Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting
aa r X i v : . [ qu a n t - ph ] F e b Novel one-shot inner bounds for unassisted fully quantumchannels via rate splitting
Sayantan Charaborty ∗ Aditya Nema ∗ Pranab Sen * Abstract
We prove the first non-trivial one-shot inner bounds for sending quantum informationover an entanglement unassisted two-sender quantum multiple access channel (QMAC) andan unassisted two-sender two-receiver quantum interference channel (QIC). Previous worksonly studied the unassisted QMAC in the limit of many independent and identical uses of thechannel also known as the asymptotic iid limit, and did not study the unassisted QIC at all.We employ two techniques, rate splitting and successive cancellation , in order to obtain our in-ner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, forclassical channels in the asymptotic iid setting. Our main technical contribution is to extendrate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In theasymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yardet al. [YDH05]. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting.All our results also extend to the case where limited entanglement assistance is provided, inboth one-shot and asymptotic iid settings. The limited entanglement results for one-setting forboth QMAC and QIC are new. For the QIC the limited entanglement results are new even inthe asymptotic iid setting.
The multiple access channel (MAC), where two independent senders Alice (A) and Bob (B) haveto send their respective messages to a single receiver Charlie (C) via a communication channelwith two inputs and one output, is arguably the simplest multiterminal channel. Yet, it ab-stracts out important practical situations like several independent users transmitting their respec-tive messages to a base station. Ahlswede [Ahl71], and independently Liao [Lia72], obtainedthe first optimal rate region for the classical MAC in the asymptotic iid setting, using a pow-erful method called simultaneous decoding . Their region looks like the one in Figure 1, where I ( A : B ) : = H ( A ) + H ( B ) − H ( AB ) denotes the mutual information between two jointly dis-tributed random variables A , B . Simultaneous decoding means that Charlie is able to decode anypoint in the rate region, e.g. point P in Figure 1, by a one-step procedure. Later on, other authorsobtained the same rate region in a computationally less intensive fashion by using successive can-cellation and time sharing . In successive cancellation decoding Charlie first decodes Alice’s messageand then uses it as an additional channel output in order to next decode Bob’s message, or viceversa. In other words, Charlie can either decide to decode point S or point T in Figure 1. In order * School of Technology and System Science, Tata Institute of Fundamental Research, Mumbai, India. Email: {kings-bandz, aditya.nema30, pranab.sen.73}@gmail.com n → ∞ P ( I ( B : AC )) ( I ( A : BC ) , 0 ) S = ( I ( A : BC ) , I ( B : C )) T = ( I ( A : C ) , I ( B : AC )) x + y = I ( AB : C ) Figure 1: Achievable rate region per channel use for the classical MAC in the asymptotic iid set-ting.to decode another point in the rate region, e.g. point P in Figure 1, Charlie first figures out theconvex combination ( α , 1 − α ) of points S and T that would give point P. Out of n iid channel uses,Charlie then decodes the first α n uses according to point S’s decoding strategy and the remaining ( − α ) n channel uses according to point T’s decoding strategy. This idea is called time sharing.The interference channel is another important channel where sender Alice wants to send hermessage to receiver Charlie and sender Bob, whose message is independent of that of Alice, wantsto send his message to receiver Damru via a communication channel with two inputs and twooutputs. It abstracts out the important practical situation where independent sender-receiver pairsare communicating simultaneously via a noisy medium. Han and Kobayashi [HK81] (see also[CMGE08]) obtained the best known inner bound for this channel in the classical asymptotic iidsetting.The multiple access and interference channels can be defined in the context of quantum in-formation theory also. Early work studied the sending of classical information over a quantumMAC, without [Win01] or with [HDW08] entanglement assistance, in the asymptotic iid setting.These works obtained the natural quantum analogues of the optimal classical rate regions usingsuccessive cancellation and time sharing. Later, Fawzi et al. [FHS +
12] and Sen [Sen12] studied thesending of classical information over a quantum interference channel in the asymptotic iid settingby first obtaining a simultaneous decoder for the quantum MAC. The latter paper managed toobtain the natural quantum analogue of the Han-Kobayashi inner bound.For a variety of reasons recent research in Shannon theory has studied in depth the one-shot set-ting where the channel can be used only once. This is the most general setting and subsumes theasymptotic iid, asymptotic non-iid aka information spectrum, and finite block length settings. Ide-ally, the one-shot inner bounds should match or supersede the best inner bounds for the respectivechannels in the asymptotic iid setting. Sen [Sen18a] obtained the natural one-shot quantum ana-logues of best known classical rate regions for sending classical information over entanglementunassisted and assisted quantum MACs and quantum interference channels. His one-shot innerbounds, obtained by simultaneous decoding, approach the optimal inner bounds known earlierfor the classical and quantum asymptotic iid settings.Note that presence of shared randomness does not affect the rates of sending classical orquantum information over channels. Also the rates of sending quantum information and clas-sical information over entanglement assisted quantum channels are related by a factor of twobecause of quantum teleportation. So the main setting left unstudied in the above works is the2etting of sending quantum information over an entanglement unassisted quantum channel i.e.the senders and the receivers do not share any entanglement prior to the beginning of the pro-tocol. The first works to address this setting looked at a point-to-point quantum channel in theasymptotic iid setting [Llo97, Sho02], culminating in the work of Devetak [Dev05] which showedwith full rigour that the regularised coherent information from sender A to receiver B defined by I ∗ ( A > B ) : = lim k → ∞ I ( A k > B k ) / k , I ( A k > B k ) : = H ( B k ) − H ( A k B k ) where A k B k is defined bythe channel action ( N A ′ → B ) ⊗ k on an arbitrary (in general entangled) pure state | σ i A k ( A ′ ) k , is thecapacity of an unassisted quantum channel in the asymptotic iid limit. Hayden et al. [HHYW07].showed that one can recover Devetak’s result using a technique called decoupling These works naturally lead one to consider unassisted multiterminal quantum channels. Tothe best of our knowledge, the only inner bound known for the unassisted QIC is what one wouldobtain by treating the channel as two independent unassisted point to point channels. For theunassisted QMAC more is known. Yard et al. [YDH05] showed that the natural quantum analogueof the classical rate region, with mutual information replaced by regularised coherent informationas in Figure 2, is an inner bound for the unassisted quantum MAC (QMAC) in the asymptotic iidsetting. They proved their inner bound by time sharing and a suitable adaptation of successivecancellation. Bob Alice n → ∞ P ( I ∗ ( B > AC )) ( I ∗ ( A > BC ) , 0 ) S = ( I ∗ ( A > BC ) , I ∗ ( B > C )) T = ( I ∗ ( A > C ) , I ∗ ( B > AC )) x + y = I ∗ ( AB > C ) Figure 2: Achievable rate region for the unassisted quantum MAC per channel use in the asymp-totic iid setting.The above works behoove one to consider the problem of sending quantum information overan unassisted quantum channel in the one-shot setting. Buscemi and Datta [BD10] proved thefirst one-shot achievability result for the unassisted point-to-point channel in terms of smoothmodified Rényi entropies. Their result was generalised by Dupuis [Dup10] to the case where thereceiver has some side information about the sender’s message. In the asymptotic iid limit, theseone-shot results approach the regularised coherent information obtained in earlier works.It is thus natural to study inner bounds for the unassisted QMAC in the one-shot setting. Inthis paper we take the first steps towards this problem. Observe that successive cancellation canonly give the two endpoints S and T of the dominant line of the pentagonal rate region in Figure 2.Since time sharing cannot be used in the one-shot setting, it is not clear how to obtain other ratetuples like the point P . An alternative would be to develop a simultaneous decoder for the QMACwhich can obtain a point like P directly, but that is a major open problem with connections to thenotorious simultaneous smoothing open problem [CNS21].Instead in this paper, we take inspiration from another powerful classical channel coding tech-nique called rate splitting . Grant, et al. [GRUW01] showed that it is possible to ‘split’ Alice into two3enders Alice and Alice , each sending disjoint parts of Alice’s original message, such that anypoint in the pentagonal rate region of Figure 2 like P can be obtained without time sharing by asuccessive cancellation process where Charlie first decodes Alice ’s message, then Bob’s messageusing Alice ’s message as side information and finally Alice ’s message using Bob’s and Alice ’smessages as side information. Though Grant et al.’s rate splitting technique was developed for theclassical MAC in the asymptotic iid setting, in this paper we show how it can be adapted to the one-shot quantum setting . This is a non-trivial task, which we tackle in two steps. In the first step we useideas from Yard et al. [YDH05] and Dupuis [Dup10] and suitably adapt successive cancellation tothe one-shot unassisted quantum setting. In the second step, we adapt the rate splitting functionof Grant et al. [GRUW01] to the one-shot quantum setting. Our one-shot rates are in terms ofthe smooth coherent Rényi-2 information defined in Section 2. Since the smooth coherent Rényi-2information is not known to possess a chain rule with equality, we get an achievable rate regionof the form in Figure 3. Our achievable rate region is a subset of the ‘ideal’ pentagonal rate re-gion shown by the dashed line. Nevertheless, using a quantum asymptotic equipartition result ofTomamichel et al. [TCR09], we show that this ‘subpentagonal’ achievable rate region approachesthe ‘pentagonal’ region of Yard et al. [YDH05] (equal to the region demarcated by the dashed line)in the iid limit. The reason why splitting of Alice into Alice and Alice allows one to obtain aBob Alice n = n → ∞ S = ( I ǫ /8002 ( A > BC ) , I ǫ /8002 ( B > C )) T = ( I ǫ /8002 ( A > C ) , I ǫ /8002 ( B > AC ))( I √ ǫ ( AB > C ) , 0 )( I √ ǫ ( AB > C )) P Figure 3: One-shot achievable rate region for the unassisted QMAC (for single channel use only),contained inside the ‘ideal’ pentagonal region demarcated by the dashed line, and approaching itin the asymptotic iid limit. O ( log ǫ ) additive factors have been ignored in the figure.