CCatalytic quantum teleportation
Patryk Lipka-Bartosik
1, 2 and Paul Skrzypczyk H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre,Faculty of Mathematics, Physics and Informatics, University of Gda´nsk, Wita Stwosza 57, 80-308 Gda´nsk, Poland (Dated: February 24, 2021)Quantum catalysis is an intricate feature of quantum entanglement. It demonstrates that in certain situationsthe very presence of entanglement can improve one’s abilities of manipulating other entangled states. At thesame time, however, it is not clear if using entanglement catalytically can provide additional power for any ofthe existing quantum protocols. Here we show, for the first time, that catalysis of entanglement can provide agenuine advantage in the task of quantum teleportation. More specifically, we show that extending the standardteleportation protocol by giving Alice and Bob the ability to use entanglement catalytically, allows them toachieve fidelity of teleportation at least as large as the regularisation of the standard teleportation quantifier, theso-called average fidelity of teleportation. Consequently, we show that this regularised quantifier surpasses thestandard benchmark for a variety of quantum states, therefore demonstrating that there are quantum states whoseability to teleport can be further improved when assisted with entanglement in a catalytic way. This hints thatentanglement catalysis can be a promising new avenue for exploring novel advantages in the quantum domain.
I. INTRODUCTION
The existence of quantum entanglement is one of the moststriking consequences implied by the laws of quantum me-chanics [1–3]. It is manifested when correlations between dif-ferent particles are strong enough so that the action on oneof them affects the other, in a way so subtle that it cannot beexplained by any classical mechanism. Although entangle-ment was initially recognized as a bizarre property separatingquantum from classical physics, it is nowadays viewed as anindispensable resource with an enormous number of modernapplications.Arguably one of the most important applications of quan-tum entanglement is the protocol of quantum teleportation[4]. In its standard form it is a communication task whichinvolves transferring an unknown quantum state to a remoterecipient using classical communication and shared entangle-ment. This is also perhaps the best evidence for the resourcenature of quantum entanglement, as the laws of quantum me-chanics rule out transferring quantum states without its pres-ence. The significance of the protocol can be best evidencedby its wide applicability in various areas of quantum informa-tion [5, 6], computation [7–9] and even the theory of relativity[10, 11]. Quantum teleportation is also a building block formany quantum information processing tasks and has been re-alised in laboratories using variety of different technologies,including photonic qubits [12–17], optical modes [18–20], nu-clear magnetic resonance (NMR) [21], atomic ensembles [22–24], trapped atoms [25–27] or solid-state systems [28–30].It is also a folklore knowledge that quantum entanglementis a resource which must be consumed in order to outperformclassical systems in information processing tasks. What isless known, however, is that in certain cases the very pres-ence of entanglement can provide advantage, without it beingconsumed or degraded. This surprising and yet not clearlyunderstood phenomenon is called quantum catalysis and wasintroduced in [31], further analysed in [32–38] and subse-quently adapted to other physical settings like quantum ther-modynamics [39–49], resource theory of coherence [50, 51], purity [52], asymmetry [53] or to the study of quantum refer-ence frames [54]. In the particular case of entanglement the-ory, quantum catalysis demonstrates that access to a specialentangled state (the catalyst) can sometimes allow two distantparties to manipulate their entanglement in a way that wouldotherwise be impossible. Importantly, the ancillary entangledstate used in this process is not consumed, so that the partiescan repeat their task or use the catalyst for some other task.This makes catalysis a particularly interesting extension of thestandard paradigm of local operations and classical commu-nication (LOCC). However, despite its