿ u,v P Z | W p z | u ` v q ¯Λ Au b ¯Λ Bv , @ z P Z , where W “ U ` V , U “ V “ W “ F p , with F p being a prime finite field, and Z is a finite set. Further, ifthe mapping P Z | W is a deterministic function, then the POVM is said to have a separable decomposition withdeterministic modulo-sum integration.The following theorem provides an inner bound to the achievable rate region, which is proved in Section VI. Theorem 2.
Given a density operator ρ AB P D p H A b H B q , and a POVM M AB “ t Λ ABz u z P Z acting on H A b H B having a separable decomposition with stochastic modulo sum integration with respect to a prime finite field F p (as in Definition 8), yielding POVMs ¯ M A “ t ¯Λ Au u u P U and ¯ M B “ t ¯Λ Bv u v P V and a stochastic map P Z | W : F p Ñ Z for U “ V “ F p and W “ U ` V , a triple p R , R , C q is achievable if the following inequalities are satisfied: R ě I p U ; R, B q σ ` S p U ` V q σ ´ S p U q σ , (27a) R ě I p V ; R, A q σ ` S p U ` V q σ ´ S p V q σ , (27b) R ` C ě I p U ; R, Z, V q σ ` S p U ` V q σ ´ S p U q σ , (27c) R ` C ě I p V ; R, Z q σ ` S p U ` V q σ ´ S p V q σ , (27d) R ` R ` C ě I p U, V ; R, Z q σ ` S p U ` V q σ ´ S p U, V q σ , (27e) where Ψ ρ AB RAB is a purification of ρ AB , U, V, W are defined on F p , and the above information quantities arecomputed for the auxiliary states σ RUB “ ∆ p id R b ¯ M A b id B qp Ψ ρ AB RAB q , σ RAV “ ∆ p id R b id A b ¯ M B qp Ψ ρ AB RAB q , and RUV Z “ ∆ ř u,v,z ? ρ AB ` ¯Λ Au b ¯Λ Bv ˘ ? ρ AB b P Z | W p z | u ` v q | u yx u | b | v yx v | b | z yx z | , for some orthonormal sets t | u yu u P U , t | v yu v P V , and t | z yu z P Z . Note that the rate-region obtained in Theorem 6 of [8] contains the constraint R ` R ` C ě I p U, V ; R, Z q σ .Hence when S p U ` V q σ ă S p U, V q σ , the above theorem gives a lower sum rate constraint. As a result, therate-region above contains points that are not contained within the rate-region provided in [8]. To illustrate this factfurther, consider the following example. Example 1.
Suppose the composite state ρ AB is described using one of the Bell states on H A b H B as ρ AB “ p | y AB ` | y AB q px | AB ` x | AB q . Since π A “ Tr B ρ AB and π B “ Tr A ρ AB , Alice and Bob would perceive each of their particles in maximally mixedstates π A “ I A and π B “ I B , respectively. Upon receiving the quantum state, the two parties wish to independentlymeasure their states, using identical POVMs M A and M B , given by M A “ ∆ (cid:32) Λ Au ( u P F , M B “ ∆ (cid:32) Λ Bv ( v P F , where F is a binary field with elements and , and Λ A “ Λ B “ „ . . ` i . . ´ i . . , Λ A “ Λ B “ „ . ´ . ´ i . ´ . ` i . . . Alice and Bob together with Charlie are trying to simulate the action of M AB “ ∆ (cid:32) Γ ABz ( z P F using the classicalcommunication and common randomness as the resources available to them, where Γ ABz “ ∆ λ z ` Λ A b Λ B ` Λ A b Λ B ˘ ` p ´ λ z q ` Λ A b Λ B ` Λ A b Λ B ˘ , for z P F . (28)Note that the above POVM M AB admits a separable decomposition as given in Definition 8 with respect to theprime finite field F , and hence the above theorem can be employed. This gives S p U ` V q σ “ . , S p U q σ “ S p V q σ “ . , S p U, V q σ “ . , I p U, V q σ “ . , (29)where σ is as defined in the statement of Theorem 2. Since S p U q σ ´ S p U ` V q σ “ S p V q σ ´ S p U ` V q σ “ I p U,V q σ , the constraints on R , R , R ` C and R ` C are the same as obtained in Theorem 6 of [8]. However, with S p U ` V q σ ´ S p U, V q σ “ ´ . ă , the constraint on R ` R ` C in the above theorem (27e) is strictlyweaker than the constraint obtained in Theorem 6 of [8]. Therefore, the rate-region obtained above in Theorem(2) is strictly larger than the rate-region in Theorem 6 of [8]. For further discussion on the comparison of the tworate-regions see [12]. VI. P ROOF OF T HEOREM ¯ M A “ ∆ t ¯Λ Au u u P U and ¯ M B “ ∆ t ¯Λ Bv u v P V and a stochastic map P Z | W : F p Ñ Z , suchthat M AB can be decomposed as Λ ABz “ ÿ u,v P Z | W p z | u ` v q ¯Λ Au b ¯Λ Bv , @ z, (30)where W is defined as W “ U ` V . The coding strategy used here is based on Unionized Coset Codes, similarto the one employed in the point-to-point proof (Section IV-B), but extended to a distributed setting. Further, thestructure in these codes provide a method to exploit the structure present in the stochastic processing appliedby Charlie on the classical bits received, i.e., P Z | U ` V . Using this technique, we aim to strictly reduce the rateconstraints compared to the ones obtained in Theorem VII of [10]. Also note that, the results in [10] are based on16he assumption that approximating POVMs are all mutually independent. However, since the structured constructionof the POVMs only guarantees pairwise independence among the operators of the POVM, the proofs below becomesignificantly different from [10].We start by generating the canonical ensembles corresponding to ¯ M A and ¯ M B , defined as λ Au “ ∆ Tr t ¯Λ Au ρ A u , λ Bv “ ∆ Tr t ¯Λ Bv ρ B u , λ ABuv “ ∆ Tr tp ¯Λ Au b ¯Λ Bv q ρ AB u , ˆ ρ Au “ ∆ λ Au ? ρ A ¯Λ Au ? ρ A , ˆ ρ Bv “ ∆ λ Bv ? ρ B ¯Λ Bv ? ρ B , ˆ ρ ABuv “ ∆ λ ABuv ? ρ AB p ¯Λ Au b ¯Λ Bv q? ρ AB . (31)With this notation, corresponding to each of the probability distributions, we can associate a δ -typical set. Let usdenote T p n q δ p U q , T p n q δ p V q and T p n q δ p U V q as the δ -typical sets defined for t λ Au u , t λ Bv u and t λ ABuv u , respectively.Let Π ρ A and Π ρ B denote the δ -typical projectors (as in [19]) for marginal density operators ρ A and ρ B ,respectively. Also, for any u n P U n and v n P V n , let Π Au n and Π Bv n denote the conditional typical projectors (as in[19]) for the canonical ensembles t λ Au , ˆ ρ Au u and t λ Bv , ˆ ρ Bv u , respectively. For each u n P T p n q δ p U q and v n P T p n q δ p V q define ˜ ρ Au n “ ∆ Π ρ A Π Au n ˆ ρ Au n Π Au n Π ρ A , ˜ ρ Bv n “ ∆ Π ρ B Π Bv n ˆ ρ Bv n Π Bv n Π ρ B , and ˜ ρ Au n “ , and ˜ ρ Bv n “ for u n R T p n q δ p U q and v n R T p n q δ p V q , respectively, with ˆ ρ Au n “ ∆ Â i ˆ ρ Au i and ˆ ρ Bv n “ ∆ Â i ˆ ρ Bv i . A. Construction of Structured POVMs
In what follows, we construct the random structured POVM elements. Fix a block length n ą , a positiveinteger N, and a finite field F p . Let µ P r , N s denote the common randomness shared between the first encoderand the decoder, and let µ P r , N s denote the common randomness shared between the second encoder and thedecoder, with log( N ) + log( N ) ď log( N ). Further, let U and V be random variables defined on the alphabets U and V , respectively, where U “ V “ F p . In building the code, we use the Unionized Coset Codes (UCCs) [16] asdefined above in Definition 5.For every µ “ ∆ p µ , µ q , consider two UCCs p G, h p µ q q and p G, h p µ q q , each with parameters p n, k, l , p q and p n, k, l , p q , respectively. Note that, for every µ P r , N s , they share the same generator matrix G. For each p µ , µ q , the generator matrix G along with the function h µ and h µ generates p k ` l and p k ` l codewords, respectively. Each of these codewords are characterized by a triple p a i , m i , µ i q , where a i P F kp and m i P F l i p corresponds to the coarse code and the fine code