mgr Piotr Kopszak

Promotor: dr hab. Marek Mozrzymas, prof. UWr, dr Michał Studziński

Multi port-based teleportation - a group theoretic approach

I recenzent: dr hab. Marcin Marciniak, prof. UG, Uniwersytet Gdański

II recenzent: dr hab. Michał Oszmaniec, Centrum Fizyki Teoretycznej PAN

III recenzent: prof. dr hab. Andrzej Grudka, Uniwersytet im. Adama Mickiewicza w Poznaniu

The aim of this dissertation is the analysis of the performance of the variations of so-called Port-Based Teleportation protocols, enabling transmission of multiple (k, to set the notation) quantum states. The techniques are based primarily on the representation theory of symmetric group S(n). The Chapters 1-4 are introductory and present already known material, necessary either for derivation of further results, or for their better understanding. The main technical result is the introduction of matrix basis (and thus irreducible representa- tion) in a certain ideal of the algebra of partially transposed permutation operators At(k) (d). This ideal is spanned by the operators of the form V(k) i.e. permutation operators acting on Cd⊗n correspondingtothepermutation(n−2k+1,n)...(n−k,n−k+1),transposedonk last systems. This construction enables the analysis of the objects of main interest in the study of Multi Port-based Teleportation (MPBT) protocols, in various forms. Moreover, the new or- thogonality relations and other identities for the irreducible representations of the symmetric group S(n) are provided, together with the formula for an arbitrary partial trace of the Young projector on a given irrep of S(n). This results are provided in the Chapter 5. As for the results concerning the ability of quantum systems to teleport multiple quantum states they are as follows. To begin with, the MPBT protocol is postulated in two forms, prob- abilistic and deterministic, together with the possible optimisation procedures of both. The first important result is the derivation of the lower bound of the entanglement fidelity in the deterministic MPBT scheme. Despite having provided the derivation of the exact expression as well, this lower bound is an important result for two reasons. On one hand it relies on purely combinatorial reasoning and the correspondence between port-based teleportation and the state-discrimination task, without employing the representation theoretic formalism. More- over, the expression for the bound (one of two bounds, in fact) due to its simplicity enables the direct study of the behaviour of deterministic MPBT (dMPBT) in the asymptotic regime (not only when k is fixed and N → ∞, but also when k is function of N, i.e. the aim is to teleport arbitrarily large amount of quantum states). This result is followed by the comprehensive treatment of the quantities determining the per- formance of MPBT protocols, deterministic and probabilistic, non optimal and optimal. It heav- ily relies on the technical results concerning the representation theory of the algebra of partially transposed permutation operators and, in some cases require solving SDP problems. Thanks to the presence of the symmetries in those problems the strong duality holds for each of them, enabling the derivation of the exact solutions. The results are given in terms of the parameters describing the irreducible representations of the symmetric group S(n) and allow to establish the numerical values of the quantities in question (either entanglement fidelity in terms of the deterministic protocol or the probability of success in the probabilistic one) for arbitrary num- ber of quantum states to be teleported k, the local dimension d and the number of employed ports N. Moreover, the ability to teleport arbitrary large number of quantum states k as a function of the number of ports N that goes to infinity is compared against the same ability of so-called packaged PBT, the protocol that is based on standard PBT and enables teleporta- tion of k quantum systems, performing k PBT protocols in parallel. It turns out that requiring k = aNα the MPBT protocols outperform packaged protocols qualitatively, allowing for the faster growth of k in relation to N. All this results are presented in the Chapter 6. The last result is a different the take on the ability of PBT protocols to teleport multiple states. It concerns so-called entanglement recycling, that is the multiple uses of the resource state in subsequent rounds of PBT. In particular, the study of the resource state degradation, measured as the fidelity F(Prec(N, d, 1)) between the actual resource state after the first round of PBT and so-called idealised state, that perfectly suits to be used for the subsequent teleportation is de- rived. This quantity relies on the spectral properties of the measurements used in the protocol. In case of the deterministic scheme the measurement in question is the square-root measure- ment, meaning that the derivation of this results involved the detailed study of it structure. It particular the result quantifying whether or not SRM is projective, depending on its form in respective irreducible subspaces is provided. It turns out that in an asymptotic regime the degradation of the resource state vanishes, implying that the state can be used for the second round. Moreover, the lower bound for the cumulative fidelity after k rounds is provided which is given by F(Prec(N, d, k)) ≥ 1 − 2k(1 − F(Prec(N, d, 1))). This analysis was already provided in [1], however here it is carried for general local dimension d, as opposed to the original result, and involves the analysis of the optimal deterministic scheme as well. The similar study in the probabilistic case however does not provide definite answer - the fidelity after one round converges to a finite number between 0 and 1, making it impossible to deduce the usefulness of the remaining resource in the probabilistic scheme. This results are presented in the Chap- ter 7. Finally, the aim of this dissertation is to provide the unified and accessible description of the re- sults that partly have already been presented and concern the formalism of the representation theory of the algebra of partially transposed permutation operators, including so-called par- tially reduced irreducible representations (PRIRs) and to highlight their connection to the repre- sentation theory of symmetric group, including the induced representation indS(n−k) on the on S(n−2k) hand, and the application in the problems relevant in quantum information on the other.