fundamental signifi-cance for the theory of quantum entanglement, it is still notclear whether the subtle improvement of manipulation abili-ties provided by catalysis can ever lead to any advantages forgenuine quantum task. Finding protocols whose performancecan be improved by utilising catalysis is therefore an impor-tant open problem, both from a theoretical and experimentalpoint of view.In this work we solve this problem in positive by extend-ing the task of quantum teleportation by allowing Alice andBob use an arbitrary amount of entanglement that has to bereturned intact. We then show that this extension of the stan-dard protocol can sometimes significantly improve their opti-mal fidelity of teleportation. In particular, we show that in acatalytic setting the optimal fidelity of teleportation is lower-bounded by a regularisation of the standard fidelity of telepor-tation. This new quantity is then shown to be strictly largerthan the standard fidelity of teleportation for a wide range ofpure states, meaning that catalysis allows for a generic im-provement over the standard teleportation protocol. To thebest of our knowledge, this is the first time when catalysis isused to provide a quantitative advantage in a quantum infor-mation processing task. Our proof is based on a techniquewhich uses a catalyst to increase entanglement fraction of an-other state, a realisation which we believe to be of independentinterest. Since entanglement fraction determines performanceis many relevant information-processing tasks, we believe thatthis could shed more light on catalytic advantages that couldbe observed in other information-processing tasks. a r X i v : . [ qu a n t - ph ] F e b II. FRAMEWORK
In what follows we denote a discrete and finite-dimensionalHilbert space associated with a quantum system S with H S .We also denote the space of all density operators in H S with D ( H S ) . We will be interested in scenarios involving two dis-tant parties (Alice and Bob) who are allowed to use local op-erations and classical communication (LOCC). A quantumchannel E is a completely positive and trace-preserving linearmap acting between spaces of density operators. We say that E ∈
LOCC ( A : B ) if it can be written as a sequence of quan-tum channels applied locally by A and B , intertwined withclassical communication between the two parties. An impor-tant entanglement quantifier which we use extensively here isthe so-called entanglement fraction of a state, defined as thebest overlap with a maximally entangled state [55]. More pre-cisely, f ( ρ ) := max E (cid:104) φ + AB |E ( ρ AB ) | φ + AB (cid:105) s.t. E ∈
LOCC ( A : B ) , (1)where (cid:12)(cid:12) φ + AB (cid:11) = (cid:80) di =1 | i (cid:105) A | i (cid:105) B / √ d denotes a maximally-entangled state shared between A and B . A. Standard quantum teleportation
Before presenting our main results let us briefly recall thetask of quantum teleportation [4]. In its most general form theprotocol involves two spatially separated parties, Alice andBob, who share an arbitrary quantum state ρ AB of local di-mension d . A third party, often called the Referee, providesAlice with a quantum state ϕ R of dimension d R which is un-known to both parties. The goal set before Alice and Bob isto transfer the unknown state from one party to another, us-ing only local operations and classical communications, i.e.quantum channels T ∈
LOCC ( RA : B ) , and shared entan-glement. Under this conditions all possible states which canbe achieved in Bob’s lab can be written as: ρ (cid:48) B = tr RA T ( ϕ R ⊗ ρ AB ) , (2)where tr S denotes a partial trace over subsystem S . The aboveprotocol can be equivalently viewed as a process of establish-ing a quantum channel between Alice and Bob that maps theinput state ϕ R to the output ρ (cid:48) B . The quality of a teleportationprotocol or, equivalently, the fidelity of the resulting telepor-tation channel, can be quantified using the optimal fidelity ofteleportation [56], which for a density operator ρ is defined as: (cid:104) F (cid:105) ρ := max T (cid:90) (cid:104) ϕ | tr RA T ( ϕ R ⊗ ρ AB ) | ϕ (cid:105) d ψ s.t. T ∈