‘middle’ rate point like P , in addition to the ‘corner’ points S and T , is as follows. The rate point P is the projection onto the (Alice, Bob) plane of the ‘corner’ rate point P ′ in the (Alice , Bob, Alice )space where the rates of Alice and Alice are summed to obtain Alice’s rate. The point P ′ can beobtained by a 3-step successive cancellation decoding. Note that the split of Alice depends on therate point P to be attained.We now state our result for the unassisted QIC. The trivial inner bound treats the QIC as twoindependent unassisted point to point channels from Alice to Charlie and Bob to Damru. Ratesplitting and successive cancellation can be similarly used to obtain non-trivial rate regions forthe unassisted QIC where one party, say Alice, sacrifices her rate in order to boost Bob’s rate withrespect to the trivial inner bound. The situation is summarised in Figure 5. Though the discussionabove only involved unassisted QMAC and QIC, our actual results also hold for the QMAC andQIC with limited entanglement assistance. However they seem to be inferior to the known resultswhen entanglement assistance is unlimited [Sen18b].4lice BobAlice n = ( I ǫ /8002 ( A > C ) , I ǫ /8002 ( B > A C ) , I ǫ /8002 ( A > A BC )) = P ′ Figure 4: The ‘corner’ point P ′ can be obtained by successive cancellation following the orderAlice → Bob → Alice with splitting of Alice followed by one use of the unassisted QMAC. Point P ′ projects down to point P in Figure 3. Only the ‘dominant face’ of the rate region is shown.Successive cancellation can only obtain the corner points of the dominant face and all ‘sub-points’by ‘resource wasting’. It cannot obtain ‘middle’ points of the ‘dominant’ face. O ( log ǫ ) additivefactors have been ignored in the figure. We will use the following conventions throughout the rest of the paper :1. Suppose that | ω ( U ) i XA ′ B ′ be a generic intermediary state (defined in Section 2.6), where X is a placeholder for other systems involve din the protocol. Suppose we are given a channel N A ′ B ′ → C and its corresponding Stinespring dilation U A ′ B ′ → CE N . Then, we denote the state U N | ω ( U ) i XA ′ B ′ by the symbol | ω ( U ) i XCE . Although the two states are denoted using thesame greek letter, we differentiate them by the systems on which they are defined. Thesesystems will always be explicitly mentioned whenever we make use of this convention.2. We will use the same rule for control states, For example, suppose | σ i A ” A ′ B ” B ′ is a controlstate for some channel coding protocol. Suppose we are given the channel N A ′ B ′ → C Thenwe use the following convention σ A ” B ” C : = N · σ A ” A ′ B ” B ′ We will use this convention while specifying entropic quantities. It will be clear from contextwhich state we refer to. For example, consider the expressions H ǫ min ( A ” ) σ and I ǫ min ( A ” | C ) σ .It is clear from the arguments of the entropic expressions that in the first case σ = σ A ” A ′ B ” B ′ and in the second case σ = σ A ” B ′ C .3. We will, on several occasions use the operator op X → YA ′ B ′ ( | ω ( U ) i XYA ′ B ′ ) . To lessen the bur-den on notation, whenever we use this operator, we will not mention the systems on whichthe argument of the op operator is defined. It will however always mention the domain andrange of the op operator in these cases to avoid any confusion.5ob Alice ( I ǫ /8002 ( A > C ) , I ǫ /8002 ( B > D ))( I ǫ /8002 ( A > C ) , I ǫ /8002 ( B > A D ))( I ǫ /8002 ( A > B C ) , I ǫ /8002 ( B > D )) n = Figure 5: One-shot achievable rate region (for single channel use only) for the unassisted QIC.The trivial region is shown dotted. Alice can sacrifice her rate in order to boost Bob’s rate withrespect to the trivial region, as shown by the solid rectangle. The dashed rectangle can be similarlyobtained by Bob sacrificing his rate in order to boost Alice’s. O ( log ǫ ) additive factors have beenignored in the figure. For a pair of subnormalised density matrices ρ and σ in the same Hilbert space their purified dis-tance is denoted by P ( ρ , σ ) : = p − F ( ρ , σ ) where F ( ρ , σ ) : = (cid:13)(cid:13) √ ρ √ σ (cid:13)(cid:13) + p ( − Tr [ ρ ]) · ( − Tr [ σ ]) is the generalised fidelity and k·k is the Schatten 1-norm. We use σ ≈ ǫ ρ as a shorthand for P ( σ , ρ ) ≤ ǫ .The Shannon (aka von Neumann) entropy for a normalised quantum state ρ A is defined by H ( A ) ρ : = − Tr [ ρ log ρ ] . For a bipartite quantum state ρ AB , the ǫ -smooth sandwiched Rényi-2 coherentinformation is defined as I ǫ ( A > B ) ρ : = H ǫ ( A | E ) ρ where | ρ i ABE is a purification of ρ AB . Above,the ǫ -smooth sandwiched Rényi-2 conditional entropy is defined as H ǫ ( A | E ) ρ : = − ( ρ ′ ) AE ≈ ǫ ρ AE min σ E (cid:13)(cid:13)(cid:13) ( A ⊗ ( σ E ) − ) · ( ρ ′ ) AE (cid:13)(cid:13)(cid:13) ,where σ E ranges over non-singular normalised states over E , k·k is the Schatten 2-norm akaFrobenius norm and M · N : = MNM † for operators M , N in the same Hilbert space. The ǫ -smoothsandwiched Rényi-2 coherent information is now defined by I ǫ ( A > B ) ρ : = H ǫ ( A | E ) ρ , where | ρ i ABE is a purification of ρ AB .The ǫ -smooth conditional min-entropy is given by H ǫ min ( A | E ) ρ : = − log min ( ρ ′ ) AE ≈ ǫ ρ AE min σ E : ( ρ ′ ) AE ≤ A ⊗ σ E Tr [ σ E ] ,where σ E ranges over positive semidefinite operators on E . Then the ǫ -smooth coherent min-information aka the negative of the ǫ -smooth conditional max-entropy is given by I ǫ min ( A > B ) ρ : = − H ǫ max ( A | B ) ρ : = H ǫ min ( A | E ) ρ ,where again | ρ i ABE is a purification of ρ AB . The unconditional smooth entropies are now definedfrom the conditional ones by taking the conditioning system to be one dimensional.6s shown in [Dup10], the smooth sandwiched Rényi-2 conditional entropy upper bounds thesmooth conditional min-entropy. The smooth conditional min-entropy is further lower boundedby the familiar conditional Shannon entropy in the amortised sense in the asymptotic iid limit[TCR09], a result that is sometimes referred to as the fully quantum asymptotic equipartitionproperty. To summarise, the smooth sandwiched Rényi-2 coherent information upper boundsthe Shannon coherent information in the amortised sense in the asymptotic iid limit.We will now state some properties on the smooth conditional min entropy that we will usethroughout the rest of the paper. Fact 2.1 (Chaining for Smooth min-entropy [VDTR13, DBWR14])
Let ǫ > and ǫ ′ , ǫ ” ≥ and let ρ ABC be a quantum state. ThenH ǫ + ǫ ′ + ǫ ”min ( AB | C ) ρ ≥ H ǫ ′ min ( A | BC ) ρ + H ǫ ”min ( B | C ) ρ − log 2 ǫ Fact 2.2 (Unitary Invariance of Smooth min-entropy)
Given ǫ ≥ , a quantum state ρ AB and isome-tries U : H A → H C and V : H B → H D , define the state σ CD : = ( U ⊗ V ) ρ AB ( U † ⊗ V † ) . ThenH ǫ min ( A | B ) ρ = H ǫ min ( C | D ) σ Fact 2.3 (Continuity of Smooth min-entropy)
Given two quantum states ρ AB and σ AB such that P ( ρ AB , σ AB ) ≤ δ and ǫ > , then | H ǫ min ( A | B ) ρ − H ǫ min ( A | B ) σ | ≤ c · δ ′ where c is an absolute constant and depends on the dimensions of system A and B and δ ′ = √ δ + ǫδ The proofs of both Theorem 2.2 and Theorem 2.3 can be found in [TCR10].
Fact 2.4 (Quantum Asymptotic Equipartition Property [TCR09])
Given a bipartite quantum state ρ AB on the system H A ⊗ H B , ǫ > , an integer n ∈ N and the iid extension of the state ρ nAB it holdsthat lim ǫ → lim n → ∞ n H ǫ min ( A n | B n ) ρ n = H ( A | B ) ρ One of the main technical tools we use in this paper, which is a workhorse in most of our proofs, isthe notion of mapping a vector into an operator. This operation is denoted simply by ’op’ and wecompile some of its properties in this section for completeness. The interested reader is referredto [Dup10] for further details.
Definition 2.5 (’The op operator’)
Given the systems A and B, fix the standard bases | a i i A and | b j i B .Then we define op A → B : A ⊗ B → L ( A , B ) as op A → B ( | a i i | b j i ) : = | b j i h a i | ∀ i , jNotice that this definition is basis dependant and hence whenever we use this operator a choice of bases isimplied, although not always explicitly mentioned. act 2.6 Let | ψ i AB and | ϕ i AC be vectors on the systems AB and AC respectively. Then op A → C ( | ϕ i AC ) | ψ i AB = op A → B ( | ψ i AB ) | ϕ i AC Fact 2.7
Given a vector | ψ i AB , let | Φ i AA ′ be an EPR state, where A ∼ = A ′ . Then, q | A | op A → B ( | ψ i AB ) | Ψ i AA ′ = | ψ i A ′ B Fact 2.8
For all vectors | ψ i AB and any M A → C , op C → B ( M | ψ i ) = op A → B ( | ψ i ) M T Fact 2.9
For all | ψ i AB , Tr B [ ψ AB ] = op B → A ( | ψ i ) op B → A ( | ψ i ) † Fact 2.10 Decoupling Theorem
Given ǫ > a density matrix ρ AE and any completely positive operator T A → R , define ω A ′ R : = (cid:0) T ⊗ I A ′ (cid:1) Φ AA ′ . Then Z U ( A ) (cid:13)(cid:13)(cid:13) T (cid:0) U · ρ (cid:1) − ω E ⊗ ρ R (cid:13)(cid:13)(cid:13) ≤ − H ǫ ( A ′ | R ) ω − H ǫ ( A | E ) ρ + ǫ where the integration is over the Haar measure on the set of all unitaries on the system A, denoted by U ( A ) . The single sender decoupling theorem implies the following channel coding theorem.
Fact 2.11 [Dup10, Theorem 3.14] Let | ψ i ABR be a pure state, N A ′ → C be any CPTP superoperator withStinespring dilation U A ′ → CE N , N and complementary channel ¯ N A ′ → E , let ω A ” CE : = U N · σ A ” A ′ , where σ A ” A ′ is any pure state and A ” ∼ = A ′ , and let ǫ > . Then, there exists an encoding partial isometryV A → A ′ and a decoding superoperator D CB → AB such that: k ¯ N ( V · ψ AR ) − ω E ⊗ ψ R k ≤ p δ + δ and k ( D ◦ N ◦ E ) ψ ABR − ψ ABR k ≤ q ( p δ + δ ) where δ : = × H ǫ max ( A ) ψ − H ǫ ( A ” ) ω + ǫ , δ : = · − H ǫ ( A ” | E ) ω − H ǫ ( A | R ) ψ + ǫ Fact 2.12
Given states ρ ABC , σ A , η C , σ AB , ω BC such that (cid:13)(cid:13)(cid:13) ρ ABC − σ A ⊗ ω BC (cid:13)(cid:13)(cid:13) ≤ ǫ (cid:13)(cid:13)(cid:13) ρ ABC − σ AB ⊗ η C (cid:13)(cid:13)(cid:13) ≤ ǫ it holds that (cid:13)(cid:13)(cid:13) ρ ABC − σ A ⊗ σ B ⊗ η C (cid:13)(cid:13)(cid:13) ≤ ǫ + ǫ Fact 2.13
For any two density matrices ρ and σ and any real c ∈ R , the following holds true: k ρ − σ k ≤ k c ρ − σ k .6 Almost CPTP Maps In this section we will precisely define the two main technical tools used in the paper: intermediarystates and almost CPTP maps. We describe the objects in this section in the context of a QMAC N A ′ B ′ → C . We assume that we ar given the control state | σ i A ” A ′ B ” B ′ = | Ω i A ” A ′ | ∆ i B ” B ′ and the taskis for the 2 senders Alice and Bob to transmit the A and B systems of the states | ψ i AR and | ϕ i BR via the channel after some suitable encoding. At the very outset we use the isometric embeddings W A → A ”1 and W B → B ”2 to map the systems A and B to A ” and B ”. To ease notation, we use thesymbols | ψ i A ” R and | ϕ i B ” R to denote the states W | ψ i and W | ϕ i . First, suppose U is a Haarrandom unitary and the U is a random unitary sampled from the distribution µ . Definition 2.14 (Intermediary State)
We define the intermediary state with respect to the unitary U as | ω ( U ) i A ” A ′ B ′ R : = √ B ” (cid:16) op B ” → A ” A ′ B ′ (cid:0) σ (cid:1) U B ”2 | ϕ i B ” R (cid:17) We will need one more definition, that of the almost
CPTP map.
