TY - BOOK
T1 - Tensors and the Entanglement of Pure Quantum States
AU - Jensen, Asger Kjærulff
PY - 2019
Y1 - 2019
N2 - Entanglement constitutes one of the important resources in quantum information theory. Accordingly, characterizing the entanglement content of a given quantum state is an important concept in quantum information. A natural approach to quantifying entanglement is to consider how an entangled quantum state can be transformed, by transformations that cannot generate entanglement in a system. A central class of such operations are so-called Local Operations and Classical Communication (LOCC). This thesis deals with asymptotic conversion rates for pure quantum states under exact LOCC transformations, in particular by expressing such rates through various kinds of entanglement monotones. Firstly, the hierarchy of multipartite states under stochastic LOCC (SLOCC) is studied via its equivalence to restrictions of tensors. The tensor rank is of particular interest, as it describes the cost of creating the associated state by Greenberger–Horne–Zeilinger (GHZ) states. The fact that the GHZ-cost is non-linear is equivalent to the non-multiplicativity of tensor rank under the Kronecker product. In order to better understand how and why this non-linearity occurs, we consider whether strict sub-multiplicativity of tensor rank stems entirely from the joining of tensor legs, or if it can happen without joining tensor legs. It is shown that strict sub-multiplicativity happens for both tensor rank and border rank when just taking the tensor product.A tool for working with asymptotic tensor rank is that of an asym ptotic spectrum of a preordered semiring. This theory reduces the question of asymptotic restrict-ability to majorization on the set of order preserving homomorphisms into the reals. This concept will be introduced and an example will be computed in the tripartite case, for the sub-semiring generated by the W and GHZ state together with an Einstein–Podolsky–Rosen (EPR) pair shared between two fixed parties. As the ultimate goal should be characterizing multipartite entanglement through nonstochastic LOCC, the asymptotic spectrum method is applied to a refinement of the tensor semiring. This refinement keeps some control on the probability of successful outcomes of LOCC protocols, specifically yielding a set of monotones, describing asymptotic conversion rates given any converse error exponent, r. While this is still some distance from the ideal asymptotic regime, we see that in the bipartite case, in fact the conversion rate for success probability going to 1 is also captured by these monotones. Finally, a formula for the pure, bipartite, exact, deterministic conversion rate is presented, as derived through type class arguments.
AB - Entanglement constitutes one of the important resources in quantum information theory. Accordingly, characterizing the entanglement content of a given quantum state is an important concept in quantum information. A natural approach to quantifying entanglement is to consider how an entangled quantum state can be transformed, by transformations that cannot generate entanglement in a system. A central class of such operations are so-called Local Operations and Classical Communication (LOCC). This thesis deals with asymptotic conversion rates for pure quantum states under exact LOCC transformations, in particular by expressing such rates through various kinds of entanglement monotones. Firstly, the hierarchy of multipartite states under stochastic LOCC (SLOCC) is studied via its equivalence to restrictions of tensors. The tensor rank is of particular interest, as it describes the cost of creating the associated state by Greenberger–Horne–Zeilinger (GHZ) states. The fact that the GHZ-cost is non-linear is equivalent to the non-multiplicativity of tensor rank under the Kronecker product. In order to better understand how and why this non-linearity occurs, we consider whether strict sub-multiplicativity of tensor rank stems entirely from the joining of tensor legs, or if it can happen without joining tensor legs. It is shown that strict sub-multiplicativity happens for both tensor rank and border rank when just taking the tensor product.A tool for working with asymptotic tensor rank is that of an asym ptotic spectrum of a preordered semiring. This theory reduces the question of asymptotic restrict-ability to majorization on the set of order preserving homomorphisms into the reals. This concept will be introduced and an example will be computed in the tripartite case, for the sub-semiring generated by the W and GHZ state together with an Einstein–Podolsky–Rosen (EPR) pair shared between two fixed parties. As the ultimate goal should be characterizing multipartite entanglement through nonstochastic LOCC, the asymptotic spectrum method is applied to a refinement of the tensor semiring. This refinement keeps some control on the probability of successful outcomes of LOCC protocols, specifically yielding a set of monotones, describing asymptotic conversion rates given any converse error exponent, r. While this is still some distance from the ideal asymptotic regime, we see that in the bipartite case, in fact the conversion rate for success probability going to 1 is also captured by these monotones. Finally, a formula for the pure, bipartite, exact, deterministic conversion rate is presented, as derived through type class arguments.
UR - https://rex.kb.dk/permalink/f/h35n6k/KGL01012069468
M3 - Ph.D. thesis
BT - Tensors and the Entanglement of Pure Quantum States
PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
ER -