LOCC ( RA : B ) . (3)In what follows we will refer to the above quantity as simply“fidelity of teleportation”. The integral in (3) is computed overa uniform distribution of all possible input states ϕ = | ϕ (cid:105)(cid:104) ϕ | according to a normalised Haar measure (cid:82) d ψ = 1 . It can be easily verified that ≤ (cid:104) F (cid:105) ρ ≤ for all density operators ρ ∈ D ( H A ⊗ H B ) . Furthermore, the case (cid:104) F (cid:105) ρ = 1 corre-sponds to perfect teleportation and is possible if and only if ρ is a maximally-entangled state. In practice, teleportation cannever be perfect and the optimal fidelity of teleportation willgenerally be less than one. If Alice and Bob do not share anentangled state, then the corresponding teleportation protocolis said to be “classical”. In that case the optimal fidelity cannever exceed the threshold value (cid:104) F (cid:105) c := 2 / ( d + 1) . There-fore, if (cid:104) F (cid:105) ρ > (cid:104) F (cid:105) c can be demonstrated, then the associatedstate ρ is necessarily entangled. Computing the raw expres-sion ( ) is generally a difficult task. However, it was shownin Ref. [55] that fidelity of teleportation (3) is related withentanglement fraction (1) via: (cid:104) F (cid:105) ρ = f ( ρ ) d + 1 d + 1 . (4)This important realisation allows to easily compute fidelityof teleportation for many relevant cases. In the next sectionwe focus on this quantity and show that using catalysts in aproper way allows to increase entanglement fraction of thestate, without consuming additional entanglement. III. RESULTSA. Catalytic quantum teleportation
Let us now describe the catalytic extension of the standardquantum teleportation protocol. We assume that Alice andBob, in addition to their shared state ρ AB , have also access toa quantum system CC (cid:48) which is prepared in some state ω CC (cid:48) .This additional system is then distributed to both parties, sothat Alice has access only to C , and Bob only to its C (cid:48) part.Alice is then given an unknown quantum state ϕ R and theparties perform a protocol T ∈
LOCC ( RAC : BC (cid:48) ) whichnow acts on both systems they share and the input system.Moreover, for the protocol to be catalytic we demand that theaction of T does not modify the state of the catalyst. Notably,we do allow the catalyst to become correlated with other sys-tems during this protocol. Later in the Appendix we show thatthese correlations can be made arbitrarily small, at the expanseof using larger catalysts. The final state of Bob’s subsystem atthe end of the catalytic teleportation protocol reads: ρ (cid:48) B = tr RACC (cid:48) [ T ( ϕ R ⊗ ρ AB ⊗ ω CC (cid:48) )] (5)The quality of this protocol can be quantified similarly as inthe case of standard teleportation, i.e. using the fidelity ofteleportation (3). Notice, however, that now we also have free-dom to choose any entangled state to be used as the catalyst.In order to benchmark the quality of the arising teleportationprotocol, we now define a new teleportation quantifier thatcaptures this additional freedom. Therefore, we define the fi-delity of catalytic teleportation (cid:104) F cat (cid:105) ρ as the the solution ofthe following optimisation problem: (cid:104) F cat (cid:105) ρ = max T , ω (cid:90) (cid:104) ψ | tr RACC (cid:48) T ( ϕ R ⊗ ρ AB ⊗ ω CC (cid:48) ) | ψ (cid:105) d ψ s.t. tr RAB T ( ρ RAB ⊗ ω CC (cid:48) ) = ω CC (cid:48) , T ∈
LOCC ( RAC : BC (cid:48) ) ω CC (cid:48) ∈ D ( H C ⊗ H C (cid:48) ) . (6)By definition we again have ≤ (cid:104) F cat (cid:105) ρ ≤ and the reducedstate of the catalyst ω CC (cid:48) is left unchanged, as we discussed inthe Introduction. Before we present our main result let us firstdefine a regularisation of the entanglement fraction from Eq.(1), a quantity whose significance will soon become evident.Therefore, the regularised entanglement fraction f reg ( ρ ) willbe defined as: f reg ( ρ ) := lim n →∞ f n ( ρ ⊗ n ) n , (7)where f n ( ρ ) is the solution to: f n ( ρ ) := max E n (cid:88) i =1 (cid:104) φ + | tr /i E ( ρ AB ) | φ + (cid:105) , s.t. E ∈
LOCC ( A . . . A n : B . . . B n ) , (8)where tr /i ( · ) is the partial trace performed over particles . . . i − , i + 1 . . . n . Notice that by taking a sub-optimalguess E = E ⊗ E = . . . E n with E = E = . . . = E n we caninfer that f reg ( ρ ) ≥ f ( ρ ) for all density operators ρ . With theabove definitions we are now ready to present our main result. Theorem 1.