Definition 2.15 (Almost CPTP)
We define the linear map T B ” → R as an almost CPTP if T has the fol-lowing properties:1. T is CP.2. Tr [ T ( π B ” )] ∈ [ − δ , 1 + δ ] for some small δ ≥ .3. R T ( U B ”2 · ξ ) d µ = Tr [ ξ ] T ( π B ” ) Lemma 2.16
When the measure µ is set to be the Haar measure on the unitary group on B ” , there existsan almost CPTP T B ” → R such that, with constant probability,H ǫ min ( A ” | R ) ω ( U ) ≥ H ǫ min ( A ” | B ” ) σ − O ( ) Proof: [Proof of Theorem 2.16] Let T B ” → R ( ξ ) : = | B ” | (cid:0) op B ” → R ( ϕ ) · ξ (cid:1) Firstly, it is clear that T is CP. Next, we see thatTr [ T ( π B ” )] = Tr [ op B ” → R ( ϕ ) op B ” → R ( ϕ ) † ]= Tr [ Tr B ” ( ϕ )]= Z T ( U · ξ ) d µ = Tr [ ξ ] T ( π B ” ) T is indeed an almost CPTP. Again, using the properties of the op operator wesee that T (cid:0) ( U B ”2 ) T · σ (cid:1) = | B ” | (cid:16) op B ” → R ( ϕ ) · (cid:0) ( U B ”2 ) T · σ A ” B ” A ′ B ′ (cid:1)(cid:17) = | B ” | (cid:16) op B ” → R ( U B ”2 ϕ ) · σ A ” B ” A ′ B ′ (cid:17) = | B ” | (cid:16) op B ” → A ” A ′ B ′ ( σ ) · (cid:0) U B ”2 · ϕ B ” R (cid:1)(cid:17) = ω ( U ) Now, suppose that ˜ σ is the optimiser in the definition of H ǫ min ( A ” | B ” ) σ and that k ˜ σ − σ k ≤ ǫ .Suppose also that λ B ” be a positive semidefinite matrix such that Tr [ λ B ” ] = − H ǫ min ( A ” | B ” ) σ and˜ σ A ” B ” ≤ I A ” ⊗ λ B ” Then, using the fact that T is a CP map, we see that T (cid:16)(cid:0) U B ”2 (cid:1) T · ˜ σ A ” B ” (cid:17) ≤ I A ” ⊗ T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” (cid:17) R = ⇒ ω ( U ) A ” R ≤ I A ” ⊗ T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” (cid:17) R First notice that, by properties 2 and 3 of almost CPTP maps, Z T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” (cid:17) dU = Z T (cid:16) U B ”2 · λ B ” (cid:17) dU = Tr [ λ B ” ] T ( π B ” ) Taking trace on both sidesTr [ Z T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” (cid:17) dU ] = − H ǫ min ( A ” | B ” ) σ where the last equality stems from the fact that for T , property 2 holds with δ =
0. Next, from thefact that ˜ σ − σ is Hermitian, we can write ˜ σ − σ = ∆ + − ∆ − where ∆ ± are positive semidefinitematrices with disjoint support. This implies that k ˜ σ − σ k = Tr [ ∆ + ] + Tr [ ∆ − ] ≤ ǫ then Z (cid:13)(cid:13)(cid:13) T (cid:16)(cid:0) U B ”2 (cid:1) T · ˜ σ (cid:17) − T (cid:16)(cid:0) U B ”2 (cid:1) T · σ (cid:17)(cid:13)(cid:13)(cid:13) dU = Z (cid:13)(cid:13)(cid:13) T (cid:16) U T · (cid:0) ∆ + − ∆ − (cid:1)(cid:17)(cid:13)(cid:13)(cid:13) dU ≤ Z Tr h T (cid:16) U T · (cid:0) ∆ + (cid:17)i dU + Z Tr h T (cid:16) U T · (cid:0) ∆ − ) (cid:17)i dU = (cid:0) Tr [ ∆ + ] + Tr [ ∆ − ] (cid:1) Tr [ T ( π B ” )] ≤ ǫ Z P (cid:16) T (cid:16)(cid:0) U B ”2 (cid:1) T · ˜ σ (cid:17) , T (cid:16)(cid:0) U B ”2 (cid:1) T · σ (cid:17)(cid:17) dU ≤ √ ǫ Define the random variables X : = Tr [ T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” (cid:17) ] X : = P (cid:16) T (cid:16)(cid:0) U B ”2 (cid:1) T · ˜ σ (cid:17) , T (cid:16)(cid:0) U B ”2 (cid:1) T · σ (cid:17)(cid:17) Then, by a union bound and Markov’s inequality,Pr [ X ≥ · − H ǫ min ( A ” | B ” ) σ ∪ X ≥ √ ǫ ] ≤ B ” of probability at least suchthat, for choices of U in this set, H ǫ min ( A ” | R ) ω ( U ) ≥ H ǫ min ( A ” | B ” ) σ − log 3 (cid:3) As a demonstration of how we will use Theorem 2.16, consider the followin lemma.
Lemma 2.17
Given the intermediary state | ω ( U ) i A ” A ′ B ′ R and the channel N A ′ B ′ → C with Stinespringdilation U A ′ B ′ → CE , and setting the measure µ to be the Haar measure over the unitary group correspondingto the system B ” , the following holds true with constant probability over the choices of U and U (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R ( ω ( U )) U · ψ A ” R (cid:17) R R E − ψ R ⊗ ω ( U ) R E (cid:13)(cid:13)(cid:13)(cid:13) ≤ H ǫ k ( A ” | B ” E ) σ − H ǫ min ( A ” | R ) ψ + log k + k ǫ where k is a some positive integer. Proof: [Proof of Theorem 2.17] We will first apply the smooth single sender decoupling theorem tothe quantity on the left in the theorem statement: Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R ( ω ( U )) U · ψ A ” R (cid:17) R R E − ψ R ⊗ ω ( U ) R E (cid:13)(cid:13)(cid:13)(cid:13) dU ≤ − H ǫ min ( A ” | R E ) ω ( U ) − H ǫ min ( A ” | R ) ψ + ǫ We will now invoke the arguments of Theorem 2.16, with some tweaks. We work with the samealmost CPTP map T B ” → R . Carrying forward the notation of Theorem 2.16 in the natural way,notice that Z T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” E (cid:17) dU = Z T (cid:16) U B ”2 · λ B ” E (cid:17) dU = T ( π B ” ) ⊗ λ E [ Z T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” E (cid:17) dU ] = Tr [ λ E ]= Tr [ λ B ” E ]= − H ǫ min ( A ” | B ” E ) σ As in Theorem 2.16, define the random variables1. X : = Tr [ T (cid:16)(cid:0) U B ”2 (cid:1) T · λ B ” E (cid:17) ] X : = P (cid:16) T (cid:16)(cid:0) U B ”2 (cid:1) T · ˜ σ (cid:17) , T (cid:16)(cid:0) U B ”2 (cid:1) T · σ (cid:17)(cid:17) X : = (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R ( ω ( U )) U · ψ A ” R (cid:17) R R E − ψ R ⊗ ω ( U ) R E (cid:13)(cid:13)(cid:13)(cid:13) Let k ∈ N be some positive integer ≥
4. Then, repeating the arguments in Theorem 2.16, weconclude that there exists a set of probability at least 1 − k over choices of U and U such that thefollowing holds: (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R ( ω ( U )) U · ψ A ” R (cid:17) R R E − ψ R ⊗ ω ( U ) R E (cid:13)(cid:13)(cid:13)(cid:13) ≤ H ǫ k ( A ” | B ” E ) σ − H ǫ min ( A ” | R ) ψ + log k + k ǫ (cid:3) A special case which will be of importance to us is when the measure µ is a product of 2 Haarmeasures. Specifically, suppose we are given the control state | σ i A ” A ” B ” A ′ B ′ G . Here the register G is present to ensure that σ is pure. We will see later that G will not make an appearance in ourprotocol. Given (Stinespring dilation of) the channel U A ′ B ′ → CE N , let | ψ i R A ” | η i R A ” and | ϕ i R B ” bethe transmission states. We define the intermediary state | ω ( U , U ) i A ” A ′ B ′ R R G : = p A ” B ” (cid:16) op A ” B ” → A ” A ′ B ′ G ( σ ) (cid:0) U A ” ⊗ U A ” (cid:1) | ψ i A ” R | ϕ i A ” R (cid:17) where U and U are independent Haar random unitaries. Lemma 2.18
Given the intermediary state | ω ( U , U ) i A ” A ′ B ′ R R G and the channel U A ′ B ′ → CE , and set-ting the measure µ to be the product of Haar measures over the systems A ” and B ” , the following holdstrue with constant probability over the choices of U , U and U (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R R G ( ω ( U , U )) U · η R A ” (cid:17) R R R GE − η R ⊗ ω ( U , U ) R R GE (cid:13)(cid:13)(cid:13)(cid:13) ≤ k · − H ǫ min ( A ” | R ) η − H ǫ k ( A ” | A ” B ” GE ) σ + k ǫ where U is a Haar random unitary and k is a some positive integer. roof: [Proof of Theorem 2.18] As before, the single sender decoupling theorem tells us that (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) | A ” | Tr C U N op A ” → A ′ B ′ R R G ( ω ( U , U )) U · η R A ” (cid:17) R R R GE − η R ⊗ ω ( U , U ) R R GE (cid:13)(cid:13)(cid:13)(cid:13) ≤ − H ǫ min ( A ” | R ) η − H ǫ min ( A ” | R R GE ) ω ( U U ) + ǫ Define the map T A ” B ” → R R ( ξ ) : = | A ” B ” | (cid:0) op A ” B ” → R R ( | ψ i | ϕ i ) · ξ (cid:1) First, recall the following properties of Haar integration1. R U A ⊗ U B · ρ AB dU dU = Tr [ ρ AB ] π AB R U A ⊗ U B ⊗ I C · ρ ABC dU dU = π AB ⊗ ρ C It is now easy to verify that T is indeed an almost CPTP. The first two properties can be shown tobe true using reasoning similar to that used in Theorem 2.16. Finally, using property 1 of doubleHaar integration above, one can immediately see that Z T ( U ⊗ U · ξ A ” B ” ) dU dU = Tr [ ξ ] T ( π A ” B ” ) Next suppose ˜ σ A ” A ” B ” CEG be a state such that H ǫ min ( A ” | A ” B ” GE ) σ = H min ( A ” | A ” B ” GE ) ˜ σ where k ˜ σ − U N · σ k ≤ ǫ . Let λ A ” B ” GE be a positive semidefinite matrix such thatTr [ λ ] = − H ǫ min ( A ” | A ” B ” GE ) σ and ˜ σ A ” A ” B ” GE ≤ I A ” ⊗ λ A ” B ” GE Then, it holds thatTr [ Z T (cid:16) ( U A ” ⊗ U B ”2 ) T · λ A ” B ” GE dU dU ] = Tr [ T ( π A ” B ” ) ⊗ λ GE ]= − H ǫ min ( A ” | A ” B ” GE ) σ Similarly, one can show that Z P (cid:16) T (cid:16)(cid:0) U A ” ⊗ U B ”2 (cid:1) T · ˜ σ (cid:17) , T (cid:16)(cid:0)(cid:0) U A ” ⊗ U B ”2 (cid:1) T U N · σ (cid:17)(cid:17) dU dU ≤ √ ǫ Finally, noting that T (cid:16)(cid:0) U A ” ⊗ U B ”2 (cid:1) T · σ (cid:17) = ω ( U , U ) and using the arguments in Theorem 2.17we conclude the proof. (cid:3) Rate Splitting for Point to Point Channels
In this section we briefly review the idea of rate splitting, as detailed in [GRUW01]. Considerthe classical point to point channel ( A , P B | A , B ) between Alice and Bob and let P A be the inputdistribution that maximises I ( A : B ) . The idea is to split Alice into two independent senders,Alice and Alice and then have Bob decode their messages via a successive cancellation strategy.To do this, one shows the existence of a family of triples ( P θ U , P θ V , f ) where θ is a parameter in [
0, 1 ] , P θ U and P θ V are independent distributions on A and f is a deterministic function, such that f ( U θ , V θ ) ∼ P A . Furthermore, appealing to the properties of the mutual information one canshow that I ( A : B ) = I ( U θ V θ : B ) = I ( U θ : B ) + I ( V θ : BU θ ) From the above discussion it is clear that a simple encoding-decoding strategy is as follows :Alice uses a code of rate I ( U θ : B ) regarding Alice as noise and Alice uses a code of rate I ( V θ : BU θ ) regarding Alice as side information at the receiver. Bob decodes via successivecancellation. Finally, one can show that ( I ( U θ : B ) , I ( V θ : BU θ )) is a continuous function in θ ∈ [