indices, respectively, for i P r , s . Let U n, p µ q p a , i q and V n, p µ q p a , j q denote the codewords associated with Alice and Bob, generated using the above procedure,respectively, where U n, p µ q p a , i q “ a G ` h p µ q p i q and V n, p µ q p a , j q “ a G ` h p µ q p j q . Now, construct the operators ¯ A p µ q u n “ ∆ α u n ˆ ? ρ A ´ ˜ ρ Au n ? ρ A ´ ˙ and ¯ B p µ q v n “ ∆ β v n ˆ ? ρ B ´ ˜ ρ Bv n ? ρ B ´ ˙ , (32)where α u n “ ∆ p ` η q p n nS λ Au n , and β v n “ ∆ p ` η q p n nS λ Bv n , (33)17ith η P p , q being a parameter to be determined, and S i “ k ` l i n log p , i P r , s . Having constructed the operators ¯ A p µ q u n and ¯ B p µ q v n , we normalize these operators, so that they constitute a valid sub-POVM. To do so, we first define Σ p µ q A “ ∆ ÿ u n γ p µ q u n ¯ A p µ q u n and Σ p µ q B “ ∆ ÿ v n ζ p µ q v n ¯ B p µ q v n , where γ p µ q u n and ζ p µ q v n are defined as γ p µ q u n “ ∆ |tp a , i q : U n, p µ q p a , i q “ u n u| and ζ p µ q v n “ ∆ |tp a , j q : V n, p µ q p a , j q “ v n u| . Now, we define Π µ A and Π µ B as pruning operators for Σ p µ q A and Σ p µ q B , with respect to Π ρ A and Π ρ B , respectively(see Definition 2). Note that, these pruning operators Π µ A and Π µ B depend on the triple p G, h p µ q , h p µ q q . For easeof analysis, the subspace of Π µ A is restricted to Π ρ A and that of Π µ B to Π ρ B . Using these pruning operators, foreach µ P r , N s and µ P r , N s , construct the sub-POVMs M p n,µ q and M p n,µ q as M p n,µ q “ ∆ t γ p µ q u n A p µ q u n : u n P U n u , and M p n,µ q “ ∆ t ζ p µ q v n B p µ q v n : v n P V n u . (34)where A p µ q u n “ Π µ A ¯ A p µ q u n Π µ A and B p µ q v n “ Π µ B ¯ B p µ q v n Π µ B . Further, using these operators Π µ A and Π µ A , we have ř u n γ p µ q u n A p µ q u n “ Π µ A Σ p µ q A Π µ A ď I and ř v n ζ p µ q v n B p µ q v n “ Π µ B Σ p µ q B Π µ B ď I, and thus M p n,µ q and M p n,µ q are valid sub-POVMs for all µ P r , N s and µ P r , N s . Further, these collections M p n,µ q and M p n,µ q arecompleted using the operators I ´ ř u n P U n γ p µ q u n A p µ q u n and I ´ ř v n P V n ζ p µ q v n B p µ q v n . B. Binning of POVMs
We next proceed to binning the above constructed collection of sub-POVMs. Since, UCC is already a union ofseveral cosets, we associate a bin to each coset, and hence place all the codewords of a coset in the same bin. Foreach i P F l p and j P F l p , let B p µ q p i q “ ∆ C p G, h p µ q p i qq and B p µ q p j q “ ∆ C p G, h p µ q p j qq denote the i th and the j th bins, respectively. Formally, we define the following operators: Γ A, p µ q i “ ∆ ÿ u n P U n ÿ a P F kp A p µ q u n t a G ` h p µ q p i q“ u n u , Γ B, p µ q j “ ∆ ÿ v n P V n ÿ a P F k p B p µ q v n t a G ` h p µ q p j q“ v n u , for all i P F l p and j P F l p . Using these operators, we form the following collection: M p n,µ q A “ ∆ t Γ A, p µ q i u i P F l p , M p n,µ q B “ ∆ t Γ B, p µ q j u j P F l p . (35)Note that if M p n,µ q and M p n,µ q are sub-POVMs, then so are M p n,µ q A and M p n,µ q B , which is due to the relations ÿ i P F l p Γ A, p µ q i “ ÿ u n P U n γ p µ q u n A p µ q u n ď I, and ÿ j P F l p Γ B, p µ q j “ ÿ v n P V n ζ p µ q v n B p µ q v n ď I. (36)To make M p n,µ q A and M p n,µ q B complete, we define Γ A, p µ q and Γ B, p µ q as Γ A, p µ q “ I ´ ř i Γ A, p µ q i and Γ B, p µ q “ I ´ ř j Γ B, p µ q j , respectively . Now, we intend to use the completions r M p n,µ q A s and r M p n,µ q B s as the POVMs forencoders associated with Alice and Bob, respectively. Also, note that the effect of the binning is in reducing thecommunication rates from p S , S q to p R , R q , where R i “ l i n log p, i P t , u . Now, we move on to describingthe decoder. Note that Γ A, p µ q “ I ´ ř i Γ A, p µ q i “ I ´ ř u n P T p n q δ p U q A p µ q u n and Γ B, p µ q “ I ´ ř j Γ B, p µ q j “ I ´ ř v n P T p n q δ p V q B p µ q v n . . Decoder mapping We create a decoder that takes as an input a pair of bin numbers and produces a sequence W n P F np . Moreprecisely, we define a mapping F p µ ,µ q , for µ “ p µ , µ q , acting on the outputs of r M p n,µ q A sbr M p n,µ q B s as follows.On observing µ and the classical indices p i, j q P F l p ˆ F l p communicated by the encoder, the decoder constructs D p µ ,µ q and F p µ ,µ q p¨ , ¨q as, D p µ ,µ q i,j “ ∆ ! ˜ a P F kp : ˜ aG ` h p µ q p i q ` h p µ q p j q P T p n q ˆ δ p W q ) ,F p µ ,µ q p i, j q “ ∆ ˜ aG ` h p µ q p i q ` h p µ q p j q if D p µ ,µ q i,j ” t ˜ a u w n otherwise , (37)where ˆ δ “ pδ and w n is an additional sequence added to F np . Further, F p µ ,µ q p i, j q “ w n for i “ or j “ .Given this, we obtain the sub-POVM ˜ M AB with the following operators. ˜Λ ABw n “ ∆ N N N ÿ µ “ N ÿ µ “ ÿ p i,j q : F p µ ,µ q p i,j q“ w n Γ A, p µ q i b Γ B, p µ q j , @ w n P F np Ť t w n u . Now, we use the stochastic mapping P Z | W to define the approximating sub-POVM ˆ M p n q AB “ ∆ t ˆΛ z n u as ˆΛ ABz n “ ∆ ÿ w n ˜Λ ABw n P nZ | W p z n | w n q , @ z n P Z n . Note that ˜Λ ABw n “ for w n R T p n q δ p W q Ť t w n u . Code Ensemble:
The generator matrix G and the functions h p µ q and h p µ q are chosen randomly uniformly andindependently, for µ P r N s and µ P r N s . D. Trace Distance
In what follows, we show that ˆ M p n q AB is (cid:15) -faithful to M b nAB with respect to ρ b nAB (according to Definition 1), where (cid:15) ą can be made arbitrarily small. More precisely, using (30), we show that, E r K s ď (cid:15), where K “ ∆ ÿ z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ÿ u n ,v n b ρ b nAB ¯Λ Au n b ¯Λ Bv n b ρ b nAB P nZ | W p z n | u n ` v n q ´ b ρ b nAB ˆΛ ABz n b ρ b nAB (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (38)and the expectation is with respect to the codebook generation. Step 1: Isolating the effect of error induced by not covering
Consider the second term within K , which can be written as ÿ w n b ρ b nAB ˜Λ ABw n b ρ b nAB P nZ | W p z n | w n q“ N N ÿ µ ,µ ÿ i,j b ρ b nAB ´ Γ A, p µ q i b Γ B, p µ q j ¯ b ρ b nAB P nZ | W p z n | F p µ ,µ q p i, j qq ÿ w n t F p µ ,µ q p i,j q“ w n u looooooooooomooooooooooon “ “ T ` r T , where T “ ∆ N N ÿ µ ,µ ÿ t i ą u Ş t j ą u b ρ b nAB ´ Γ A, p µ q i b Γ B, p µ q j ¯ b ρ b nAB P nZ | W p z n | F p µ ,µ q p i, j qq , r T “ ∆ N N ÿ µ ,µ ÿ t i “ u Ť t j “ u b ρ b nAB ´ Γ A, p µ q i b Γ B, p µ q j ¯ b ρ b nAB P nZ | W p z n | w n q . K ď S ` r S, (39)where S “ ∆ ÿ z n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ÿ u n ,v n b ρ b nAB ´ ¯Λ Au n b ¯Λ Bv n P nZ | W p z n | u n ` v n q ¯ b ρ b nAB ´ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (40)and r S “ ∆ ř z n } r T } . Note that r S captures the error induced by not covering the state ρ b nAB . For the term correspondingto r S , we prove the following result. Proposition 4.
There exist functions (cid:15) r S p δ q , and δ r S p δ q , such that for all sufficiently small δ and sufficiently large n ,we have E r r S s ď (cid:15) r S p δ q , if S ą I p U ; RB q σ ´ S p U q σ ` log p ` δ r S and S ą I p V ; RA q σ ´ S p V q σ ` log p ` δ r S , where σ and σ are auxiliary states defined in the theorem and (cid:15) r S Œ , δ r S Œ as δ Œ .Proof. The proof is provided in Appendix B-D.
Remark 3.