Let ρ ∈ D ( H A ⊗ H B ) . The fidelity of catalyticteleportation of ρ satisfies: (cid:104) F cat (cid:105) ρ ≥ f reg ( ρ ) d + 1 d + 1 (9) In other words, there is a protocol
T ∈
LOCC ( RAC : BC (cid:48) ) and a catalyst ω CC (cid:48) ∈ D ( H C ⊗ H C (cid:48) ) which achieves thebound in (9).Proof. Here we will only sketch the proof of Theorem 1 andpostpone its formal derivation to the Appendix. The structureof the proof is as follows: we start by constructing the catalystand a special protocol T which increases entanglement frac-tion of the shared state and then use this state to perform anoptimal teleportation protocol T .Let n ≥ be a finite natural number and let C := C . . . C n M and C (cid:48) := C (cid:48) . . . C (cid:48) n M , where M is a classicalregister. Moreover, let E ∈
LOCC ( AC : BC (cid:48) ) be a channelperformed by Alice and Bob (yet to be determined) and de-note σ n − i := tr ...i E ( ρ ⊗ n ) , where tr ...i ( · ) denotes the traceover the first i copies of ρ ⊗ n . Consider the following catalyst: ω CC (cid:48) = 1 n n (cid:88) i =1 ρ ⊗ i ⊗ σ n − i (cid:124) (cid:123)(cid:122) (cid:125) C C (cid:48) ...C n C (cid:48) n ⊗ | i (cid:105)(cid:104) i | M . (10)To the best of our knowledge, this family of states was intro-duced for the first time in the context of entanglement trans-formations by Duan in [57]. Therefore, in what follows, wewill refer to the family of states from Eq. (10) as Duan states . Let us label for clarity A ≡ A and A i ≡ C i for ≤ i ≤ n and similarly for B i and BC (cid:48) . The joint state of the sharedsystem and the catalyst, ρ AB ⊗ ω CC (cid:48) , is presented in Fig. 1afor the exemplary case when n = 5 . The initial protocol T can be summarised as follows:1. Apply E ∈
LOCC ( AC : BC (cid:48) ) to the n -th pair usingthe classical register as the control (see Fig. 1b).2. Relabel the register M using | i (cid:105) M → | i + 1 (cid:105) M for ≤ i < n and | n (cid:105) M → | (cid:105) M (see Fig. 1c).3. Relabel quantum systems according to the value in theregister: A B → A i B i for ≤ i ≤ n (see Fig. 1d).4. Discard the catalyst CC (cid:48) This results in the system and the catalyst being transformedas: ρ AB → ρ ( n ) AB = tr CC (cid:48) T ( ρ AB ⊗ ω CC (cid:48) )= 1 n n (cid:88) i =1 tr /i E ( ρ ⊗ nAB ) , (11) ω CC (cid:48) → ω (cid:48) CC (cid:48) = tr AB T ( ρ AB ⊗ ω CC (cid:48) ) = ω CC (cid:48) . (12)Let us now describe the protocol T , which is a standardteleportation scheme for noisy states [58]. Let { U A a } for a ∈ { , . . . , d } be a set of generalised Pauli operators withrespect to the basis {| i (cid:105) A } . The protocol T reads as follows:1. Twirl the shared state into an isotropic state:T WIRL ( ρ ( n ) AB ) = f ( ρ ( n ) AB ) φ + AB + (1 − f ( ρ ( n ) AB )) φ ⊥ AB , (13)where φ ⊥ = ( − φ + ) / ( d − and tr (cid:0) φ + φ ⊥ (cid:1) = 0 .2. Perform standard teleportation on RA → B :(a) Alice measures RA using a POVM with elements: M RAa = ( ⊗ U a ) φ + RA ( ⊗ U † a ) , (14)(b) Alice communicates outcome a to Bob,(c) Bob applies U † a ( · ) U a to his share of the state.The optimal fidelity of teleportation that can be achieved inthe above process reads: f ( ρ ( n ) AB ) d + 1 d + 1 . (15)Notice that so far the channel E was arbitrary. Let us nowoptimize the protocol T = T ◦ T over all feasible channels E ∈