0, 1 ] and traces out the straight line joining the points ( I ( A : B )) and ( I ( A : B ) , 0 ) due tothe chain rule of mutual information with equality. With this construction in hand, one can designan encoding and decoding scheme for the classical MAC without appealing to time sharing orjointly typical simultaneous decoding. Firstly, split Alice into the two users Alice and Alice by the construction above. Then Charlie does a successive cancellation decoding for this 3 senderMAC: first decode Alice ’s message treating the other senders as noise, then decode Bob’s messageregarding Alice ’s message as side information and Alice as noise, and finally decode Alice ’smessage regarding Bob’s and Alice ’s message as side information. Thus three point to pointchannel decodings are done by Charlie in order to decode the sent messages at the rate triple ( I ( U θ : C ) , I ( C : BU θ ) , I ( V θ : CBU θ )) . Notice that, all points in the dominant face of the achievableregion in Figure 1 can be achieved in this way due to continuity as θ varies from 0 to 1. Alsoobserve that the split of Alice depends on θ .The triple ( f , P θ U , P θ V ) with respect to the distribution P A is called a split of P A . That such atriple exists is given by the following fact: Fact 3.1
Given a distribution P A on the set A , there exist two distributions P θ U and P θ V (both defined on A ), parameter θ ∈ [
0, 1 ] and a function f : A × A → A such that the following hold true :1. f ( U , V ) ∼ P A
2. For fixed values of x and u, P θ f ( U , V ) | U ( a | u ) is a continuous function of θ .3. For θ = , P θ f ( U , V ) | U ( a | u ) = P A ( a ) .4. For θ = , and all u ∈ A , P θ f ( U , V ) | U ( a | u ) puts all its mass on one element. Proof:
We demonstrate an explicit construction, as shown in [GRUW01]. Assume that A is anordered set. We describe the distribution in terms of distribution functions, for which we use the14etter F along with the appropriate subscript. Then, define, for all i ∈ A : F θ U ( i ) : = θ F A + − θ F θ V ( i ) : = F A ( i ) F θ U ( i ) f ( u , v ) : = max { u , v } ∀ u , v ∈ A It is easy to check the triple defined above satisfies all the properties in Theorem 3.1. The interestedreader may look at [GRUW01] for details. (cid:3)
We first describe the task of entanglement transmission over a quantum channel. Unlike its classi-cal counterpart, there are multiple seemingly disparate information transmission tasks that onecan consider to define the quantum capacity of a quantum channel. However, it was shownin [KW03], that these definitions are equivalent. In this paper, we will consider the task of en-tanglement transmission.Alice has access to the A system of the EPR state | Φ i RA , where | R | = | A | . Alice possess adecoding CPTP map E A → A ′ such that, after acting this map on the system A , Alice sends the A ′ part of the pure state (cid:0) E · Φ (cid:1) RA ′ through the channel N A ′ → B . Finally, Bob applies a decoding map D B → ˆ A on his system B , such that the final state he shares with R is close to the EPR state | Φ i RA .Formally, a ( Q , ǫ ) entanglement generation code consists of an encoder E A → A ′ and a decodingCPTP D B → ˆ A such that | Φ i RA = √ Q Q ∑ i = | i i R | i i A F (cid:0) | Φ i RA , I R ⊗ ( D ◦ N ◦ E ) ( Φ RA )) ≥ − ǫ A rate Q is achievable for the channel N is there exists a ( Q , ǫ ) code for N .In fact we consider the following more general situation: Alice and Bob share the pure state ψ ABR where Alice holds A , Bob holds B and R is the reference. Alice wants to send her share toBob through a single use of the channel N A ′ → C . To do this, one needs to show the existence of anencoder decoder pair E A → A ′ and D CB → ˆ AB such that (cid:13)(cid:13) D ◦ N ◦ E (cid:0) ψ (cid:1) − ψ (cid:13)(cid:13) ≤ ǫ Note that in this situation Bob can potentially utilise the correlations present in the system B for decoding, to boost the rate of transmission.To derive an entanglement transmission protocol out of this, we set ψ ABR to be the state Φ RM ⊗ Φ ˜ AB , where the registers M ˜ A play the role of A in Theorem 2.11. Notice that Φ ˜ AB is essentiallyshared entanglement between Alice and Bob, and changing the rank of ˜ A essentially allows us tocontrol the amount of shared entanglement available for use in the protocol. Thus, we call thissituation entanglement transmission with rate limited entanglement assistance or partial entanglementassisted transmission protocol . [Dup10]. 15e say that a rate pair ( Q , E ) ( where Q denotes the rank of the EPR state to be transmitted,in this case | R | = | M | and E denotes the amount of pre-shared entanglement available for useduring the protocol) is ǫ -achievable if there exists an encoder decoder pair such that (cid:13)(cid:13)(cid:13) D ◦ N ◦ E ( Φ RM ⊗ Φ ˜ AB ) − Φ RM (cid:13)(cid:13)(cid:13) ≤ ǫ The rate Q is achievable for transmission with entanglement assistance if there exists E ≥ ( Q , E ) is ǫ achievable, and it is achievable for unassisted transmission if ( Q , 0 ) is ǫ achievable.The idea behind proving that such an encoder decoder pair exist is to consider the comple-mentary channel ¯ N A ′ → E . Suppose one can show that¯ N A ′ → E ◦ E A → A ′ ( ψ AR ) ≈ ¯ N ◦ E ( ψ A ) ⊗ ψ R Intuitively this would imply that the original channel preserves correlations perfectly, and thenthe existence of a decoder can be inferred by appealing to Uhlmann’s theorem. The main workthen goes towards proving the existence of an encoder which has this property.To do this, consider the state | ω A ” CE i : = U A ′ N | σ i A ” A ′ . The state ω will act as our control state.We will consider a randomized encoder, augmented by a Haar random unitary. This encoder takesthe follwoing form : E A → A ′ RAND _ ENC ( ψ AR ) : = op A ” → A ′ ( | σ i A ” A ′ ) U A ” RAND W A → A ” · ψ AR where W is some isometric embedding from the space A to A ” and U RAND is a Haar randomunitary.It can be shown that averaging over all unitaries, this randomized encoder satisfies the follow-ing properties :1. There exists a fixed isometric encoder which does almost as well as the randomized encoder.The error in the performance of the fixed isometric encoding and the randomized encoder ismeasure by the quantity H ǫ max ( A ) ψ − H ǫ ( A ” ) Ω i.e. how close the input distribution ψ A is tothe target distribution ω A ” .2. The action of the complementary channel and the randomized encoder nearly decouples thesystems E and R . The accuracy of the decoupling is measured by the quantity − H ǫ ( A ” | E ) Ω − H ǫ ( A | R ) ψ .Collating the two properties together one can infer the existence of an isometric encoder, and by asubsequent application of Uhlmann’s theorem, a decoder such that Alice can send the register A through the channel with high fidelity, given that the entropic constraints are satisfied. One shouldnote that both desired properties above, along with the entropic conditions are given by directapplications of the single user decoupling theorem. The above discussion can be encapsulated bythe following fact: Fact 3.2
For any quantum channel N A ′ → C and pure state σ A ” A ′ where A ” and A ′ are isomorphic, the ratepair ( Q , E ) is ǫ achievable for quantum transmission with rate limited assistance through N ifQ + E < H ǫ ( A ” ) Ω Q − E < H ǫ ( A ” | E ) Ω here we have suppressed some additive error terms in terms of log ǫ in the bounds above. A more preciseversion of this theorem is stated as Theorem 2.11. Notice that, to get the unassisted entanglement trans-mission rate, one need only set E to . This implies that the unassisted rate is ≈ H ǫ ( A ” | E ) U N · Ω aftersuppressing the additive log terms. To describe rate splitting in the entanglement transmission scenario, we will define an abstract splitting scheme with some properties of interest:
Definition 3.3
Splitting Scheme
Given a control state | σ i A ” A ′ B ” B ′ = | Ω i A ” A ′ | ∆ i B ” B ′ and systems A ” and A ” such that A ” ∼ = A ” ∼ = A ” , we define a splitting scheme to be a family of isometric embeddingsU θ A ” → A ”0 A ”1 parametrized by a variable θ ∈ [
0, 1 ] , such that:1. For all θ , θ ′ ∈ [
0, 1 ] and ǫ > there exists δ > such that whenever | θ − θ ′ | ≤ δ , k U θ · σ − U θ ′ · σ k ≤ ǫ
2. Given any channel N A ′ B ′ → C and its Stinespring dilation U A ′ B ′ → CE N ,I ( A ” B ” > C ) σ = I ( A ” B ” > C ) σ = I ( A ” B ” > C ) σ where σ : = U · σ and σ : = U · σ . Notice that we have define the splitting scheme with respect to the more general control state | σ i A ” A ′ B ” B ′ = | Ω i A ” A ′ | ∆ i B ” B ′ . This will be useful when we describe the splitting protocol for moregeneral multiterminal channels. For the purposed of this section, where we only demonstratesplitting for the point to point channel, one may simply ignore the state | ∆ i . Also note that theinvariants in the splitting scheme are specified in terms of the coherent information. A moregeneral definition would be to specify the invariants in terms of the smooth min entropy. Wework with this more general definition.We will first give an overview of the strategy for the unassisted case. We will then state andprove the main technical lemma of this section, Theorem 3.4. The ideas in used in proving thislemma will generalise easily to the setting of the multiterminal channels such as the QMAC andthe QIC.We will emulate the strategy outlined in Section 3.1 for a bipartite pure quantum state | Ω i A ′′ A ′ : = ∑ a ′′ ∈A ′′ p P A ′′ ( a ) | a i A ′′ | ζ a i A ′ , where | a ′′ i runs over the computational basis of A ′′ and P A ′′ is aprobability distribution on the basis set A ′′ . We split the system A ′′ into two registers A ′′ and A ′′ corresponding to the two senders Alice and Alice . Let the split ( P θ U , P θ V , f ) be as in the previoussubsection. Define the isometric embedding U SPLIT ( θ ) A ′′ → A ′′ A ′′ as follows: q P A ( a ) | a i A ′′ U SPLIT ( θ ) ∑ ( u , v ) ∈ f − ( a ′′ ) q P θ U ( u ) P θ V ( v ) | u i A ′′ | v i A ′′ and | Ω ( θ ) i A ′′ A ′′ A ′ : = U SPLIT ( θ ) | Ω i A ′′ A ′ .We now pass the system A ′ through a point to point channel N A ′ → B and obtain the quantumstate | Ω ( θ ) i A ′′ A ′′ B . By unitary invariance, I ǫ ( A ′′ > B ) Ω = I ǫ ( A ′′ A ′′ > B ) Ω ( θ ) . From the worksof [Dup10, SDTR13] applied to transmission of quantum information over one-shot unassisted17oint to point quantum channels, we first realise that Bob can decode Alice ’s quantum messageat the rate of I ǫ /8002 ( A ′′ > B ) Ω ( θ ) − O ( log ǫ − ) with error at most O ( √ ǫ ) . Then, employing thesuccessive cancellation methods of Yard et al. [YDH05] Bob can decode Alice ’s quantum