The terms corresponding to the operators that complete the sub-POVMs M p n,µ q A and M p n,µ q B , i.e., I ´ ř u n P T p n q δ p U q γ p µ q u n A p µ q u n and I ´ ř v n P T p n q δ p V q ζ p µ q v n B p µ q v n are taken care in r T . The expression T excludes thesecompleting operators. Step 2: Isolating the effect of error induced by binning
We begin by simplifying T as T “ N N ÿ µ ,µ ÿ u n ,v n ÿ i ą ,j ą b ρ b nAB ¨˝ ÿ a P F kp A p µ q u n t a G ` h p µ q p i q“ u n u b ÿ a P F kp B p µ q v n t a G ` h p µ q p j q“ v n u ˛‚b ρ b nAB P nZ | W p z n | F p µ ,µ q p i, j qq . Note that the p u n , v n q that appear in the above summation is confined to p T p n q δ p U q ˆ T p n q δ p V qq , however for ease ofnotation, we do not make this explicit. We substitute the above expression into S as in (40), and add and subtractan appropriate term within S and apply the triangle inequality to isolate the effect of binning as S ď S ` S , where S “ ∆ ÿ z n ››››› ÿ u n ,v n b ρ b nAB ˜ ¯Λ Au n b ¯Λ Bv n ´ N N ÿ µ ,µ γ p µ q u n A p µ q u n b ζ p µ q v n B p µ q v n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,S “ ∆ ÿ z n ››››››› N N ÿ µ ,µ ÿ i ą j ą ÿ a ,a ÿ u n ,v n b ρ b nAB ´ A p µ q u n b B p µ q v n ¯b ρ b nAB t a G ` h p µ q p i q“ u n ,a G ` h p µ q p j q“ v n u ´ P nZ | W p z n | u n ` v n q ´ P nZ | W ´ z n | F p µ ,µ q p i, j q ¯¯››› . Note that the term S characterizes the error introduced by approximation of the original POVM with the collectionof approximating sub-POVMs M p n,µ q and M p n,µ q , and the term S characterizes the error caused by binning ofthese approximating sub-POVMs. In this step, we analyze S and prove the following proposition.20 roposition 5 (Mutual Packing) . There exist (cid:15) S p δ q , such that for all sufficiently small δ and sufficiently large n ,we have E r S s ď (cid:15) S p δ q , if S ´ R ă log p ´ S p W q σ , or equivalently, S ´ R ă log p ´ S p W q σ , where σ isthe auxiliary state defined in the theorem and (cid:15) S Œ as δ Œ .Proof. The proof is provided in Appendix B-E
Step 3: Isolating the effect of Alice’s approximating measurement
In this step, we separately analyze the effect of approximating measurements at the two distributed parties in theterm S . For that, we split S as S ď Q ` Q , where Q “ ∆ ÿ z n ››››› ÿ u n ,v n b ρ b nAB ˜ ¯Λ Au n b ¯Λ Bv n ´ N N ÿ µ “ γ p µ q u n A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,Q “ ∆ ÿ z n ››››› N N ÿ µ “ ÿ u n ,v n b ρ b nAB ´ γ p µ q u n A p µ q u n b ¯Λ Bv n ´ N N ÿ µ “ γ p µ q u n A p µ q u n b ζ p µ q v n B p µ q v n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› . With this partition, the terms within the trace norm of Q differ only in the action of Alice’s measurement. Andsimilarly, the terms within the norm of Q differ only in the action of Bob’s measurement. Showing that these twoterms are small forms a major portion of the achievability proof. Analysis of Q : To prove Q is small, we characterize the rate constraints which ensure that an upper bound to Q can be made to vanish in an expected sense. In addition, this upper bound becomes lucrative in obtaining asingle-letter characterization for the rate needed to make the term corresponding to Q vanish. For this, we define J as J “ ∆ ÿ z n ,v n ›››››ÿ u n b ρ b nAB ˜ ¯Λ Au n b ¯Λ Bv n ´ N N ÿ µ “ γ p µ q u n A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› . (41)By defining J and using triangle inequality for block operators (which holds with equality), we add the sub-system V to RZ , resulting in the joint system RZV , corresponding to the state σ as defined in the theorem. Then weapproximate the joint system RZV using an approximating sub-POVM M p n q A producing outputs on the alphabet U n . To make J small for sufficiently large n, we expect the sum of the rate of the approximating sub-POVM andcommon randomness, i.e., S ` n log N , to be larger than I p U ; RZV q σ . We prove this in the following.Note that from the triangle inequality, we have Q ď J. Further, we add and subtract appropriate terms within J and again use the triangle inequality obtain J ď J ` J , where J “ ∆ ÿ z n ,v n ›››››ÿ u n b ρ b nAB ´ ¯Λ Au n b ¯Λ Bv n ´ γ p µ q u n ¯ A p µ q u n b ¯Λ Bv n ¯ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,J “ ∆ ÿ z n ,v n ››››› N N ÿ µ “ ÿ u n b ρ b nAB ´ γ p µ q u n ¯ A p µ q u n b ¯Λ Bv n ´ γ p µ q u n A p µ q u n b ¯Λ Bv n ¯ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› . Now we use the following proposition to bound the term corresponding to J . Proposition 6.
There exist (cid:15) J p δ q , δ J p δ q such that for all sufficiently small δ and sufficiently large n , we have E r J s ď (cid:15) J if S ` n log N ě I p U ; RZV q σ ` log p ´ S p U q σ ` δ J , where σ is the auxiliary state defined inthe theorem and (cid:15) J , δ J Œ as δ Œ . roof. The proof of proposition is provided in Appendix B-F.Now, we move on to bounding the term corresponding to J . We start by applying triangle inequality followedby Lemma 1 on J to obtain J ď ÿ z n ÿ u n ,v n P nZ | W p z n | u n ` v n q ››››› N N ÿ µ “ b ρ b nA ´´ γ p µ q u n ¯ A p µ q u n ´ γ p µ q u n A p µ q u n ¯ b ¯Λ Bv n ¯ b ρ b nA ››››› “ ÿ u n ,v n ››››› N N ÿ µ “ b ρ b nA ´´ γ p µ q u n ¯ A p µ q u n ´ γ p µ q u n A p µ q u n ¯ b ¯Λ Bv n ¯ b ρ b nA ››››› ď N N ÿ µ “ ÿ u n γ p µ q u n ››››b ρ b nA ´ ¯ A p µ q u n ´ A p µ q u n ¯ b ρ b nA ›››› . (42)Now we use the following proposition to bound the term corresponding to J . Proposition 7.
There exist (cid:15) J p δ q , δ J p δ q such that for all sufficiently small δ and sufficiently large n , we have E r J s ď (cid:15) J if S ě I p U ; RB q σ ` log p ´ S p U q σ ` δ J , where σ , σ are the auxiliary state defined in thetheorem and (cid:15) J , δ J Œ as δ Œ .Proof. The proof is provided in Appendix B-G.Since Q ď J ď J ` J , hence J , and consequently Q , can be made arbitrarily small for sufficiently large n,if S ` n log N ą I p U ; RZV q σ ´ S p U q σ ` log p ` δ J and S ě I p U ; RB q σ ´ S p U q σ ` log p ` δ J . Now wemove on to bounding Q . Step 4: Analyzing the effect of Bob’s approximating measurement
Step 3 ensured that the sub-system
RZV is close to a tensor product state in trace-norm. In this step, we approximatethe state corresponding to the sub-system RZ using the approximating POVM M p n q B , producing outputs on thealphabet V n . We proceed with the following proposition. Proposition 8.
There exist functions (cid:15) Q p δ q and δ Q p δ q , such that for all sufficiently small δ and sufficiently large n , we have E r Q s ď (cid:15) Q , if S ` n log N ě I p U ; RZV q σ ´ S p U q σ ` log p ` δ Q , S ` n log N ě I p V ; RZ q σ ´ S p V q σ ` log p ` δ Q , S ě I p U ; RB q σ ´ S p U q σ ` log p ` δ Q , and S ě I p V ; RA q σ ´ S p V q σ ` log p ` δ Q where σ , σ , σ are the auxiliary states defined in the theorem and (cid:15) Q , δ Q Œ as δ Œ .Proof. The proof is provided in Appendix B-H.
E. Rate Constraints
To sum-up, we showed E r K s ď (cid:15) holds for sufficiently large n if the following bounds hold: S ě I p U ; RB q σ ´ S p U q σ ` log p, (43a) S ě I p V ; RA q σ ´ S p V q σ ` log p, (43b) S ` C ě I p U ; RZV q σ ´ S p U q σ ` log p, (43c) S ` C ě I p V ; RZ q σ ´ S p V q σ ` log p, (43d) S ´ R “ S ´ R , (43e) S ´ R ď log p ´ S p U ` V q σ , (43f)22 ě R ě , S ě R ě , (43g) C ` C ď C, C ě , (43h)where C i “ ∆ 1 n log N i , i P t , u and C “ ∆ 1 n log N . Therefore, there exists a distributed protocol with parameters p n, nR , nR , nC q such that its overall POVM ˆ M AB is (cid:15) -faithful to M b nAB with respect to ρ b nAB . Lastly, we completethe proof of the theorem using the following lemma. Lemma 8.
Let R denote the set of all p R , R , C q for which there exists p S , S q such that the septuple p R , R ,C, S , S , C , C q satisfies the inequalities in (43) . Let, R denote the set of all triples p R , R , C q that satisfiesthe inequalities in (27) given in the statement of the theorem. Then, R “ R .Proof. This follows from Fourier-Motzkin elimination [20].VII. C
ONCLUSION
We developed a technique of randomly generating structured POVMs using algebraic codes. Using this technique,we demonstrated a new achievable information-theoretic rate-region for the task of faithfully simulating a distributedquantum measurement and function computation. We further devised a Pruning Trace inequality which is a tighterversion of the known operator Markov inequality, and a covering lemma which is independent of the operatorChernoff inequality, so as to analyse pairwise-independent POVM elements. Finally, combining these techniques,we demonstrated rate gains for this problem over traditional coding schemes, and provided a multi-party distributedfaithful simulation and function computation protocol.
Acknowledgement:
We thank Arun Padakandla for his valuable discussion and inputs in developing the prooftechniques. A
PPENDIX AP ROOF OF L EMMAS
A. Proof of Lemma 3Proof.
Note that if P prunes X , then P also prunes η p X ´ p ´ η q I A q with respect to I A . Using Lemma 2, weobtain Tr t I A ´ P u ď η Tr t X ´ p ´ η q I A u . Applying expectation and using the assumption on E r X s , we get E r Tr t I A ´ P us ď η E r Tr t X ´ E r X sus ď η E r} X ´ E r X s} s . (44) B. Proof of Lemma 4Proof.
We begin by defining the ensemble t λ x , ˜ σ x u x P X where ˜ σ x “ ΠΠ x σ x Π x Π for all x P X . Further, let S bedefined as S “ ∆ ››› ÿ x P X λ x σ x ´ M ÿ x P X M ÿ m “ λ x µ x σ x t C m “ x u ››› .