LOCC ( AC : BC (cid:48) ) . Taking the limit n → ∞ and using lim n →∞ f ( ρ ( n ) AB ) = f reg ( ρ AB ) leads to Eq. (9).The regularised entanglement fraction can be, in general,difficult to compute, both because of the limit n → ∞ andthe optimisation over all LOCC protocols in (7). However,it turns out that in the limit of large n we can make use oftypicality arguments to determine a wide range of states forwhich the inequality in Eq. (9) is strict . In other words, thereare many entangled states whose performance in teleportationcan be improved by utilising the effect of catalysis. FIG. 1. The main part of the protocol which uses a noisy entangledstate as a catalyst to improve the fidelity of teleportation. Subplots ( a ) − ( e ) describe different steps of the protocol and ( f ) describesthe final state of the main system and the catalyst. In particular, thecatalyst remains unchanged by the protocol as the system is trans-formed into a state with a higher entanglement fraction. This state isthen used as a basis for the standard teleportation protocol. B. Demonstrating catalytic advantage in teleportation
Our reasoning so far was valid for arbitrary bipartite densityoperators ρ AB ∈ D ( H A ⊗ H B ) . In this section we will re-strict our attention to pure states ρ AB = | ψ AB (cid:105)(cid:104) ψ AB | and usetypicality arguments to infer that the presented protocol forcatalytic teleportation leads to a generic advantage over thestandard teleportation protocol. Interestingly, this is a con-sequence of an essential property of catalysis: that certaincatalysts (Duan states) “amplify” typical properties of states,even at the level of a single copy. This remarkable propertyof catalysis was first used in the resource-theoretic formula-tion of thermodynamics [39, 44] and more recently in severalother contexts [45, 46, 59]. Interestingly, all of these works re-veal some interesting properties of catalysis and suggest thatcatalysts of the form (10) allow to access the power of multi-copy transformations, while effectively consuming only a sin-gle copy of the state. A similar behavior can be observed inthe setting of catalytic teleportation. To see this, consider thefollowing lemma: Lemma 1.
The regularised entanglement fraction f reg ( ψ AB ) for pure states ψ AB ∈ H A ⊗ H B can be bounded as: f reg ( ψ AB ) ≥ max ψ (cid:48) f ( ψ (cid:48) AB ) (16)s.t. S ( ρ A ) ≥ S ( ρ (cid:48) A ) , (17) where ρ A = tr B ψ AB and ρ (cid:48) A = tr B ψ (cid:48) AB and S ( ρ ) = − tr ρ log ρ is the Shannon entropy. The state ψ (cid:48) AB from the above lemma can be interpreted asproduced in the first part of the catalytic teleportation protocol T , that is ψ (cid:48) AB = tr CC (cid:48) T ( ψ AB ⊗ ω CC (cid:48) ) , where ω CC (cid:48) is theDuan state (10) for a sufficiently large n . Importantly, Theo-rem 1 along with Lemma 1 tells us that, at least for pure states,the fidelity of catalytic teleportation can be strictly larger thestandard fidelity of teleportation. To see this more clearly,consider the following example. Example.
Let us consider teleporting a three-dimensionalquantum system ( d R = 3 ) using the singlet state. In thiscase the state shared between Alice and Bob can be writ-ten as ψ AB = (cid:80) i =1 √ λ i | i (cid:105) A | i (cid:105) B , with Schmidt coefficients λ = 1 / , λ = 1 / and λ = 0 . Its entanglement fraction isequal to f ( ψ AB ) = ( (cid:80) i =1 √ λ i ) / / and therefore itsfidelity of teleportation is given by: (cid:104) F (cid:105) ψ = 0 . , (18)which is also larger than the classical threshold (cid:104) F c (cid:105) = 1 / .Let us now analyse the analogous protocol for catalytic tele-portation. In this case the relevant benchmark is the fidelity ofcatalytic teleportation (6) whose lower bound can be foundusing Lemma 1. To compute it, let us choose the optimizerin (16) to be the state ψ ∗ AB with Schmidt coefficients λ ∗ = x and λ ∗ = λ ∗ = (1 − x ) / , where x is the unique solutionto h ( x ) = x log 2 (which is approximately x ≈ . ) and h ( x ) = − x log x − (1 − x ) log(1 − x ) is the binary entropy.It can be easily verified that this is a feasible choice since theentropy of marginals of both states ψ AB and ψ ∗ AB is equal to log 2 . According