messageat the rate of I ǫ /8002 ( A ′′ > BA ′′ ) Ω ( θ ) − O ( log ǫ − ) with error at most O ( √ ǫ ) .Doing both the steps above requires us to overcome a few technical challenges which we doby defining a notion of almost CPTP maps (see Section 2.6) that should be useful in other situa-tions too, and combining it with another proof technique by Dupuis for the unassisted quantumbroadcast channel [Dup10].We have thus operationally shown the chain rule inequality I ǫ /800min ( A ′′ > BA ′′ ) Ω ( θ ) + I ǫ /800min ( A ′′ > B ) Ω ( θ ) ≤ I ǫ min ( A ′′ A ′′ > B ) Ω ( θ ) . (suppressing the log factors). One can prove this fact indepen-dently using the chain rule for smooth min entropies Theorem 2.1. We now see that as θ variesfrom 0 to 1, the point ( R ( θ ) , R ( θ )) = ( I ǫ /800min ( A ′′ > B ) Ω ( θ ) , I ǫ /800min ( A ′′ > BA ′′ ) Ω ( θ ) ) traces out acontinuous curve that lies on or below the line segment joining the point ( I ǫ min ( A > B ) , 0 ) to thepoint ( I ǫ min ( A > B )) and meets it at its endpoints. The continuity of the curve follows from thecontinuity of the states and the functionals involved. Continuity of the functionals is implied byTheorem 2.3 whereas continuity of the states is implied by Theorem A.1. These arguments alsoimply that U SPLIT ( θ ) is indeed a valid splitting scheme.This rate splitting and successive cancellation idea can now be easily generalised to the unas-sisted QMAC.We will now consider the general case, when Bob has side information available at the decoder.Suppose Alice and Alice wish to transmit the systems A and A of the states | η i A B R and | ψ i A B R to Bob. We wish to prove there exists an encoder E A A → A ′ and a decoder C BC C → A A such that F (cid:0) C ◦ N ◦ E ( η ⊗ ψ ) , η ⊗ ψ (cid:1) ≥ − ǫ Given that such an encoder decoder pair exist, set | η i A B R ← | Φ i M R | Φ i ˜ A B and | ψ i A B R ←| Φ i A M | Φ i ˜ A B . Let Q A = log | M | , Q A = log | M | . and E A = log | B | , E A = log | B | . The rates Q A , Q A are the entanglement transmission rates of Alice and Alice and E A and E A quantifythe amount of pre-shared entanglement available to them before the protocol begins.We will consider the simpler case, when Alice does not share any entanglement with Bob, butAlice does, i.e. the register C is trivial. We quantify the rates in the following lemmas: Proposition 3.4
Given the control state | Ω i A ” A ′ , the point to point quantum channel N A ′ → B and thesplitting scheme U A ” θ , suppose Alice has to send states | η i A R ⊗ | ψ i A B R to Bob, where A and A arethe message registers and B models the side information Bob has about the A . R and R are referencesystems. We define | Ω ′ ( θ ) i A ” A ” A ′ : = U A ” θ | Ω i A ” A ′ and | Ω ′ ( θ ) i A ” A ” BE : = U A ′ → BE N | Ω ′ ( θ ) i A ” A ” A ′ Then there exist an encoder E A A → A ′ and a decoder C BB → A A such that (cid:13)(cid:13)(cid:13) C ◦ N ◦ E ( η A R ⊗ ψ A B R ) − η A R ⊗ ψ A B R (cid:13)(cid:13)(cid:13) ≤ δ here δ = p δ dec ( ) + p δ dec ( ) + q δ enc(0) + δ enc ( ) and δ dec ( ) = · − H ǫ ( A | R ) η − H ǫ ( A ” | A ” E ) Ω ′ ( θ ) + ǫδ dec ( ) = · − H ǫ ( A | R ) ψ − H ǫ ( A ” | E ) Ω ′ ( θ ) + ǫδ enc ( ) = · H ǫ max ( A ) η − H ǫ ( A ” | A ” ) Ω ′ ( θ ) + ǫδ enc ( ) = H ǫ max ( A ) ψ − H ǫ min ( A ” ) Ω ′ ( θ ) + ǫ where ǫ = ǫ for some positive ǫ . An easy corollary of Theorem 3.4 is the following:
Corollary 3.5
Given the control state | Ω i A ” A ′ and the point to point channel N A ′ → B , and the splittingscheme U A ” → A ” A ” θ , Alice can transmit EPR states to Bob at the rate Q A + Q A given E A bits of pre-shared entanglement, with error at most √ ǫ wheneverQ A < H ǫ max ( A ” | A ” ) Ω ′ ( θ ) + log 4 ǫ Q A < I ǫ min ( A ” > B ) U N · Ω ′ ( θ ) + log 4 ǫ Q A + E A < H ǫ max ( A ” ) Ω ′ ( θ ) + log 4 ǫ Q A − E A < I ǫ min ( A ” > A ” B ) U N · Ω ′ ( θ ) + log 4 ǫ where ǫ = ǫ and | Ω ′ i A ” A ” A ′ = U A ” θ | Ω i A ” A ′ . Proof: [Proof of Theorem 3.5] We initialise the states | η i A R and | ψ i A R B as follows | η i A R ← | Φ i A R | ψ i A R B ← | Φ i R M | Φ i ˜ A B Here, the registers M ˜ A play the roles of A , and the notation Φ is used generically to mean anEPR state. Let | R | = Q A | R | = Q A and | B | = E A Note that Alice’s actual rate Q A is Q A + Q A . The following relatons are easy to check: H max ( A ) η = Q A = ⇒ H ǫ max ( A ) η ≤ Q A H max ( M ˜ A ) ψ = Q A + E A = ⇒ H ǫ max ( M ˜ A ) ψ ≤ Q A + E A H min ( A | R ) η = Q A = ⇒ H ǫ min ( A ) η ≥ Q A H min ( M ˜ A | R ) ψ = E A − Q A = ⇒ H ǫ min ( M ˜ A | R ) ψ ≥ E A − Q A δ dec ( ) < ǫδ dec ( ) < ǫδ enc ( ) < ǫδ enc ( ) < ǫ Pluggin in these numbers in the bounds shown in Theorem 3.4 completes the proof. (cid:3)
Proof: [Proof of Theorem 3.4]Consider the randomised encoder E A A → A ′ RAND ≡ q | A ” || A ” | op A ” A ” → A ′ ( Ω ′ ( θ )) (cid:0) U A ” W A → A ” ⊗ U A ” W A → A ” (cid:1) where W and W are isometric embeddings and | Ω ( θ ) i A ” A ” A ′ = U A ” → A ” A ” θ | Ω i A ” A ′ .We will define two intermediary states, which will allow us to randomise over one input at atime, keeping the other fixed. To that end, define the states | ω Alice ( U ) i A ” A ′ B R : = q | A ” | (cid:0) op A ” → A ” A ′ ( Ω ′ ( θ )) U W | ψ i A B R (cid:1) | ω Alice ( U ) i A ” A ′ R : = q | A ” | (cid:0) op A ” → A ” A ′ ( Ω ′ ( θ )) U W | η i A R (cid:1) Also define | ω Alice ( U ) i A ” B R CE : = U A ′ → CE N | ω Alice ( U ) i A ” A ′ B R and | ω Alice1 ( U ) i A ” R CE : = U A ′ → CE N | ω Alice1 ( U ) i A ” A ′ R The Decoupling Step :
As promised, these states will help us randomise over one input,while holding the other fixed, as can be seen via two applications of the single sender decouplingtheorem: E U h (cid:13)(cid:13)(cid:13) | A ” | Tr C U N op A ” → A ′ B R ( ω Alice ( U )) U W · η R A − η R ⊗ ω B R E Alice ( U ) (cid:13)(cid:13)(cid:13) i ≤ − H ǫ ( A | R ) η − H ǫ min ( A ” | B R E ) ω Alice ( U ) + ǫ (dec_Alice ) E U h (cid:13)(cid:13)(cid:13) | A ” | Tr CR U N op A ” → A ′ R ( ω Alice1 ( U )) U W · ψ R A − ψ R ⊗ ω E Alice1 ( U ) (cid:13)(cid:13)(cid:13) i ≤ − H ǫ ( A | R ) ψ − H ǫ min ( A ” | E ) ω Alice ( U ) + ǫ (dec_Alice )Notice that in the second expression, we traces out both C and R . This is in anticipation of thefact that Alice will use the correlations in R to decode Alice ’s input.20he inequalities above will go towards proving the the existence of a good decoder. To showthat a good encoder exists, we use the single sender decoupling theorem twice more : E U h (cid:13)(cid:13)(cid:13) | A ” | Tr A ′ op A ” → A ′ B R ( ω Alice ( U )) U W · η R A − η R ⊗ ω B R Alice ( U ) (cid:13)(cid:13)(cid:13) i ≤ H ǫ max ( A ) η − H ǫ min ( A ” | B R ) ω Alice ( U ) + ǫ (enc_Alice ) E U h (cid:13)(cid:13)(cid:13) | A ” | Tr A ′ op A ” → A ” A ′ ( Ω ′ ( θ )) U W · ψ R B A − ψ R B (cid:13)(cid:13)(cid:13) i ≤ H ǫ max ( A ) ψ − H ǫ min ( A ” ) Ω ′ ( θ ) + ǫ (enc_Alice )First, note that by Theorem 2.6, the following holds: E A A → A ′ RAND | η i AR | ψ i BB R = q | A ” | op A ” → A ′ B R ( ω Alice ( U )) U W | η i R A = q | A ” | op A ′ R ( ω Alice ( U )) U W | ψ i A R B : = | GLOBAL i A A ′ R R Also, we observe that the first term in the LHS of Eq. (enc_Alice ) is, by definition, equal to ω B R Alice ( U ) .Next, we need to derandomise the four inequalities, i.e. we will show that there exist fixed U and U such that all four inequalities hold. The issue is that the RHS of each inequality, asidefrom Eq. (enc_Alice ), has min entropy terms which are functions of ω Alice ( U ) and ω Alice ( U ) ,when we want bounds in terms of the control state Ω ′ ( θ ) . For example, consider the expression H ǫ min ( A ” | B R E ) ω Alice ( U ) . To get the correct upper bounds, we would have to prove an inequalityof the following kind: H ǫ min ( A ” | B R E ) ω Alice ( U ) ≥ H ǫ min ( A ” | A ” E ) Ω ′ ( θ ) for every U . Unfortunately this is not true. The way we get around this issue is that, we showthat on average over the choice of U , the following inequality holds true H ǫ min ( A ” | B R E ) ω Alice ( U ) ≥ H O ( ǫ ) min ( A ” | A ” E ) Ω ′ ( θ ) − O ( ) (average_dpi)The way we prove the above statement is by via the almost CPTP maps defined in Section 2.6,specifically, Theorem 2.18. The state of Theorem 2.18, adapted to the current setting, implies that,there exist with constant probability, two fixed unitaries U and U such that Eq. (dec_Alice ) andEq. (average_dpi) hold simultaneously.To show that a similar results holds for Eq. (dec_Alice ) and Eq. (enc_Alice ), note that theproof of Theorem 2.18 requires a union bound over 3 bad events, each with probability at most k ,where k is some adjustable integer parameter ≥
4. In this case, we will need to apply Theorem 2.18to Eq. (dec_Alice ), Eq. (dec_Alice ) and Eq. (enc_Alice ), each of which will contribute 3 badevents. Additionally, we will also need to show that, for our fixed U and U , Eq. (enc_Alice )21olds. Thus, we have to take a union bound over 10 bad events. Thus, a good choice for theadjustable parameter k is 20.Set ǫ : = ǫ . The above discussion then implies that there exist fixed unitaries U and U suchthat Eq. (dec_Alice ), Eq. (dec_Alice ), Eq. (enc_Alice ) and Eq. (enc_Alice ) can be written as: k Tr C U N · GLOBAL − η R ⊗ ω B R E Alice ( U ) (cid:13)(cid:13)(cid:13) ≤ · − H ǫ ( A | R ) η − H ǫ ( A ” | A ” E ) UN Ω ′ ( θ ) + ǫ : = δ dec ( ) k Tr CR U N · Tr B GLOBAL − ψ R ⊗ ω E Alice1 ( U ) (cid:13)(cid:13)(cid:13) ≤ · − H ǫ ( A | R ) ψ − H ǫ ( A ” | E ) Ω ′ ( θ ) + ǫ : = δ dec ( ) k Tr A ′ GLOBAL − η R ⊗ ψ R B (cid:13)(cid:13)(cid:13) ≤ · H ǫ max ( A ) η − H ǫ ( A ” | A ” ) Ω ′ ( θ ) + ǫ + H ǫ max ( A ) ψ − H ǫ min ( A ” ) Ω ′ ( θ ) + ǫ : = δ enc ( ) + δ enc ( ) : = δ enc where we have use the triangle inequality on the derandomised versions of Eq. (dec_Alice )and Eq. (dec_Alice ) to get the last inequality. One technical issue is that since E RAND is not a TPmap, we cannot apply Uhlmann’s theorem directly, since | GLOBAL i may not be a unit vector. Tocircumvent this we use Theorem 2.13. Suppose c : = [ GLOBAL ] . Then, by Theorem 2.13, k c Tr C U N · GLOBAL − η R ⊗ ω B R E Alice ( U ) (cid:13)(cid:13)(cid:13) ≤ δ dec ( ) We can make similar statements for the other two inequalities.