23y adding an subtracting appropriate terms within the trace norm of S and using the triangle inequality we obtain, S ď S ` S ` S , where S “ ∆ ›› ÿ x P X λ x σ x ´ ÿ x P X λ x ˜ σ x ›› , S “ ∆ ›› M M ÿ m “ λ C m µ C m ˜ σ C m ´ M M ÿ m “ λ C m µ C m σ C m ›› , and S “ ∆ ›› ÿ x P X λ x ˜ σ x ´ M ÿ x P X M ÿ m “ λ x µ x ˜ σ x t C m “ x u ›› . We begin by bounding the term corresponding to S and S as follows: S ď ÿ x P X λ x } σ x ´ ΠΠ x σ x Π x Π } ď ÿ x P X λ x } σ x ´ Π σ x Π ›› ` ÿ x P X λ x } Π σ x Π ´ ΠΠ x σ x Π x Π } ď ? (cid:15) ` ÿ x P X λ x } Π } } σ x ´ Π x σ x Π x } } Π } ď ? (cid:15) “ δ p (cid:15) q , (45)where the first two inequality uses the triangle inequality, the third uses the gentle measurement lemma (given theassumption (5a) from the statement of the Lemma) for the first term, and operator Holder’s inequality (Exercise12.2.1 in [18]) for the second term. The last inequality follows again from the gentle measurement given theassumption (5b). Similarly, for S we have E C r S s ď E C « M M ÿ m “ ÿ x P X λ x µ x t C m “ x u } σ x ´ ˜ σ x } ff “ M M ÿ m “ ÿ x P X λ x } σ x ´ ˜ σ x } ď ? (cid:15) “ δ p (cid:15) q (46)where we use the fact that E C r t c m “ x u s “ µ x , and the last inequality uses similar arguments as in (45). Finally, weproceed to bound the term corresponding to S . Firstly, note that, E C r M ř m λ Cm µ Cm ˜ σ C m s “ ř x P X λ x ˜ σ x . This gives E C r S s “ E C «››› M ÿ m λ C m µ C m ˜ σ C m ´ E C „ M ÿ m λ C m µ C m ˜ σ C m ››› ff ď Tr $’&’%gfffe E C »–˜ M ÿ m λ C m µ C m ˜ σ C m ´ E C „ M ÿ m λ C m µ C m ˜ σ C m ¸ fifl,/./- “ Tr $’&’%gfffe E C »–˜ M ÿ m λ C m µ C m ˜ σ C m ¸ fifl ´ ˜ E C „ M ÿ m λ C m µ C m ˜ σ C m ¸ ,/./- “ Tr $’’’&’’’%gffffe M ÿ m E C «ˆ λ C m µ C m ˜ σ C m ˙ ff ` M ÿ m,m m ‰ m E C „ λ C m ˜ σ C m µ C m λ C m ˜ σ C m µ C m ´ ˜ M ÿ m E C „ λ C m ˜ σ C m µ C m ¸ ,///.///- “ Tr $&%gffe M E C «ˆ λ C ˜ σ C µ C ˙ ff ´ M ˆ E C „ λ C ˜ σ C µ C ˙ ,.- ď Tr $&%gffe M E C «ˆ λ C ˜ σ C µ C ˙ ff,.- , (47)where the first inequality follows from concavity of operator square-root function (L¨owner-Heinz theorem, seeTheorem . in [21]). The last equality uses the fact that codewords of the random code C are pairwise independent,and the last inequality follows from monotonicity of the operator square-root function (Theorem . in [21]).24oving on, we now bound the operator within the square root of (47) as E C «ˆ λ C ˜ σ C µ C ˙ ff “ ÿ x P X λ x µ x ˜ σ x ď ÿ x P X κλ x ˜ σ x “ κ ÿ x P X λ x Π p Π x σ x Π x q Π p Π x σ x Π x q Π , where we use the assumption λ x µ x ď κ for all x P X . Further since, Π ď I , we have p Π x σ x Π x q Π p Π x σ x Π x q ďp Π x σ x Π x q , which gives E C «ˆ λ C ˜ σ C µ C ˙ ff ď κ ÿ x P X λ x Π p Π x σ x Π x q Π . Moreover, using the assumption 5d, i.e., Π x σ x Π x ď d Π x ď d I , we get p Π x σ x Π x q “ a Π x σ x Π x p Π x σ x Π x q a Π x σ x Π x ď d Π x σ x Π x , for all x P X . Thus, E C «ˆ λ C ˜ σ C µ C ˙ ff ď κd Π ˜ ÿ x P X λ x Π x σ x Π x ¸ Π ď κd Π σ Π , (48)where the second inequality uses the assumption (5e) from the statement of the Lemma. Substituting the simplifi-cation obtained in (48) into (47) and using the monotonicity of square-root operator, we obtain E C r S s ď Tr "c κM d Π σ Π * ď c κDM d , (49)where the second inequality uses the assumption (5c). Combining the bounds (45), (46), and (49) we get the desiredresult. C. Proof of Lemma 5
We begin by defining L as L “ ∆ ›››››ÿ w n λ w n θ w n ´ p ` η q p n nS N N ÿ µ “ ÿ a,m λ W n, p µ q p a,m q θ W n, p µ q p a,m q ››››› . Further, let θ “ ∆ ř w P W λ w θ w and let Π θ and Π θw n denote the δ -typical projector of θ and conditional typicalprojector of θ w n , respectively. Define ˜ λ nw “ λ nw ´ ε for w n P T p n q δ p W q , and otherwise, where ε “ ř w n R T p n q δ p w q λ nw . Using triangle inequality we can bound L as L ď L ` L ` L , where L “ ∆ ›››››ÿ w n λ w n θ w n ´ ÿ w n ˜ λ w n θ w n ››››› ,L “ ∆ ›››››ÿ w n ˜ λ w n θ w n ´ p n nS N N ÿ µ “ ÿ a,m ÿ w n ˜ λ w n θ w n t W n, p µ q p a,m q“ w n u ››››› ,L “ ∆ ››››› p n nS N ÿ µ ÿ a,m ÿ w n ˆ ˜ λ w n ´ λ w n p ` η q ˙ θ w n t W n, p µ q p a,m q“ w n u ››››› . We begin by bounding the term corresponding to L as L ď ÿ w n P T p n q δ p W q λ w n ε ´ ε } θ w n } loomoon “ ` ÿ w n R T p n q δ p W q λ w n } θ w n } loomoon “ “ ε. (50)25ow consider the term corresponding to L , for which we employ Lemma 4. Toward this, we consider thefollowing identification: λ x with ˜ λ w n , σ x with r θ “ ∆ ř w n ˜ λ w n θ w n , X with T p n q δ p W q , ¯ X with F np , σ with r θ , Π with Π θ , Π x with Π θw n , and µ x “ p n for all x P ¯ X . Since the collection of random variables t W n, p µ q p a, m qu aregenerated using Unionized Coset Codes, we have P ` t W n, p µ q p a,m q“ w n u “ ˘ “ p n , for all w n P F np . Note that ˜ λ wn { p n ď ´ n p S p W q σ ´ log p ´ δ w q for all w n P F np , where δ w p δ q Œ as δ Œ . With these, we checkthe hypotheses of Lemma 4. Firstly, using the pinching arguments described in Property 15.2.7 [6], we have Tr t Π θ θ w n u ě ´ (cid:15) for all (cid:15) P p , q , δ ą and sufficiently large n . Secondly, (5b) and (5e) are satisfied from theconstruction of Π θw n . Next, we consider the hypothesis (5c). We have ››› Π θ a ˜ θ ››› “ Tr !a Π θ ˜ θ Π θ ) ď a p ´ ε q Tr ! ? Π θ θ b n Π θ ) ď n p S p R q σθ ` δ w q , where the first inequality above follows from the fact that ř w n ˜ λ w n θ w n ď p ´ ε q ř w n λ w n θ w n “ θ b n p ´ ε q and usingthe operator monotonicty of the square-root function. The second inequality follows from the property of the typicalprojector for some δ w such that δ w Œ as δ Œ . This gives D “ n p S p R q σθ ` δ w q . Finally, the hypotheses (5d) is satisfied from the property of conditional typical projectors for d “ n p S p R | W q σθ ´ δ w q ,where δ w Œ as δ Œ . Next we check the pairwise independence of W n, p µ q p a, m q and W n, p µ q p ˜ a, ˜ m q . Since theseare constructed using randomly and uniformly generated G p µ q and h p µ q , we have t W n, p µ q p a, m qu a P F kp ,m P F lp ,µ Pr N s to be pairwise independent for each (see [16] for details). Therefore, employing Lemma 4 we get E r L s ď d n p S p R q σθ ` δ w q ´ n p S p W q σθ ´ log p ´ δ w q N nS n p S p R | W q σθ ´ δ w q ` ? (cid:15) ď ´ n p S ` n log N ´ I p R ; W q σθ ´ log p ` S p W q σθ ´ δ w ´ δ w ´ δ w q q ` ? (cid:15). (51)As for L , taking expectation and using E r t W n, p µ q p a,m q“ w n u s “ p n gives E r L s ď η ` ε p ` η q ` ε p ` η q “ η ` ε ` η . (52)Combining the bounds from (50) , (51) and (52) gives the desired result. D. Proof of Lemma 6