to Lemma 1 the regularised entanglementfidelity can be lower-bounded by the entanglement fidelity of ψ ∗ AB , therefore f reg ( ψ AB ) ≥ f ( ψ ∗ AB ) ≈ / . Using Theorem1 we can then lower-bound the fidelity of catalytic teleporta-tion as: (cid:104) F cat (cid:105) ≥ . , (19)which is roughly larger than the best fidelity that couldever be obtained when using the state ψ AB alone. Interest-ingly, this simple example is not a singualar case: there arein fact many entangled states whose performance in telepor-tation can be improved using the assistance of catalysts. Toshow this in Fig. 2 we used Lemma 1 and numerically com-puted the lower-bound on the catalytic advantage η ( ψ ) de-fined as η ( ψ ) := ( (cid:104) F cat (cid:105) − (cid:104) F (cid:105) ) / (cid:104) F (cid:105) . It is easy to show thata similar improvement can be demonstrated for higher dimen-sional entangled states as well, e.g. using existing numericaloptimisation packages to solve (16) . FIG. 2. The advantage η ( ψ ) achieved by using entanglement cataly-sis in the protocol of quantum teleportation. The triangle correspondsto the space of all three-dimensional pure biparitite quantum states.In particular, each point λ = ( λ , λ , λ ) in the plot corresponds toa unique (up to local unitaries) pure bipartite state with Shmidt coef-ficients { λ i } for ≤ i ≤ . The red point corresponds to the explicitexample presented in the main text. IV. DISCUSSION
We have studied an extension of the standard teleportationprotocol to the case when Alice and Bob are allowed to useancillary entangled states in a catalytic way. We have shownthat when arbitrary catalysts are allowed the fidelity of tele-portation is replaced with a more general quantity which wereferred to as the fidelity of catalytic teleportation. We then derived a lower bound for this quantifier which can be viewedas a regularisation of the standard fidelity of teleportation.By employing typicality arguments we then showed a simplelower bound of this regularised quantifier valid for pure states.Finally, we have shown that this lower bound is tight enoughto demonstrate a genuine catalytic advantage for a wide rangeof quantum states.We emphasise that quantum teleportation is one of manyquantum protocols whose performance depends directly onthe entanglement fraction of the shared state. Our protocolfor catalytic teleportation involves a preprocessing step whosesole purpose is to increase the entanglement fraction of theshared state. Therefore this preprocessing can be readily ap-plied to other scenarios and we expect that a similar type ofcatalytic advantage can be demonstrated for other relevantinformation-processing tasks as well.Furthermore, our definition of catalytic fidelity of telepor-tation (6) a priori allows for using arbitrary large-dimensionalstates as catalysts. It would be interesting to see the effect ofconstraining the dimension of the catalyst and, perhaps, quan-tifying the trade-off between the size of the catalyst and theimprovement it can offer in quantum teleportation.
NOTE ADDED
After completing this work, an interesting and independentwork of Kondra et. al. appeared on the arXiv [59]. In thatwork, the authors propose an extension of the task of quan-tum state merging and show that using catalysts, also in thatsetting, allows for an improved performance.
ACKNOWLEDGMENTS