The Successive Cancellation Step:
Roughly, the idea is as follows : Bob first decodes forAlice ’s input, and recovers a state close to | η i A R . That such an isometric decoder exists is in-ferred from the normalised version of decoding condition for Alice and Uhlmann’s theorem.After this step, Bob possesses the registers A and F , where F contains correlations with the restof the parties involved in the protocol.Bob keeps the A system aside, and brings in the locally prepared state | η i ◦ A ◦ R . He then invertsthe decoding procedure on the systems ◦ A F and maps to a state which is close to GLOBAL , withthe important difference that now the register ◦ R is in Bob’s possession.22inally, from from the decoding condition for Alice and Uhlmann’s theorem, we infer theexistence of a decoder which uses not only the registers BB for decoding but also utilises thecorrelations that ◦ R has with R . This allows Alice to transmit at a higher rate than the simplepoint to point case.To make things precise, we first define the states | Ω ′ ( θ ) i ◦ A ” A ” A ′ and | η i ◦ A ◦ R , which are allcopies of the corresponding states defined earlier, but with the registers A , A ” and R replacedby ◦ A , ◦ A ” and ◦ R , which are all local to Bob. Following this convention, we can also define thestate | ω Alice ( U ) i ◦ A ” A ′ B R analogously. Next, define | TARGET i ◦ R R B A ′ : = q | A ” | op ◦ A ” → A ′ B R ( ω Alice ( U )) (cid:0) U W | η i ◦ A ◦ R (cid:1) where the unitary U acts on the register ◦ A ” , and W maps ◦ A → ◦ A ” .By the decoding condition for Alice and Uhlmann’s theorem, we see that there exists an iso-metric decoder V B → A F Alice such that (cid:13)(cid:13)(cid:13) c · V Alice U N · GLOBAL − η A R ζ F R B E (cid:13)(cid:13)(cid:13) ≤ q δ dec ( ) where η and ζ are pure and ζ is a purification of ω Alice ( U ) B R E .Since GLOBAL and TARGET are the same states aside from renaming the R register, the aboveresult also implies that there exists an isometry V B → ◦ A F Sim_Alice such that (cid:13)(cid:13)(cid:13)(cid:13) c · V Sim_Alice U N · TARGET − η ◦ A ◦ R ζ F R B E (cid:13)(cid:13)(cid:13)(cid:13) ≤ q δ dec ( ) Simple algebra and a triangle inequality then implies that c · (cid:13)(cid:13)(cid:13)(cid:13) V − η ◦ A ◦ R V Alice U N · GLOBAL − η A R U N TARGET (cid:13)(cid:13)(cid:13)(cid:13) ≤ q δ dec ( ) Note that, with the registers A ” and R replaced by ◦ A ” and ◦ R , and making the appropriatechanges in the definition of the state ω Alice ( U ) , the decoding condition for Alice along withUhlmann’s theorem implies that, there exists a decoding isometry V BB ◦ R → A B F Alice1 such that (cid:13)(cid:13)(cid:13) c · V Alice1 U N · TARGET − ψ A R B ζ F E (cid:13)(cid:13)(cid:13) ≤ q δ dec ( ) Defining V BB ◦ R ◦ A → A A B F Bob : = V Alice1 ◦ V − ◦ V Alice and collating the above arguments, wesee that (cid:13)(cid:13)(cid:13)(cid:13) c · V Bob U N · η ◦ A ◦ R GLOBAL − η A R ψ A R B ζ F E (cid:13)(cid:13)(cid:13)(cid:13) ≤ q δ dec ( ) + q δ dec ( ) Isometric Encoding:
We finally show that there exists an isometric encoder. To do this, wesimply consider the decoupling condition k c · Tr A ′ GLOBAL − η R ⊗ ψ R B (cid:13)(cid:13)(cid:13) ≤ δ enc V A A → A ′ enc such that k c · GLOBAL − V enc η A R ⊗ ψ A R B (cid:13)(cid:13)(cid:13) ≤ p δ enc Tracing out F E and another triangle inequality then shows that the proposition is true. (cid:3) We now state our one-shot inner bounds for the QMAC with limited entanglement assistance. Butfirst we need a technical proposition akin to Theorem 3.4, from which our inner bounds followeasily.
Proposition 4.1
Consider the quantum multiple access channel N A ′ B ′ → C . Consider a pure ‘control state’ | σ i A ” B ” A ′ B ′ : = | Ω i A ” A ′ | ∆ i B ” B ′ . Let | ψ i A C R ⊗ | η i A C R and | φ i BDS be the states that are to be sentto Charlie through the channel by Alice and Bob respectively, where C , C , D model the side informationabout the respective messages A , A , B that Charlie possesses and R , R , S are reference systems that areuntouched by channel and coding operators. Let I denote the identity superoperator. For θ ∈ [
0, 1 ] , let U A ” θ be a splitting scheme. We define | σ ( θ ) i A ′′ A ′′ A ′ B ′′ B ′ : = U θ | Ω i A ” A ′ | ∆ i B ” B ′ and σ ( θ ) A ′′ A ′′ B ′′ C : = ( N A ′ B ′ → C ⊗ I A ′′ A ′′ B ′′ )( σ ( θ ) A ′′ A ′′ A ′ B ′′ B ′ ) . Then there exist encoding maps A A A → A ′ , B B → B ′ and a decoding map C CC C D → A C A C BD such that (cid:13)(cid:13) ( C ⊗ I R R S )(( N ⊗ I C R C R DS )(( A ⊗ B ⊗ I C R C R DS )(( η ⊗ ψ ) ⊗ φ ))) − η ⊗ ψ ⊗ φ k ≤ δ , where δ : = δ enc + δ dec and δ enc : = ( q min { δ enc (
0, 1 ) , δ enc (
1, 0 ) } + q δ enc ( )) δ dec : = ( q δ dec ( ) + q δ dec ( ) + q δ dec ( )) δ enc (
0, 1 ) = · ( H ǫ max ( A ) η − H ǫ ( A ′′ | A ′′ ) σ ( θ ) ) + · ( H ǫ max ( A ) ψ − H ǫ ( A ′′ ) σ ( θ ) ) + ǫ , δ enc (
1, 0 ) = · ( H ǫ max ( A ) η − H ǫ ( A ′′ | A ′′ ) σ ( θ ) ) + · ( H ǫ max ( A ) ψ − H ǫ ( A ′′ ) σ ( θ ) ) + ǫ , δ enc ( ) = · ( H ǫ max ( B ) φ − H ǫ ( B ′′ ) σ ( θ ) ) + ǫ , δ dec ( ) : = · − ( H ǫ ( A | R ) η + I ǫ ( A ′′ > C ) σ ( θ ) ) + ǫ , δ dec ( ) : = · − ( H ǫ ( A | R ) ψ + I ǫ ( A ′′ > CA ′′ B ′′ ) σ ( θ ) ) + ǫ , δ dec ( ) : = · − ( H ǫ ( B | S ) φ + I ǫ ( B ′′ > CA ′′ ) σ ( θ ) ) + ǫ .24heorem 4.1 immediately implies the following theorem, by the arguments presented in Theorem 3.5. Theorem 4.2
Consider the setting of Proposition 4.1. Let Q A , E A , Q B , E B be the number of message qubitsand number of available ebits of Alice and Bob respectively. Let θ , ǫ ∈ [
0, 1 ] and ǫ : = ǫ . Then there existencoding and decoding maps such that any message cum ebit rate -tuple satisfying either the following setof constraints or the set obtained by interchanging A ′′ with A ′′ , Q A ( ) with Q A ( ) and E A ( ) with E A ( ) in the right hand sides of the first two inequalities, is achievable with error at most √ ǫ :Q A = Q A ( ) + Q A ( ) , E A = E A ( ) + E A ( ) , Q A ( ) + E A ( ) < H ǫ min ( A ′′ | A ′′ ) σ ( θ ) + ǫ , Q A ( ) + E A ( ) < H ǫ ( A ′′ ) σ ( θ ) + ǫ , Q A ( ) − E A ( ) < I ǫ min ( A ′′ > C ) σ ( θ ) + ǫ , Q A ( ) − E A ( ) < I ǫ min ( A ′′ > C A ′′ B ′′ ) σ ( θ ) + ǫ , Q B + E B < H ǫ ( B ′′ ) σ ( θ ) + ǫ , Q B − E B < I ǫ min ( B ′′ > C A ′′ ) σ ( θ ) + ǫ . Proof: [Proof of Theorem 4.1]
Preprocessing :
Let W A → A ” , W A → A ” and W A be isometricembeddings such that | ˜ η i : = W A → A ” | η i| ˜ ψ i : = W A → A ” | ψ i| ˜ ϕ i : = W A → A ” | ϕ i Global State : We define | A ” A ” B ” | op A ” A ” B ” → A ′ B ′ ( σ ( θ )) · (cid:0) U ⊗ U ⊗ U (cid:1) · (cid:0) ˜ η R C A ” ⊗ ˜ ψ R A ” C ⊗ ˜ ϕ SB ” D (cid:1) : = GLOBAL Intermediary States :
Encoding | ω enc ( U ) i A ” C R A ′ : = q | A ” | (cid:16) op A ” → A ” A ′ ( Ω ′ ( θ )) U A ” | ˜ ψ i A ” C R (cid:17) Decoding | ω Alice0 ( U , U ) i A ” A ′ B ′ C R DS : = q | A ” B ” | (cid:16) op A ” B ” → A ” A ′ B ′ ( σ ( θ ))( U A ” ⊗ U B ”2 ) ˜ | ψ i A ” C R | ˜ ϕ i B ” DS (cid:1) | ω Bob ( U , U ) i B ” A ′ B ′ C R C R : = q | A ” A ” | (cid:16) op A ” A ” → B ” A ′ B ′ ( σ ( θ ))( U A ” ⊗ U A ” ) | ˜ η i A ” C R ˜ | ψ i A ” C R (cid:1) | ω Alice ( U , U ) i A ” A ′ B ′ C R SD : = q | A ” B ” | (cid:16) op A ” B ” → A ” A ′ B ′ ( σ ( θ )) (cid:0) U A ” ⊗ U B ”2 (cid:1) | ˜ η i A ” C R | ˜ ϕ i B ” DS (cid:17) he Decoupling Step : We will apply Theorem 2.18 simultaneously to all 4 intermediate states.We will have to choose an appropriate value of the constant k to make the derandomisation work.To that end, notice that each application of Theorem 2.18 has a union bound over 3 bad events.Generalising over all four intermediate states implies that we will have to take a union bound over12 bad events. In addition, we have to include two additional bad events in the union bound thatoriginate from applying the single sender decoupling theorem to show that good encoders exist.Thus, setting k =