We begin by using the H´older’s inequality [18], [21] for operator norm, i.e., ( } AB } ď } A } } B } ), and defining ˆΛ w n “ a ρ b n ´ ˜ ρ w n a ρ b n ´ . This gives us ÿ w n γ p µ q w n ›››a ρ b n ´ ¯ A p µ q w n ´ A p µ q w n ¯a ρ b n ››› “ N N ÿ µ “ ÿ w n α w n γ p µ q w n ››› Π ρ a ρ b n ˆΛ w n a ρ b n Π ρ ´ Π ρ a ρ b n Π µ ˆΛ w n Π µ a ρ b n Π ρ ››› ď N N ÿ µ “ ÿ w n α w n γ p µ q w n ››› Π ρ a ρ b n ››› ››› ˆΛ w n ´ Π µ ˆΛ w n Π µ ››› ď N N ÿ µ “ ´ n p S p ρ q´ δ ρ q ÿ w n α w n γ p µ q w n c Tr ! p Π ρ A ´ Π µ q ˆΛ w n ) Tr ! ˆΛ w n ) , Π ρ and Π µ commute, the first inequality follows from the H´older’sinequality, and the second inequality uses the following bounds ››› ˆΛ w n ´ Π µ ˆΛ w n Π µ ››› ď ››› ˆΛ w n ´ Π µ ˆΛ w n ››› ` ››› Π µ ˆΛ w n ´ Π µ ˆΛ w n Π µ ››› “ Tr "ˇˇˇˇ p Π ρ ´ Π µ q b ˆΛ w n b ˆΛ w n ˇˇˇˇ* ` Tr "ˇˇˇˇ Π µ b ˆΛ w n b ˆΛ w n p Π ρ ´ Π µ q ˇˇˇˇ* ď c Tr ! p Π ρ ´ Π µ q ˆΛ w n ) Tr ! ˆΛ w n ) ` c Tr ! Π µ ˆΛ w n ) Tr ! ˆΛ w n p Π ρ ´ Π µ q ) ď c Tr ! p Π ρ ´ Π µ q ˆΛ w n ) Tr ! ˆΛ w n ) , where the second inequality uses Cauchy-Schwarz inequality (see Lemma 9.4.2 in [18]) and the last inequality usesthe arguments: (i) Π µ is a projector onto a subspace of Π ρ and (ii) Tr ! Π µ ˆΛ w n ) ď Tr ! ˆΛ w n ) . Further, using thefact that for w n P T p n q δ p W q , α w n ď p n ´ n p S p W q´ δ w q { nS and Tr t ˆΛ w n u “ } Π ρ ˆΛ w n Π ρ } ď } Π ρ a ρ b n ´ } } ˜ ρ w n } loomoon ď } Π ρ a ρ b n ´ } ď } Π ρ a ρ b n ´ } ď n p S p ρ q` δ ρ q , it follows that ÿ w n γ p µ q w n ›››a ρ b n ´ ¯ A p µ q w n ´ A p µ q w n ¯a ρ b n ››› ď N ´ n p S p ρ q´ δ ρ q N ÿ µ “ ÿ w n α w n γ p µ q w n c Tr ! p Π ρ ´ Π µ q ˆΛ w n ) ď N nδ ρ N ÿ µ “ ∆ p µ q gffeÿ w n α w n γ p µ q w n ∆ p µ q Tr tp Π ρ ´ Π µ q ˜ ρ w n u“ N nδ ρ N ÿ µ “ ´ ∆ p µ q ´ E r ∆ p µ q s ` E r ∆ p µ q s ¯ gffe Tr p Π ρ ´ Π µ q ÿ w n α w n γ p µ q w n ∆ p µ q ˜ ρ w n + ď N nδ ρ N ÿ µ “ E r ∆ p µ q s gffe Tr p Π ρ ´ Π µ q ÿ w n α w n γ p µ q w n ∆ p µ q ˜ ρ w n + ` N nδ ρ N ÿ µ “ ˇˇˇ ∆ p µ q ´ E r ∆ p µ q s ˇˇˇlooooooooomooooooooon H ď N nδ ρ N ÿ µ “ ˜ H ` a p ´ ε qp ` η q a H ` H ` H ¸ , where the second inequality above follows by defining ∆ p µ q “ ř w n P T p n q δ p W q α w n γ p µ q w n and using the concavity ofthe square-root function, the third inequality follows by using the fact that sum w n α w n γ p µ q w n ∆ p µ q Tr tp Π ρ ´ Π µ q ˜ ρ w n u ď ÿ w n α w n γ p µ q w n ∆ p µ q Tr t ˜ ρ w n u ď , (53)and defining H as above. and the last one follows by first using E r ∆ p µ q s “ p ´ ε qp ` η q and then defining H , H and H as in the statement of the lemma and using the inequality Tr t Λ p ω ´ σ qu ď } Λ p ω ´ σ q} ď } Λ } } ω ´ σ } . Thiscompletes the proof. 27 PPENDIX BP ROOF OF P ROPOSITIONS
A. Proof of Proposition 1
Applying the triangle inequality on r S gives r S ď r S ` r S , where r S “ ∆ N ÿ µ ›››››ÿ w n λ w n ˆ ρ w n ´ ÿ w n α w n γ p µ q w n ˆ ρ w n ››››› , r S “ ∆ N ÿ µ ÿ w n α w n γ p µ q w n } ˆ ρ w n ´ ˜ ρ w n } . For the first term r S , we use Lemma 5, and identify θ w n with ˆ ρ w n and N “ . Using this lemma, we obtain forall sufficiently large n , E r r S s ď (cid:15) ˜ S , given the rate S satisfies S ě I p W ; R q σ ´ S p W q σ ` log p ` δ ˜ S , where σ is as defined in the statement of the theorem and (cid:15) ˜ S follows from Lemma 5. As for the second term r S , we usethe gentle measurement lemma and bound its expected value as E « N ÿ µ ÿ w n α w n γ p µ q w n } ˆ ρ w n ´ ˜ ρ w n } ff ď ÿ w n P T p n q δ p W q λ w n p ` η q } ˆ ρ w n ´ ˜ ρ w n } ` ÿ w n R T p n q δ p W q λ w n p ` η q ď (cid:15) r S , where the second inequality is based on the repeated usage of the Average Gentle Measurement Lemma [6] and (cid:15) r S Œ as δ Œ (see (35) in [2] for more details). B. Proof of Proposition 2
To provide a bound for r S , we individually bound the terms corresponding to H and r H in an expected sense.Let us first consider r H . To provide a bound for ˜ H we use Lemma 4 with the following identification: λ x with λ wn p ´ ε q , σ x with ˆ ρ w n , X with T p n q δ p W q , ¯ X with F np , Π with Π ρ , Π x with Π w n , and µ x with p n .Firstly, we have λ wn { p n ď ´ n p S p W q σ ´ log p ´ δ w q for all w n P F np , where δ w p δ q Œ as δ Œ , which gives κ “ ´ n p S p W q σ ´ log p ´ δ w q . With these, we check the hypotheses of Lemma 4. As for the first hypothesis (5a), using the pinching argumentsdescribed in Property 15.2.7 [6], we have Tr t Π ρ ˆ ρ w n u ě ´ (cid:15) for all (cid:15) P p , q , δ ą and sufficiently large n .Then the hypotheses (5b) and (5e) are satisfied from the construction of Π w n . Next, consider the hypothesis (5c).We have ››››››› Π ρ gfffe¨˝ ÿ w n P T p n q δ p W q λ w n p ´ ε q ˆ ρ w n ˛‚››››››› ď Tr !b Π ρ ρ b n Π ρ ) ď n p S p R q σ ` δ w q , where the first inequality above follows from using ř w n P T p n q δ p W q λ wn p ´ ε q ˆ ρ w n ď p ´ ε q ρ b n and the operator monotonictyof the square-root function. The second inequality follows from the property of the typical projector for some δ w such that δ w Œ as δ Œ . This gives D “ n p S p R q σ ` δ w q , where σ is as defined in the statement of the theorem. Finally, the hypotheses (5d) is satisfied from the property ofconditional typical projectors for d “ n p S p R | W q σ ´ δ w q , where δ w Œ as δ Œ . Next we check the pairwiseindependence of W n, p µ q p a, m q and W n, p µ q p ˜ a, ˜ m q . Since these are constructed using randomly and uniformlygenerated G and h p µ q , we have t W n, p µ q p a, m qu a P F kp ,m P F lp ,µ Pr ,N s to be pairwise independent (see [16] for details).Therefore, employing inequality (7) of Lemma 4 we get E r ˜ H s ď d n p S p R q σ ` δ w q ´ n p S p W q σ ´ log p ´ δ w q N nS n p S p R | W q σ ´ δ w q ď ´ n p S ` n log N ´ I p R ; W q σ ´ log p ` S p W q σ ´ δ w ´ δ w ´ δ w q . H and perform the following simplification E r H s “ p ´ ε qp ` η q E ˇˇˇˇˇˇ ÿ w n P T p n q δ p W q λ w n p ´ ε q ´ p n nS ÿ w n P T p n q δ p W q ÿ a,i λ w n p ´ ε q t W n, p µ q p a,i q“ w n u ˇˇˇˇˇˇ “ λ w n p ´ ε q E ›››››› ÿ w n P T p n q δ p W q λ w n p ´ ε q ω b n ´ p n nS ÿ w n P T p n q δ p W q ÿ a,i λ w n p ´ ε q t W n, p µ q p a,i q“ w n u ω b n ›››››› , (54)where ω P D p H q is any state independent of W . We again apply Lemma 4 to the above simplification with thefollowing identification: λ x with λ wn p ´ ε q , σ x with ω b n , X with T p n q δ p W q , ¯ X with F np , Π and Π x with Identity operator I , and µ x with p n . κ remains as above, κ “ ´ n p S p W q σ ´ log p ´ δ w q and D “ d “ . Hence, we obtain E r H s ď (cid:15) H if S ě log p ´ S p W q σ ` δ w , where (cid:15) H p δ q Œ as δ Œ . This completes the proof.