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Consider two spatially separated parties A = A . . . A n and B = B . . . B n composed of n identical and independentlydistributed (i.i.d) entangled states. Let ρ A B be a state with E D, Ω ( ρ A B ) > and let us denote A = A A . . . A n and B = B B . . . B n . We will also use the shorthand i := A B . . . A i B i . Le us consider a CPTP map E ∈ Ω and denote: E ( ρ ⊗ nAB ) = σ nAB st. tr n − m ( σ nAB ) = ( φ ⊗ m + ) A (cid:48) B (cid:48) , (A1)Consider the following state of the catalyst: ω CC (cid:48) := n (cid:88) i =1 n ρ ⊗ i − ⊗ σ n − i (cid:124) (cid:123)(cid:122) (cid:125) A B ...A n B n ⊗ | i (cid:105)(cid:104) i | R . (A2)In what follows A B is the shared state between Alice and Bob and C = A . . . A n R and C (cid:48) = B . . . B n R correspond to thecatalyst they share, with R being a classical register. We also denote σ n − i := tr i ( σ nAB ) . The initial state shared between Aliceand Bob takes the form: ρ A B ⊗ ω CC (cid:48) = 1 n (cid:0) ρ A B ⊗ σ n − n ⊗ | (cid:105)(cid:104) | R + . . . + ρ ⊗ n − n − ⊗ σ A n B n ⊗ | n − (cid:105)(cid:104) n − | R + ρ ⊗ n n ⊗ | n (cid:105)(cid:104) n | R (cid:1) (A3) = n (cid:88) i =1 n ρ ⊗ i ⊗ σ n − i (cid:124) (cid:123)(cid:122) (cid:125) A B ...A n B n ⊗ | i (cid:105)(cid:104) i | R (A4)The catalytic distillation protocol P can be summarised as follows:1. Alice and Bob apply E ∈ Ω to the state in the n -th register, conditioned on the classical register R . Therefore, the mapthey apply takes the form: id( · ) ⊗ n − (cid:88) i =1 (cid:104) i | · | i (cid:105) + E ( · ) ⊗ (cid:104) n | · | n (cid:105) . (A5)2. Alice and Bob relabel their shared classical register R in the following way: | i (cid:105)(cid:104) i | R → | i + 1 (cid:105)(cid:104) i + 1 | R for i < n, (A6) | n (cid:105)(cid:104) n | R → | (cid:105)(cid:104) | R . (A7)3. Alice and Bob relabel their quantum systems conditioned on the classical register in the following way: ρ A B A B ...A i B i A i +1 B i +1 ...A n B n ⊗ | i (cid:105)(cid:104) i | R → ρ A i +1 B i +1 A B ...A i B i A B ...A n B n ⊗ | i (cid:105)(cid:104) i | R for ≤ i ≤ n (A8)The state shared between Alice and Bob during the steps of the protocol P can be written as: ρ A B ⊗ ω CC (cid:48) −−−−→ n − (cid:88) i =1 n ρ ⊗ i ⊗ σ n − i ⊗ | i (cid:105)(cid:104) i | R + 1 n E ( ρ ⊗ n ) ⊗ | n (cid:105)(cid:104) n | R (A9) = 1 n (cid:0) ρ ⊗ σ n − ⊗ | (cid:105)(cid:104) | R + . . . + σ n ⊗ | n (cid:105)(cid:104) n | R (cid:1) (A10) −−−−→ n − (cid:88) i =0 n ρ ⊗ i ⊗ σ n − i ⊗ | i + 1 (cid:105)(cid:104) i + 1 | R (A11) = 1 n (cid:16) σ n ⊗ | (cid:105)(cid:104) | R + . . . + ρ ⊗ ( n − ⊗ σ ⊗ | n (cid:105)(cid:104) n | R (cid:17) (A12) −−−−→ n σ n ⊗ | (cid:105)(cid:104) | R + 1 n n (cid:88) i =2 (cid:101) σ i − ⊗ ρ ⊗ i − ⊗ σ n − i ⊗ | i (cid:105)(cid:104) i | R (A13) = 1 n (cid:0) σ n ⊗ | (cid:105)(cid:104) | R + (cid:101) σ ⊗ ρ ⊗ σ n − ⊗ | (cid:105)(cid:104) | R + . . . + (cid:101) σ n − ⊗ ρ ⊗ n − ⊗ | n (cid:105)(cid:104) n | R (cid:1) , (A14)where we labelled a single-particle state (cid:101) σ i := tr ...i − ,i +1 ...n ( σ n ) . The reduced state of the catalyst ( CC (cid:48) = A B . . . A n B n R )after applying the above protocol and noting that Tr A B ( σ n ) = σ n − , reads: tr A B P ( ρ A B ⊗ ω CC (cid:48) ) = n (cid:88) i =1 n ρ ⊗ i − ⊗ σ n − ⊗ | i (cid:105)(cid:104) i | R = ω CC (cid:48) (A15)The reduced state of the main system ( A B ) now reads: ρ (cid:48) A B = tr CC (cid:48) P ( ρ A B ⊗ ω CC (cid:48) ) = 1 n (cid:0)(cid:101) σ + (cid:101) σ + . . . + (cid:101) σ n (cid:1) = 1 n n (cid:88) i =1 tr ...i − ,i +1 ...n E ( ρ n ) . (A16)In particular, we see that the main system ends up in a certain averaged state, while the reduced state of the catalyst remainsunchanged. The fully entangled fraction of the main system becomes: f ( ρ (cid:48) A B ) = 1 n n (cid:88) i =1 (cid:104) φ + | tr ...i − ,i +1 ...n E ( ρ ⊗ n ) | φ + (cid:105) (A17) = tr (cid:0) E ( ρ ⊗ n )Ω (cid:1) , (A18)where Ω is a positive semidefinite operator defined as: Ω := n (cid:80) ni =1 φ + A i B i ⊗ A i B i . Appendix B: Proof of Lemma 1
Let us begin the proof by recalling the following well-known fact [60]:
Lemma 2.