20 we see that there exist fixed U , U and U such that the following conditionsare satisfied simultaneously : (cid:13)(cid:13)(cid:13) | A ” | Tr C U A ′ B ′ → CE N (cid:16) op A ” → A ′ B ′ C R DS ( ω Alice0 ( U , U )) · (cid:0) U · ˜ η A ” R (cid:1)(cid:17) − η R ⊗ ω Alice0 ( U , U ) C R DSE (cid:13)(cid:13)(cid:13) ≤ · − H ǫ ( A | R ) η − H ǫ ( A ” | A ” B ” E ) σ ( θ ) + ǫ : = δ dec ( ) (dec_Alice ) (cid:13)(cid:13)(cid:13) | B ” | Tr CR C U A ′ B ′ → CE N (cid:16) op B ” → A ′ B ′ C R C R ( ω Bob ( U , U )) · (cid:0) U · ˜ ϕ A ” S (cid:1)(cid:17) − ϕ S ⊗ ω Bob ( U , U ) C ER (cid:13)(cid:13)(cid:13) ≤ · − H ǫ ( B | S ) ϕ − H ǫ ( B ” | A ” E ) σ ( θ ) + ǫ : = δ dec ( ) (dec_Bob) (cid:13)(cid:13)(cid:13) | A ” | Tr CR C SD U A ′ B ′ → CE N (cid:16) op A ” → A ′ B ′ C R SD ( ω Alice ( U , U )) · (cid:0) U · ˜ ψ R A ” (cid:1)(cid:17) − ψ R ⊗ ω Alice ( U , U ) E (cid:13)(cid:13)(cid:13) ≤ · − H ǫ ( A ” | E ) σ ( θ ) − H ǫ ( A | R ) ψ + ǫ : = δ dec ( ) (dec_Alice ) (cid:13)(cid:13)(cid:13) | A ” | Tr A ′ (cid:16) op A ” → C R A ′ ( ω enc ( U )) · (cid:0) U · ˜ η R C A ” (cid:1)(cid:17) − η C R ⊗ ω enc ( U ) C R (cid:13)(cid:13)(cid:13) ≤ · H ǫ max ( A ) η − H ǫ ( A ” | A ” ) σ ( θ ) + ǫ : = δ enc ( ) (enc_Alice ) (cid:13)(cid:13)(cid:13) | A ” | Tr A ” A ′ (cid:16) op A ” → A ” A ′ ( Ω ′ ( θ )) (cid:0) U · ˜ ψ A ” C R (cid:1)(cid:17) − ψ C R (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ω enc ( U ) C R − ψ C R (cid:13)(cid:13)(cid:13) ≤ · − H ǫ min ( A ” ) σ ( θ ) + H ǫ max ( A ) ψ + ǫ : = δ enc ( ) (enc_Alice )26 (cid:13)(cid:13) | B ” | Tr B ′ (cid:16) op B ” → B ′ ( ∆ ) · (cid:0) U · ˜ ϕ B ” SD (cid:1)(cid:17) − ϕ SD (cid:13)(cid:13)(cid:13) ≤ · H ǫ max ( B ) ϕ − H ǫ min ( B ” ) σ + ǫ : = δ enc ( ) (enc_Bob) Decoding : The Successive Cancellation Step
This is very similar to the arguments in the proof of Theorem 3.4, and involves two applicationsof the successive cancellation procedure shown there. The technical details are identical to thosedescribed in that proof, so in the interest of brevity, we provide a high level description only. First,Charlie decodes the input of Alice to recover the state | η i A C R . He keeps this state side, andintroduces a local copy | η i ◦ A ◦ C ◦ R . Charlie then inverts the decoding using these local systems, toget a state close to GLOBAL , but with the registers ◦ C ◦ R locally available to him.Charlie then uses the registers C ◦ R C D to decode Bob’s input to recover the state | ϕ i BSD . Char-lie then keeps Bob’s decoded state aside, and introduces the locally prepared state | ϕ i ◦ B ◦ S ◦ D . He theninverts the decoding again, using these local systems, to get a state close to GLOBAL , with the sys-tems ◦ C ◦ R ◦ S ◦ D locally available to him.In last step of the decoding procedure, Charlie uses the systems C ◦ R ◦ C ◦ S ◦ DC to decode Alice ’sinput. This whole procedure outputs a state close to | η i R C A ⊗ | ψ i R C A ⊗ | ϕ i BSD with an errorof at most δ dec : = ( q δ dec ( ) + q δ dec ( ) + q δ dec ( )) Encoding
Firstly, notice that, via Theorem 2.6 in Eq. (enc_Alice ), | A ” | op A ” → C R A ′ ( ω enc ( U )) · (cid:0) U · ˜ η R A ” (cid:1) = | A ” || A ” | op A ” A ” → A ′ ( Ω ′ ) (cid:0) U ⊗ U (cid:1) · (cid:0) ˜ ψ ⊗ ˜ η (cid:1) : = ALICE R R C A ′ Then, applying a triangle inequality to Eq. (enc_Alice ) and Eq. (enc_Alice ), and using Theorem 2.13and Uhlmann’s theorem, one can see that there exists an encoding isometries V A A → A ′ ENC _ ALICE and V B → B ′ ENC _ BOB such that (cid:13)(cid:13)(cid:13) c · ALICE R R C A ′ − V ENC _ ALICE · (cid:0) ψ R C A ⊗ η R A (cid:1)(cid:13)(cid:13)(cid:13) ≤ q δ enc ( ) + δ enc ( ) (cid:13)(cid:13)(cid:13) c · | B ” | op B ” → B ′ ( ∆ ) · (cid:0) U · ˜ ϕ B ” DS (cid:1) − V ENC _ BOB · ϕ BDS (cid:13)(cid:13)(cid:13) ≤ q δ enc ( ) where c and c are appropriate normalisation factors. ALICE R R C A ′ ⊗ | B ” | op B ” → B ′ ( ∆ ) · (cid:0) U · ˜ ϕ B ” DS (cid:1) = GLOBAL (cid:13)(cid:13)(cid:13) c c · GLOBAL − (cid:0) V ENC _ ALICE ⊗ V ENC _ BOB (cid:1) · (cid:0) η R A ψ R C A ϕ SDB (cid:1)(cid:13)(cid:13)(cid:13) ≤ ( + q δ enc ( )) × ( q δ enc ( ) + δ enc ( )) + q δ enc ( ) : = δ enc We can now concude the proof by the usual arguments. (cid:3)
In this section we prove inner bounds for partially entanglement assisted entanglement trans-mission through the Quantum Interference Channel (QIC) N A ′ B ′ → CD . We wish for Alice to sendEPR pairs to Charlie and for Bob to send EPR pairs to Damru. Note that, for a fixed control state | σ i A ” A ′ B ” B ′ : = | Ω i A ” A ′ | ∆ i B ” B ′ , one can consider this situation as two point to point channels, onefrom Alice to Charlie and one from Bob to Damru. In that case, the achievable region becomes arectangle of all non negative rate pairs less than (cid:0) I ǫ min ( A ” > C ) σ , I ǫ min ( B ” > D )) σ (cid:1) (suppressingthe additive log terms).In order to show that a larger region is achievable, we use splitting schemes and successivecancellation. Essentially, we split Alice into two senders, Alice and Alice , and we require Alice ’sinput to be decoded by Damru instead of Charlie. This allows Damru to treat Alice ’s input asside information while decoding Bob’s input, which allows us to boost Bob’s rate. Alice’s rate toCharlie however, takes a hit because of this. Using a splitting scheme to do this allows us to adjustthe amount of resources that Alice dedicates towards boosting Bob’s rate, with the extreme cases θ ∈ {
0, 1 } corresponding to situations when either Alice does not help bob at all (the case of thetwo point to point channels) ot when Alice dedicates all her resources to help Bob while her ownrates drops to 0.The precise statements can be found in Theorem 5.1 and Theorem 5.2. Proposition 5.1
Consider the quantum interference channel N A ′ B ′ → CD . Consider a pure ‘control state’ | σ i A ” B ” A ′ B ′ : = | Ω i A ” A ′ | ∆ i B ” B ′ . Let | ψ i A C R and | η i A R be the states that are to be sent by Alice toCharlie and Damru respectively and let | φ i BD S be the state to be sent from Bob to Damru, where C ,D model the side information about the respective messages A , B that Charlie and Damru possess andR , R , S are reference systems that are untouched by channel and coding operators. Let I denote theidentity superoperator. For θ ∈ [
0, 1 ] , let U A ” θ be a splitting scheme. We define | σ ( θ ) i A ′′ A ′′ A ′ B ′′ B ′ : = U θ | Ω i A ” A ′ | ∆ i B ” B ′ and σ ( θ ) A ′′ A ′′ B ′′ CD : = ( N A ′ B ′ → CD ⊗ I A ′′ A ′′ B ′′ )( σ ( θ ) A ′′ A ′′ A ′ B ′′ B ′ ) . Then there exist encoding maps A A A → A ′ and B B → B ′ and decoding maps C CC → A C and D DD → A BD such that (cid:13)(cid:13)(cid:0) C ⊗ D (cid:1) ◦ N ◦ (cid:0)
A ⊗ B (cid:1) · (cid:0) ψ ⊗ η ⊗ ϕ (cid:1) − ψ ⊗ η ⊗ ϕ (cid:13)(cid:13) ≤ δ ere, δ = δ enc + δ dec where, δ enc = ( + q δ enc ( )) × ( q δ enc ( ) + δ enc ( )) + q δ enc ( ) δ dec = q δ dec ( ) + q δ dec ( ) + q δ dec ( ) and δ enc ( ) = · H ǫ max ( A ) η − H ǫ ( A ” | A ” ) σ ( θ ) + ǫδ enc ( ) = · H ǫ max ( A ) ψ − H ǫ min ( A ” ) σ ( θ ) + ǫδ enc ( ) = · H ǫ max ( B ) ϕ − H ǫ min ( B ” ) σ ( θ ) + ǫδ dec ( ) = · − H ǫ ( A | R ) η − I ǫ ( A ” > D ) σ ( θ ) + ǫδ dec ( ) = · − H ǫ ( B | R ) ϕ − I ǫ ( B ” > DA ” ) σ ( θ ) + ǫδ dec ( ) = · − H ǫ ( A | R ) ψ − I ǫ ( A ” > C ) σ ( θ ) + ǫ where ǫ = ǫ . We are now ready to state our main one shot coding theorem. In this case, we denote by Q the number of qubits available to Alice for sending to Damru, to use as side information to boostBob’s rate. The quantities of interest however are ( Q A , E A , Q B , E B ) which denote, in order, thenumber of message qubits and ebits available to Alice, and the analogous quantities for Bob . Theorem 5.2
Consider the setting of Theorem 5.1. Let Q A , E A , Q B , E B be the number of message qubitsand number of available ebits of Alice and Bob respectively. Additionally, let Q denote the number messagequbits available to Alice for transmission to Damru. Let θ , ǫ ∈ [
0, 1 ] and ǫ : = ǫ . Then there existencoding and decoding maps such that any message cum ebit rate -tuple satisfying the following inequal-ities, is achievable with error at most √ ǫ achievable for partial entanglement assisted entanglementtransmission Q < I ǫ min ( A ” > D ) σ ( θ ) + log ǫ Q < H ǫ min ( A ” | A ” ) σ ( θ ) + log ǫ Q A + E A < H ǫ min ( A ” ) σ ( θ ) + log ǫ Q A − E A < I ǫ min ( A ” > C ) σ ( θ ) + log ǫ Q B + E B < H ǫ min ( B ” ) σ ( θ ) + log ǫ Q B − E B < I ǫ min ( B ” > A ” D ) σ ( θ ) + log ǫ The proofs of Theorem 5.1 and Theorem 5.2 are essentially the same as their counterparts forthe QMAC, with some technical changes due to the presence of two decoders. These changes canbe dealt with easily via standard techniques, and hence we omit these proofs.