C. Proof of Proposition 3
We begin using the definition of A p µ q w n and applying triangle inequality to S to obtain S ď p ` η q N ÿ µ ÿ a,i ą ÿ w n ,z n λ w n p n nS t aG ` h p µ q p i q“ w n u ›››a ρ b n Π µ a ρ b n ´ ˜ ρ w n a ρ b n ´ Π µ a ρ b n ››› ˇˇˇ P nZ | W p z n | w n q ´ P nZ | W ´ z n | F p µ q p i q ¯ˇˇˇ ď nδ ρ p ` η q N ÿ µ ÿ a,i ą ÿ w n ,z n λ w n p n nS t aG ` h p µ q p i q“ w n u ˇˇˇ P nZ | W p z n | w n q ´ P nZ | W ´ z n | F p µ q p i q ¯ˇˇˇ ď nδ ρ p ` η q N ÿ µ ÿ a,i ą ÿ w n λ w n p n nS t aG ` h p µ q p i q“ w n u p µ q p w n , i q , (55)where the second inequality above uses the following arguments ›››a ρ b n Π µ a ρ b n ´ ˜ ρ w n a ρ b n ´ Π µ a ρ b n ››› “ ›››a ρ b n Π ρ Π µ a ρ b n ´ Π ρ ˜ ρ w n Π ρ a ρ b n ´ Π µ Π ρ a ρ b n ››› ď ›››a ρ b n Π ρ ››› ››› Π µ a ρ b n ´ Π ρ ˜ ρ w n Π ρ a ρ b n ´ Π µ ››› ›››a ρ b n Π ρ ››› ď ´ n p S p ρ q´ δ ρ q } Π µ } ›››a ρ b n ´ Π ρ ˜ ρ w n Π ρ a ρ b n ´ ››› } Π µ } ď nδ ρ } ˜ ρ w n } ď nδ ρ , (56)where the above inequalities follow from the H´olders inequality, and the last inequality in (55) follows by defining p µ q p w n , i q as p µ q p w n , i q “ ∆ " Dp ˜ w n , ˜ a n q : ˜ w n “ ˜ a n G ` h p µ q p i q , ˜ w n P T p n q δ p W q , ˜ w n ‰ w n * . Observe that E r p µ q p w n , i q t aG ` h p µ q p i q“ w n u s ď ÿ ˜ a P F kp ÿ ˜ w P T p n q δ p W q ˜ w ‰ w n p n p n . Using this, we obtain E r S s ď nδ ρ p ` η q n p S ´ R q p n ÿ ˜ w n P T p n q δ p W q ÿ w n P T p n q δ p W q λ w n ď n p S ´ R ´ log p ` S p W q σ ` δ S q , where δ S Œ as δ Œ , and σ is as defined in the statement of the theorem. Therefore, if S ´ R ă log p ´ S p W q σ ´ δ S , then E r S s ď (cid:15) S , where (cid:15) S Œ , and the proof is complete.29 . Proof of Proposition 4 We bound r S as r S ď r S ` r S ` r S , where r S “ ∆ ››››› N N ÿ µ ,µ ÿ i ą b ρ b nAB ´ Γ A, p µ q i b Γ B, p µ q ¯ b ρ b nAB P nZ | U ` V p z n | w n q ››››› , r S “ ∆ ››››› N N ÿ µ ,µ ÿ j ą b ρ b nAB ´ Γ A, p µ q b Γ B, p µ q j ¯ b ρ b nAB P nZ | U ` V p z n | w n q ››››› , r S “ ∆ ››››› N N ÿ µ ,µ b ρ b nAB ´ Γ A, p µ q b Γ B, p µ q ¯ b ρ b nAB P nZ | U ` V p z n | w n q ››››› . Analysis of r S : We have r S ď N N ÿ µ ,µ ÿ i ą ÿ z n P nZ | U ` V p z n | w n q ››››b ρ b nAB ´ Γ A, p µ q i b Γ B, p µ q ¯ b ρ b nAB ›››› ď N N ÿ µ ,µ ››››b ρ b nB Γ B, p µ q b ρ b nB ›››› ď N ÿ µ ›››››ÿ v n λ Bv n ˆ ρ Bv n ´ ÿ v n b ρ b nB ζ p µ q v n ¯ B p µ q v n b ρ b nB ››››› ` N ÿ µ ›››››ÿ v n b ρ b nB ζ p µ q v n ´ ¯ B p µ q v n ´ B p µ q v n ¯ b ρ b nB ››››› ď N ÿ µ ›››››ÿ v n λ Bv n ˆ ρ Bv n ´ p ` η q | V | n nS ÿ v n ÿ a ,j λ Bv n ˆ ρ Bv n V n, p µ q p a ,j q ››››› ` N ÿ µ ÿ v n β v n ζ p µ q v n ›› ˆ ρ Bv n ´ ˜ ρ Bv n ›› looooooooooooooooooomooooooooooooooooooon r S ` N ÿ µ ÿ v n ››››b ρ b nB ζ p µ q v n ´ ¯ B p µ q v n ´ B p µ q v n ¯ b ρ b nB ›››› loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon r S , (57)where the first inequality uses triangle inequality. The next inequality follows by using Lemma 1 where we use thefact that ř i ą Γ A, p µ q i ď I. Finally, the last two inequalities follows again from triangle inequality.Regarding the first term in (57), using Lemma 5 we claim that for sufficiently large n , the term can be madearbitrarily small in the expected sense, given the rate S satisfies S ě I p V ; RA q σ ´ S p V q σ ` log p ` δ ˜ S , where σ , σ are as defined in the statement of the theorem and δ ˜ S Œ as δ Œ . As for the second term, we use thegentle measurement lemma (as in (70)) and bound its expected value as E « N ÿ µ ÿ v n ζ p µ q v n ›› ˆ ρ Bv n ´ ˜ ρ Bv n ›› ff “ ÿ v n P T p n q δ p V q λ Bv n p ` η q ›› ˆ ρ Bv n ´ ˜ ρ Bv n ›› ď (cid:15) r S , where the inequality is based on the repeated usage of the Average Gentle Measurement Lemma and (cid:15) r S Œ as δ Œ (see (35) in [2] for more details ). Finally, consider the last term. To simplify this term, we appeal toLemma 6 in Section IV-B. This gives us ˜ S ď N nδ N ÿ µ “ ˜ H B ` a p ´ ε B qp ` η q b H B ` H B ` H B ¸ , (58)where H B “ ∆ ˇˇˇ ∆ p µ q B ´ E r ∆ p µ q B s ˇˇˇ , H B “ ∆ Tr p Π ρ B ´ Π µ B q ÿ v n λ Bv n ˜ ρ Bv n + , B “ ∆ ›››››ÿ v n λ Bv n ˜ ρ Bv n ´ p ´ ε B q ÿ v n β v n ζ p µ q v n E r ∆ p µ q B s ˜ ρ Bv n ››››› , H B “ ∆ p ´ ε B q ›››››ÿ v n β v n ζ p µ q v n ∆ p µ q B ˜ ρ Bv n ´ ÿ v n α v n ζ p µ q v n E r ∆ p µ q B s ˜ ρ Bv n ››››› , (59)and ∆ p µ q B “ ř v n P T p n q δ p V q β v n ζ p µ q v n and ε B “ ř v n R T p n q δ p V q λ Bv n .Further, using the simplification performed in (21), (23), and (24), and the concavity of the square-root function,we obtain, E r ˜ S s ď N nδ ρB N ÿ µ “ ˜ E r H B s ` p ´ ε B qp ` η q dˆ nδ ρB η ` ˙ E r r H B s ` d p ´ ε B qp ` η q b E r H B s ¸ , (60)where r H B “ ∆ ›››››ÿ v n λ Bv n ˜ ρ Bv n ´ p n nS ÿ v n ÿ a ,j ą λ Bv n ˜ ρ Bv n p ` η q t V n, p µ q p a ,j q“ v n u ››››› . (61)Using Proposition 2, for all sufficiently small δ and sufficiently large n , we have E ” ˜ S ı ď (cid:15) ˜ S if S ě I p V ; RA q σ ` log p ´ S p V q σ ` δ ˜ S , where σ , σ are the auxiliary state defined in the theorem and (cid:15) ˜ S , δ ˜ S Œ as δ Œ . Analysis of r S : Due to the symmetry in r S and r S , the analysis of r S follows very similar arguments as that of r S and hence we obtain E ” ˜ S ı ď (cid:15) ˜ S if S ě I p U ; RB q σ ` log p ´ S p U q σ ` δ ˜ S , where σ , σ are the auxiliarystate defined in the theorem and (cid:15) ˜ S , δ ˜ S Œ as δ Œ . Analysis of r S : We have r S ď N N ÿ µ ,µ ÿ z n P nZ | U ` V p z n | w n q ››››b ρ b nAB ´ Γ A, p µ q b Γ B, p µ q ¯ b ρ b nAB ›››› ď N N ÿ µ ,µ ››››b ρ b nAB ´ Γ A, p µ q b I ¯ b ρ b nAB ›››› ` N N ÿ µ ,µ ÿ v n ››››b ρ b nAB ´ Γ A, p µ q b B p µ q v n ¯ b ρ b nAB ›››› , (62)where the inequalities above are obtained by a straight forward substitution and use of triangle inequality. Further,since ď Γ A, p µ q ď I and ď Γ B, p µ q ď I , this simplifies the first term in (62) as N N ÿ µ ,µ ››››b ρ b nAB ´ Γ A, p µ q b I ¯ b ρ b nAB ›››› “ N ÿ µ ››››b ρ b nA ´ Γ A, p µ q ¯ b ρ b nA ›››› . Similarly, the second term in (62) simplifies using Lemma 1 as N N ÿ µ ,µ ÿ v n ››››b ρ b nAB ´ Γ A, p µ q b B p µ q v n ¯ b ρ b nAB ›››› ď N ÿ µ ››››b ρ b nA ´ Γ A, p µ q ¯ b ρ b nA ›››› . Using these simplifications, we have r S ď N ÿ µ ››››b ρ b nA ´ Γ A, p µ q ¯ b ρ b nA ›››› . The above expression is similar to the one obtained in the simplification of r S and hence we can bound r S usingsimilar constraints as r S , for sufficiently large n . 31 . Proof of Proposition 5 Recalling S , we have S ď p ` η q N N ÿ µ ,µ ÿ i,j ÿ a ,a ÿ u n ,v n p n p n λ Au n λ Bv n n p S ` S q Ω u n ,v n p µ q p u n ` v n , i, j q t a G ` h p µ q p i q“ u n u t a G ` h p µ q p j q“ v n u , where Ω u n ,v n and p µ q p w n , i, j q are defined as Ω u n ,v n “ ∆ Tr ! „` Π µ A b Π µ B ˘ b ρ b nA b ρ b nB ´ p ˜ ρ Au n b ˜ ρ Bv n q b ρ b nA b ρ b nB ´ ` Π µ A b Π µ B ˘ ρ b nAB ) , p µ q p w n , i, j q “ ∆ " Dp ˜ w n , ˜ a n q : ˜ w n “ ˜ a n G ` h p µ q p i q ` h p µ q p j q , ˜ w n P T p n q ˆ δ p U ` V q , ˜ w n ‰ w n * . Note that E r p µ q p u n ` v n , i, j q t a G ` h p µ q p i q“ u n u t a G ` h p µ q p j q“ v n u s ď ÿ ˜ a P F kp ˜ a ‰ a ÿ ˜ w P T p n q ˆ δ p U ` V q ˜ w ‰ u n ` v n p n p n p n . Using this, we obtain E r S s ď p ` η q n p S ´ R q p n ÿ ˜ w n P T p n q ˆ δ p U ` V q ÿ u n P T p n q δ p U q ÿ v n P T p n q δ p V q λ Au n λ Bv n Ω u n ,v n ď p ` η q n p S ´ R ´ log p ` S p U ` V q σ ` δ ρAB ` ˆ δ W q , where ˆ δ W Œ as δ Œ and the above inequality follows from the following lemma. Hence, E r S s ď (cid:15) S if theconditions in the proposition are satisfied. Lemma 9.