For any two pure states ψ AB and ψ (cid:48) AB there exists E ∈
LOCC ( A : B ) such that for all (cid:15) > and sufficiently large n : E ( ψ ⊗ nAB ) = (cid:98) ψ nAB s.t. (cid:13)(cid:13)(cid:13) (cid:98) ψ nAB − ( ψ (cid:48) AB ) ⊗ n (cid:13)(cid:13)(cid:13) ≤ (cid:15), (B1) if and only if: S ( ρ A ) ≥ S ( ρ (cid:48) A ) , (B2) where ρ A = tr B | ψ AB (cid:105)(cid:104) ψ AB | and ρ (cid:48) A = tr B | ψ (cid:48) AB (cid:105)(cid:104) ψ (cid:48) AB | . Let n be the smallest possible number of copies such that there exists a transformation E = E ∗ from Lemma 2. Using this asour educated guess for the optimisation in the definition of f reg ( ρ AB ) we have that for all sufficiently large n : f n ( ψ ⊗ nAB ) ≥ n (cid:88) i =1 (cid:104) φ + | tr /i E ∗ ( ψ ⊗ nAB ) | φ + (cid:105) (B3) = n (cid:88) i =1 (cid:104) φ + | tr /i (cid:98) ψ nAB | φ + (cid:105) (B4) ≥ n (cid:88) i =1 (cid:0) (cid:104) φ + | ψ (cid:48) AB | φ + (cid:105) − (cid:15) (cid:1) (B5) = nf ( ψ (cid:48) AB ) − n(cid:15). (B6)Using this and the fact that (cid:15) can be made arbitrarily small we can infer that the regularised entanglement fraction can belower-bounded by: f reg ( ψ AB ) ≥ f ( ψ (cid:48) AB ) (B7)for all ψ (cid:48) AB such that S ( ρ A ))
LOCC ( A : B ) such that for all (cid:15) > and sufficiently large n : E ( ψ ⊗ nAB ) = (cid:98) ψ nAB s.t. (cid:13)(cid:13)(cid:13) (cid:98) ψ nAB − ( ψ (cid:48) AB ) ⊗ n (cid:13)(cid:13)(cid:13) ≤ (cid:15), (B1) if and only if: S ( ρ A ) ≥ S ( ρ (cid:48) A ) , (B2) where ρ A = tr B | ψ AB (cid:105)(cid:104) ψ AB | and ρ (cid:48) A = tr B | ψ (cid:48) AB (cid:105)(cid:104) ψ (cid:48) AB | . Let n be the smallest possible number of copies such that there exists a transformation E = E ∗ from Lemma 2. Using this asour educated guess for the optimisation in the definition of f reg ( ρ AB ) we have that for all sufficiently large n : f n ( ψ ⊗ nAB ) ≥ n (cid:88) i =1 (cid:104) φ + | tr /i E ∗ ( ψ ⊗ nAB ) | φ + (cid:105) (B3) = n (cid:88) i =1 (cid:104) φ + | tr /i (cid:98) ψ nAB | φ + (cid:105) (B4) ≥ n (cid:88) i =1 (cid:0) (cid:104) φ + | ψ (cid:48) AB | φ + (cid:105) − (cid:15) (cid:1) (B5) = nf ( ψ (cid:48) AB ) − n(cid:15). (B6)Using this and the fact that (cid:15) can be made arbitrarily small we can infer that the regularised entanglement fraction can belower-bounded by: f reg ( ψ AB ) ≥ f ( ψ (cid:48) AB ) (B7)for all ψ (cid:48) AB such that S ( ρ A )) ≥ S ( ρ (cid:48) A ))