Remark 5.3
In Section 6, when we state the asymptotic iid extension of this theorem, we will fix a value ofQ and not mention it explicitly to highlight the region of interest. Extension to the Asymptotic IID Regime
The bounds proved in Theorem 4.2 and Theorem 5.2 can be easily extended to the case whenmany uses of the channel are allowed. We show in this section that one can extend Theorem 4.2 tothe asymptotic iid regime to recover the best known inner bounds for entanglement transmissionover the QMAC in that regime [YDH05]. Similar arguments allow us to extend Theorem 5.2 to thatregime and prove the first non trivial asymptotic iid inner bounds for entanglement transmissionover the QIC.The idea is simple and uses the quantum asymptotic equipartition theorem (QAEP) Theorem 2.4.As an example, consider that we are given the QMAC N A ′ B ′ → C and the control state | σ i as usual,but now we are allowed n ∈ N uses of the channel along with n tensored copies of the controlstate. Suppose that δ > θ ∈ [
0, 1 ] . Define | σ i as usualwith respect to this fixed θ . Then, for some large enough n , using the Theorem 2.4, and repeatingthe proof of Theorem 4.2, we see that the achievable region, for the fixed θ for partial entanglementassisted entanglement transmission is given by n ( Q A + E A ) ≤ nH ( A ” ) σ + nH ( A ” | A ” ) σ − n δ n ( Q A − E A ) ≤ nI ( A ” > C ) U N · σ + nI ( A ” > A ” B ” C ) U N · σ − n δ n ( Q B + E B ) ≤ nH ( B ” ) σ − n δ n ( Q B − E B ) ≤ nI ( B ” > A ” C ) U N · σ − n δ The usual continuity arguments with respect to θ ∈ [
0, 1 ] along with the invariance of theShannon entropy and the coherent information under isometries give the full pentagonal region: Q A + E A ≤ H ( A ” ) σ − δ Q A − E A ≤ I ( A ” > B ” C ) U N · σ − δ Q B + E B ≤ H ( B ” ) σ − δ Q B − E B ≤ I ( B ” > A ” C ) U N · σ − δ Q A − E A + Q B − E B < I ( A ” B ” > C ) U N · σ − δ Note that setting E A and E B to 0, we recover the bounds shown in [YDH05]. We state the abovearguments as a theorem: Theorem 6.1
Given a quantum multiple access channel N A ′ B ′ → C all rate points in the closure of thefollowing region are achievable for partial entanglement assisted entanglement generation : ∞ [ k = k Q ( N ⊗ k ) here Q ( N ⊗ k ) is the set of non negative rate tuples ( Q A , E A , Q B , E B ) in the setQ A + E A < H ( A ” k ) σ k Q A − E A < I ( A ” k > B ” k C k ) U N · σ k Q B + E B < H ( B ” k ) σ k Q B − E B < I ( B ” k > A ” k C k ) U N · σ k Q A − E A + Q B − E B < I ( A ” k B ” k > C k ) U N · σ k where | σ k i A ” k B ” k A ′ k B ′ k : = | Ω i A ” k A ′ k | ∆ i B ” k B ′ k . Before stating the analogous theorem for the QIC, we note that, the set of achievable points forthe QIC is actually larger than the one shown in Theorem 5.2. This is via the simple observationthat, we only allowed Alice to help Bob by taking a hit to her own rate. We would get a different setof achievable points if we allow Bob to help Alice. The union of these two regions is the completerate region.
Theorem 6.2
Given a quantum interference channel N ⊗ k , the control state | σ i A ” A ′ B ” B ′ the following reg-ularised rate region is achievable for partial entanglement assisted entanglement transmission : ∞ [ k = k Q ( N ⊗ k ) For each k ∈ N , Q ( N ⊗ k ) = [ A k θ [ [ B k θ where, for a fixed θ ∈ [
0, 1 ] , A k θ is the set of all non-negative tuples ( Q A , E A , Q B , E B ) such thatQ A + E A < H ( A ” k ) σ k ( θ ) Q A − E A < I ( A ” k > C k ) U N · σ k ( θ ) Q B + E B < H ( B ” k ) σ k ( θ ) Q B − E B < I ( B ” k > A ” k C k ) U N · σ k ( θ ) where | σ k i A ” k B ” k A ′ k B ′ k : = | Ω i A ” k A ′ k | ∆ i B ” k B ′ k and | σ k i : = U A ” k → A ” k A ” k | σ k i . We assume that U θ is a split-ting scheme. Analogously, B k θ is the set of those points which are obtained when the splitting isometry actson the system B ” k . References [Ahl71] R. Ahlswede. Multi-way communication channels. In
Proceedings of 2nd InternationalSymposium on Information Theory (ISIT) , pages 23–52, 1971.[BD10] F. Buscemi and N. Datta. The quantum capacity of channels with arbitrarily correlatednoise.
IEEE Transactions on Information Theory , 56(3):1447–1460, 2010.31CMGE08] Chong, H., Motani, M., Garg, H., and El Gamal, H. On the Han-Kobayashi region forthe interference channel.
IEEE Transactions on Information Theory , 54:3188–3195, 2008.[CNS21] Chakraborty, S., Nema, A., and Sen, P. A multi sender decoupling theorem and simul-taneous decoding for the quantum mac. Available at arXiv:2101.1000, 2021.[DBWR14] Frédéric Dupuis, Mario Berta, Jürg Wullschleger, and Renato Renner. One-shot de-coupling.
Communications in Mathematical Physics , 328(1):251–284, May 2014.[Dev05] I. Devetak. The private classical capacity and quantum capacity of a quantum channel.
IEEE Transactions on Information Theory , 51(1):44–55, 2005.[Dup10] Frédéric Dupuis.
The decoupling approach to quantum information theory . PhD thesis,Université de Montréal, 2010.[FHS +
12] O. Fawzi, P. Hayden, I. Savov, P. Sen, and M. Wilde. Classical communication over aquantum interference channel.
IEEE Transactions on Information Theory , 58:3670–3691,2012.[GRUW01] Alexander Grant, Bixio Rimoldi, Rüdiger Urbanke, and Philip Whiting. Rate-splittingmultiple access for discrete memoryless channels.
Information Theory, IEEE Transac-tions on , 47:873 – 890, 04 2001.[HDW08] M. Hsieh, I. Devetak, and A. Winter. Entanglement-assisted capacity of quantummultiple-access channels.
IEEE Transactions on Information Theory , 54(7):3078–3090,2008.[HHYW07] Patrick Hayden, Michal Horodecki, Jon Yard, and Andreas Winter. A decouplingapproach to the quantum capacity.
Open Systems & Information Dynamics , 15, 03 2007.[HK81] Han, T. and Kobayashi, K. A new achievable rate region for the interference channel.
IEEE Transactions on Information Theory , 27:49–60, 1981.[KW03] Dennis Kretschmann and Reinhard Werner. Tema con variazioni: Quantum channelcapacity.
New Journal of Physics , 6, 11 2003.[Lia72] H. Liao.
Multiple access channels . PhD thesis, University of Hawai, 1972.[Llo97] Seth Lloyd. Capacity of the noisy quantum channel.
Phys. Rev. A , 55:1613–1622, Mar1997.[SDTR13] Oleg Szehr, Frédéric Dupuis, Marco Tomamichel, and Renato Renner. Decouplingwith unitary approximate two-designs.
New Journal of Physics , 15(5):053022, may 2013.[Sen12] P. Sen. Achieving the Han-Kobayashi inner bound for the quantum interference chan-nel. In
IEEE International Symposium on Information Theory (ISIT) , pages 736–740, 2012.Full version at arXiv:1109.0802.[Sen18a] Pranab Sen. A one-shot quantum joint typicality lemma. arXiv e-prints , pagearXiv:1806.07278, June 2018. 32Sen18b] Pranab Sen. Inner bounds via simultaneous decoding in quantum network informa-tion theory. arXiv e-prints , page arXiv:1806.07276, June 2018.[Sho02] Peter Shor. The quantum channel capacity and coherent information.
Lecture Notes,MSRI workshop on quantum computation , 2002.[TCR09] M. Tomamichel, R. Colbeck, and R. Renner. A fully quantum asymptotic equipartitionproperty.
IEEE Transactions on Information Theory , 55(12):5840–5847, 2009.[TCR10] M. Tomamichel, R. Colbeck, and R. Renner. Duality between smooth min- and max-entropies.
IEEE Transactions on Information Theory , 56(9):4674–4681, 2010.[VDTR13] A. Vitanov, F. Dupuis, M. Tomamichel, and R. Renner. Chain rules for smooth min-and max-entropies.
IEEE Transactions on Information Theory , 59(5):2603–2612, 2013.[Win01] Andreas Winter. The capacity of the quantum multiple-access channel.
InformationTheory, IEEE Transactions on , 47:3059 – 3065, 12 2001.[YDH05] J. Yard, I. Devetak, and P. Hayden. Capacity theorems for quantum multiple ac-cess channels. In
Proceedings. International Symposium on Information Theory, 2005. ISIT2005. , pages 884–888, 2005.
A Proofs of Important Lemmas
Lemma A.1
Given θ , θ ′ ∈ [
0, 1 ] such that | θ − θ ′ | ≤ δ , we have thatP (cid:0) Ω ′ ( θ ) A ” A ” BE , Ω ′ ( θ ′ ) A ” A ” BE (cid:1) ≤ O ( √ δ ) Proof: [Proof of Theorem A.1] For the course of the proof we will neglect to mention the registersin the superscript to ease the notation, unless necessary. Since both Ω ′ ( θ ) and Ω ′ ( θ ′ ) are pure, wewill use the identity : P (cid:0) Ω ′ ( θ ) , Ω ′ ( θ ′ ) (cid:1) = q − |h Ω ′ ( θ ) | Ω ′ ( θ ′ ) i| Recall that since | Ω ′ i A ” A ” BE ( θ ) = U A ′ N ◦ U A ” A ” A ′ f (cid:16) ∑ u ∈A q P θ U ( u ) | u i A ” (cid:17) ⊗ (cid:16) ∑ v ∈A q P θ V ( v ) | v i A ” (cid:17) | i A ′ and similarly for | Ω ′ i ( θ ′ ) , h Ω ′ ( θ ) | Ω ′ ( θ ′ ) i = F ( P θ U , P θ ′ U ) F ( P θ V , P θ ′ V ) It is thus sufficient to show that the distributions P θ U and P θ V are close to P θ ′ U and P θ ′ V respectively.Then, recalling the explicit form of P θ U observe that : (cid:13)(cid:13)(cid:13) P θ ′ U − P θ U (cid:13)(cid:13)(cid:13) = | ( − θ + θ P A ( )) − ( − θ ′ + θ ′ P A ( )) | + ∑ i = | θ P A ( i ) − θ ′ P A ( i ) |≤ | θ − θ ′ | + | θ − θ ′ | ∑ i ∈A P A ( i ) ≤ δ i ∈ A P θ V ( i ) = F A ( i ) F θ U ( i ) − F A ( i − ) F θ U ( i − ) It holds that F A ( i ) F θ U ( i − ) − F A ( i − ) F θ U ( i ) = ( P A ( i ) + F A ( i − )) F θ U ( i − ) − F A ( i − )( θ F A ( i ) + − θ )= ( P A ( i ) + F A ( i − )) F θ U ( i − ) − F A ( i − )( F θ U ( i − ) + θ P A ( i ))= P A ( i )( θ F A ( i − ) + − θ ) − θ F A ( i − ) P A ( i )= ( − θ ) P A ( i ) Denote F θ U ( i ) F θ U ( i − ) : = g ( θ ) . Then, | g ( θ ) − g ( θ ′ ) | = | F θ U ( i ) F θ U ( i − ) − F θ ′ U ( i ) F θ ′ U ( i − ) |≤ | F θ U ( i ) F θ U ( i − ) − F θ ′ U ( i ) F θ U ( i − ) | + | F θ ′ U ( i ) F θ U ( i − ) − F θ ′ U ( i ) F θ ′ U ( i − ) |≤ δ Let p ∗ = min i ∈A P A ( i ) . Then, g ( θ ) ≥ ( − θ + θ p ∗ ) ≥ p ∗ | P θ V ( i ) − P θ ′ V ( i ) | = P A ( i ) (cid:12)(cid:12)(cid:12) − θ g ( θ ) − − θ ′ g ( θ ′ ) (cid:12)(cid:12)(cid:12) = P A ( i ) | g ( θ ) g ( θ ′ ) | · | ( − θ ) g ( θ ′ ) − ( − θ ′ ) g ( θ ) | ( a ) ≤ P A ( i ) p ∗ · c · δ where c is some constant and we have used the triangle inequality and the lower bound for g ( θ ) in ( a ) .Then, the above bound implies that: (cid:13)(cid:13)(cid:13) P θ V − P θ ′ V (cid:13)(cid:13)(cid:13) ≤ O ( δ ) Using the property that F ( P , Q ) ≥ − k P − Q k for any two distributions P and Q , we see that |h Ω ′ ( θ ) | Ω ′ ( θ ′ ) i| ≥ − O ( δ ) This concludes the proof. (cid:3)(cid:3)