For λ Au n , λ Bv n and Ω u n ,v n as defined above, we have ÿ u n P T p n q δ p U q ÿ v n P T p n q δ p V q Ω u n ,v n λ Au n λ Bv n ď nδ ρAB , for some δ ρ AB Œ as δ Œ . Proof.
Firstly, note that ÿ u n ,v n Ω u n ,v n λ Au n λ Bv n “ Tr "„ ` Π µ A b Π µ B ˘ ˜b ρ b nA ´ ´ ÿ u n λ Au n ˜ ρ Au n ¯b ρ b nA ´ b b ρ b nB ´ ´ ÿ v n λ Bv n ˜ ρ Bv n ¯b ρ b nB ´ ¸ ` Π µ A b Π µ B ˘ ρ b nAB * . (63)We know, ř u n λ Au n ˜ ρ Au n ď ´ n p S p ρ A q´ δ ρA q Π ρ A , where δ ρ A Œ as δ Œ . This implies, Π µ A b ρ b nA ´ ˜ÿ u n λ Au n ˜ ρ Au n ¸ b ρ b nA ´ Π µ A ď ´ n p S p ρ A q´ δ ρA q Π µ A b ρ b nA ´ Π ρ A b ρ b nA ´ Π µ A ď nδ ρA Π µ A Π ρ A Π µ A ď nδ ρA Π µ A , (64)where the second inequality appeals to the fact that b ρ b nA ´ Π ρ A b ρ b nA ´ ď n p S p ρ A q` δ ρA q Π ρ A . Similarly, usingthe same arguments above for the operators acting on H B , we have Π µ B b ρ b nB ´ ˜ÿ v n λ Bv n ˜ ρ Bv n ¸ b ρ b nB ´ Π µ B ď nδ ρB Π µ B , (65)32here δ ρ B Œ as δ Œ . Using (i) the simplifications in (64) and (65), and (ii) the fact that for A ě B ě and A ě B ě , p A b A q ě p B b B q in (63), gives ÿ u n ,v n Ω u n ,v n λ Au n λ Bv n ď n p δ ρA ` δ ρB q Tr (cid:32)` Π µ A b Π µ B ˘ ρ b nAB ( ď n p δ ρA ` δ ρB q Tr (cid:32) ρ b nρ AB ( “ n p δ ρA ` δ ρB q . Substituting δ ρ AB “ p δ ρ A ` δ ρ B q gives the result. F. Proof of Proposition 6
We start by applying triangle inequality to obtain J ď J ` J , where J “ ∆ ÿ z n ,v n ›››››ÿ u n b ρ b nAB ˜ ¯Λ Au n b ¯Λ Bv n ´ N N ÿ µ “ α u n γ p µ q u n λ Au n ¯Λ Au n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,J “ ∆ ÿ z n ,v n ››››› N N ÿ µ “ ÿ u n b ρ b nAB ˜ α u n γ p µ q u n λ Au n ¯Λ Au n b ¯Λ Bv n ´ γ p µ q u n ¯ A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› , Now with the intention of employing Lemma 5, we express J as J “ ››››› ÿ u n ,v n ,z n λ ABu n ,v n ˆ ρ ABu n ,v n b φ u n ,v n ,z n ´ p ` η q p n nS N ÿ µ ÿ u n ,v n ,z n ÿ a ,i ą λ Au n t U n, p µ q p a ,i q“ u n u λ ABu n ,v n λ Au n ˆ ρ ABu n ,v n b φ u n ,v n ,z n ››››› , where the equality above is obtained by defining φ u n ,v n ,z n “ P nZ | W p z n | u n ` v n q | v n yx v n | b | z n yx z n | and using thedefinitions of α u n , γ p µ q u n and ˆ ρ ABu n ,v n , followed by using the triangle inequality for the block diagonal operators. Notethat the triangle inequality in this case becomes an equality.Let us define T u n as T u n “ ∆ ÿ v n ,z n λ ABu n ,v n λ Au n ˆ ρ ABu n ,v n b φ u n ,v n ,z n . Note that in the above definition of T u n we have T u n ě and Tr t T u n u “ for all u n P F np . Further, it contains allthe elements in product form, and thus can be written as T u n “  ni “ T u i . This simplifies J as J “ ›››››ÿ u n λ Au n T u n ´ p ` η q p n nS N ÿ µ ÿ u n ÿ a ,i ą λ Au n T u n t U n, p µ q p a ,i q“ u n u ››››› . Using Lemma 5 we claim that E r J s ď (cid:15) J , for all sufficiently large n, given the constraint S ` n log N ě I p U ; RZV q σ ´ S p U q σ ` log p ` δ J , where σ is the auxiliary state defined in the theorem and (cid:15) J , δ J Œ as δ Œ .Now, we consider the term corresponding to J and prove that its expectation with respect to the Alice’scodebook is small. Recalling J , we get J ď N N ÿ µ “ ÿ u n ,v n ÿ z n P nZ | W p z n | u n ` v n q ›››››b ρ b nAB ˜ α u n γ p µ q u n λ Au n ¯Λ Au n b ¯Λ Bv n ´ γ p µ q u n ¯ A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB ››››› , “ N N ÿ µ “ ÿ u n ,v n α u n γ p µ q u n ››››b ρ b nAB ˆˆ λ Au n ¯Λ Au n ´ b ρ b nA ´ ˜ ρ Au n b ρ b nA ´ ˙ b ¯Λ Bv n ˙ b ρ b nAB ›››› , ř z n P nZ | W p z n | u n ` v n q “ for all u n P U n and v n P V n and using the definition of A p µ q u n . By applying expectation of J over theAlice’s codebook, we get E r J s ď p ` η q ÿ u n λ Au n ÿ v n ››››b ρ b nAB ˆˆ λ Au n ¯Λ Au n ´ b ρ b nA ´ ˜ ρ Au n b ρ b nA ´ ˙ b ¯Λ Bv n ˙ b ρ b nAB ›››› , where we have used the fact that E r α u n γ p µ q u n s “ λ Aun p ` η q . To simplify the above equation, we employ Lemma 1 whichcompletely discards the effect of Bob’s measurement.Since ř v n ¯Λ Bv n “ I , from Lemma 1 we have for every u n , ÿ v n ››››b ρ b nAB ˆˆ λ Au n ¯Λ Au n ´ b ρ b nA ´ ˜ ρ Au n b ρ b nA ´ ˙ b ¯Λ Bv n ˙ b ρ b nAB ›››› “ ››››b ρ b nA ˆ λ Au n ¯Λ Au n ´ b ρ b nA ´ ˜ ρ Au n b ρ b nA ´ ˙ b ρ b nA ›››› . This simplifies E r J s as E r J s ď p ` η q ÿ u n λ Au n ››››b ρ b nA ˆ λ Au n ¯Λ Au n ´ b ρ b nA ´ ˜ ρ Au n b ρ b nA ´ ˙ b ρ b nA ›››› ď p ` η q ÿ u n R T p n q δ p U q λ Au n ›› ˆ ρ Au n ›› ` p ` η q ÿ u n P T p n q δ p U q λ Au n ››` ˆ ρ Au n ´ ˜ ρ Au n ˘›› ď ε A ` (cid:15) J “ (cid:15) J , where the last inequality is obtained by repeated usage of the Average Gentle Measurement Lemma and (cid:15) J ,(cid:15) J Œ as δ Œ (see (35) in [2] for details). G. Proof of Proposition 7
Noting the similarity between J and the term ˜ S defined in the proof of Theorem 1 (see Section IV-B), webegin by further simplifying J using Lemma 6. This gives us J ď N nδ ρA N ÿ µ “ ˜ H A ` a p ´ ε A qp ` η q b H A ` H A ` H A ¸ , (66)where H A “ ∆ ˇˇˇ ∆ p µ q A ´ E r ∆ p µ q A s ˇˇˇ , H A “ ∆ Tr p Π ρ A ´ Π µ A q ÿ w n λ Au n ˜ ρ Au n + ,H A “ ∆ } ÿ u n λ Au n ˜ ρ Au n ´ p ´ ε A q ÿ u n α u n γ p µ q u n E r ∆ p µ q A s ˜ ρ Au n } , H A “ ∆ p ´ ε A q} ÿ u n α u n γ p µ q u n ∆ p µ q A ˜ ρ Au n ´ ÿ u n α u n γ p µ q u n E r ∆ p µ q A s ˜ ρ Au n } , and ∆ p µ q A “ ř u n P T p n q δ p U q α u n γ p µ q u n , ε A “ ∆ ř u n R T p n q δ p U q λ Au n , and δ ρ A p δ q Œ as δ Œ . Further, using thesimplification performed in (21), (23), and (24), and the concavity of the square-root function, we obtain, E r J s ď N nδ ρA N ÿ µ “ ˜ E r H A s ` p ´ ε A qp ` η q dˆ nδ ρA η ` ˙ E r r H A s ` d p ´ ε A qp ` η q b E r H A s ¸ , (67)where r H A “ ∆ ›››››ÿ u n λ Au n p ´ ε A q ˜ ρ Au n ´ p n nS ÿ u n ÿ a ,i ą λ Au n p ´ ε A q ˜ ρ Au n t U n, p µ q p a ,i q“ u n u ››››› . (68)The proof from here follows from Proposition 2. 34 . Proof of Proposition 8 We start by adding and subtracting the following terms within Q p i q ÿ u n ,v n b ρ b nAB ` ¯Λ Au n b ¯Λ Bv n ˘ b ρ b nAB P nZ | W p z n | u n ` v n q , p ii q ÿ u n ,v n N N ÿ µ “ b ρ b nAB ˜ ¯Λ Au n b β v n ζ p µ q v n λ Bv n ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q , p iii q ÿ u n ,v n N N ÿ µ ,µ b ρ b nAB ˜ γ p µ q u n A p µ q u n b β v n ζ p µ q v n λ Bv n ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q , p iv q ÿ u n ,v n N N ÿ µ ,µ b ρ b nAB ´ γ p µ q u n A p µ q u n b ζ p µ q v n ¯ B p µ q v n ¯ b ρ b nAB P nZ | W p z n | u n ` v n q . This gives us Q ď Q ` Q ` Q ` Q ` Q , where Q “ ∆ ÿ z n ››››› ÿ u n ,v n b ρ b nAB ˜˜ N N ÿ µ “ γ p µ q u n A p µ q u n ¸ b ¯Λ Bv n ´ ¯Λ Au n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,Q “ ∆ ÿ z n ››››› ÿ u n ,v n b ρ b nAB ˜ ¯Λ Au n b ¯Λ Bv n ´ ¯Λ Au n b ˜ N N ÿ µ “ β v n ζ p µ q v n λ Bv n ¯Λ Bv n ¸¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,Q “ ∆ ÿ z n ››››› ÿ u n ,v n b ρ b nAB ˜˜ ¯Λ Au n ´ N ÿ µ γ p µ q u n A p µ q u n ¸ b ˜ N ÿ µ β v n ζ p µ q v n λ Bv n ¯Λ Bv n ¸¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,Q “ ∆ ÿ z n ››››› ÿ u n ,v n N N ÿ µ ,µ b ρ b nAB ˜ γ p µ q u n A p µ q u n b ˜ β v n ζ p µ q v n λ Bv n ¯Λ Bv n ´ ζ p µ q v n ¯ B p µ q v n ¸¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ,Q “ ∆ ÿ z n ››››› ÿ u n ,v n N N ÿ µ ,µ b ρ b nAB ´ γ p µ q u n A p µ q u n b ´ ζ p µ q v n ¯ B p µ q v n ´ ζ p µ q v n B p µ q v n ¯¯ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› . We start by analyzing Q . Note that Q is exactly same as Q and hence using the same rate constraints as Q ,this term can be bounded. Next, consider Q . Substitution of ζ p µ q v n gives Q “ ››››› ÿ u n ,v n ,z n λ ABu n ,v n ˆ ρ ABu n ,v n b ψ u n ,v n ,z n ´ N ÿ µ ÿ u n ,v n ,z n β v n ÿ a ,j ą t V n, p µ q p a ,j q“ v n u λ ABu n ,v n λ Bv n ˆ ρ ABu n ,v n b ψ u n ,v n ,z n ››››› , where ψ u n ,v n ,z n is defined as ψ u n ,v n ,z n “ P nZ | W p z n | u n ` v n q | z n yx z n | , and the equality uses the triangle inequalityfor block operators. Now we use Lemma 5 to bound Q . Let T v n “ ÿ u n ,z n λ ABu n ,v n λ Bv n ˆ ρ ABu n ,v n b ψ u n ,v n ,z n . Note that T v n can be written in tensor product form as T v n “  ni “ T v i . This simplifies Q as Q “ ›››››ÿ v n λ Bv n T v n ´ p ` η q p n nS N ÿ µ ÿ v n ÿ a ,j ą λ Bv n T v n t V n, p µ q p a ,j q“ v n u ››››› . Lemma 5 gives us functions, (cid:15) Q p δ q , δ Q p δ q P p , q , such that if S ` n log N ě I p V ; RZ q σ ´ S p V q σ ` log p ` δ Q , (69)then E r Q s ď (cid:15) Q where (cid:15) Q , δ Q Œ as δ Œ . Q . Taking expectation with respect G, h p µ q , h p µ q gives E r Q s ď E « ÿ z n ,v n N N ÿ µ “ β v n ζ p µ q v n λ Bv n ›››››ÿ u n b ρ b nAB ` ¯Λ Au n b ¯Λ Bv n ˘ b ρ b nAB P nZ | W p z n | u n ` v n q´ ÿ u n b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ff “ E G,h »– ÿ z n ,v n N N ÿ µ “ E h | G ” β v n ζ p µ q v n | G ı λ Bv n ›››››ÿ u n b ρ b nAB ` ¯Λ Au n b ¯Λ Bv n ˘ b ρ b nAB P nZ | W p z n | u n ` v n q´ ÿ u n b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ff “ E G,h « ÿ z n ,v n p ` η q ›››››ÿ u n b ρ b nAB ` ¯Λ Au n b ¯Λ Bv n ˘ b ρ b nAB P nZ | W p z n | u n ` v n q´ ÿ u n b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b ¯Λ Bv n ¸ b ρ b nAB P nZ | W p z n | u n ` v n q ››››› ff “ E « J p ` η q ff , where the inequality above is obtained by using the triangle inequality, and the first equality follows from h p µ q and h p µ q being generated independently. The last equality follows from the definition of J as in (41). Hence, weuse the result obtained in bounding E r J s . Next, we consider Q . Q ď ÿ u n ,v n ÿ z n P nZ | W p z n | u n ` v n q ››››› N N ÿ µ ,µ b ρ b nAB ˜ γ p µ q u n A p µ q u n b β v n ζ p µ q v n λ Bv n ¯Λ Bv n ¸ b ρ b nAB ´ N N ÿ µ ,µ b ρ b nAB ´ γ p µ q u n A p µ q u n b β v n ζ p µ q v n ´ ? ρ B ´ ˜ ρ Bv n ? ρ B ´ ¯¯ b ρ b nAB ››››› ď N ÿ µ ÿ u n ,v n β v n ζ p µ q v n ›››››b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b λ Bv n ¯Λ Bv n ¸ b ρ b nAB ´ b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b ´ ? ρ B ´ ˜ ρ Bv n ? ρ B ´ ¯¸ b ρ b nAB ››››› , where the inequalities follow from the definition of ¯ B p µ q v n and using multiple triangle inequalities. Taking expectationof Q with respect to h p µ q , we get E r Q s ď E G,h « ÿ u n ,v n λ Bv n p ` η q ›››››b ρ b nAB ˜ N ÿ µ γ p µ q u n A p µ q u n b ˜ λ Bv n ¯Λ Bv n ´ ? ρ B ´ ˜ ρ Bv n ? ρ B ´ ¸¸ b ρ b nAB ff ď E G,h «ÿ v n λ Bv n p ` η q ›››››b ρ b nB ˜ λ Bv n ¯Λ Bv n ´ ? ρ B ´ ˜ ρ Bv n ? ρ B ´ ¸ b ρ b nB ››››› ff “ ÿ v n R T p n q δ p V q λ Bv n p ` η q ›› ˆ ρ Bv n ›› ` ÿ v n P T p n q δ p V q λ Bv n p ` η q ›› ˆ ρ Bv n ´ ˜ ρ Bv n ›› ď ε B ` (cid:15) Q “ (cid:15) Q , (70)where the second inequality above follows by using Lemma 1 and the fact that N ř µ ř u n γ p µ q u n A p µ q u n ď I, and thelast inequality follows by applying the Average Gentle Measurement Lemma repeated and (cid:15) Q , (cid:15) Q Œ as δ Œ Q . Finally, we move onto considering Q . 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