Port-based teleportation (PBT), introduced in 2008, is a type of quantum teleportation protocol which transmits the state to the receiver without requiring any corrections on the receiver's side. ...Evaluating the performance of PBT was computationally intractable and previous attempts succeeded only with small systems. We study PBT protocols and fully characterize their performance for arbitrary dimensions and number of ports. We develop new mathematical tools to study the symmetries of the measurement operators that arise in these protocols and belong to the algebra of partially transposed permutation operators. First, we develop the representation theory of the mentioned algebra which provides an elegant way of understanding the properties of subsystems of a large system with general symmetries. In particular, we introduce the theory of the partially reduced irreducible representations which we use to obtain a simpler representation of the algebra of partially transposed permutation operators and thus explicitly determine the properties of any port-based teleportation scheme for fixed dimension in polynomial time.
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Deterministic port-based teleportation (dPBT) protocol is a scheme where a quantum state is guaranteed to be transferred to another system without unitary correction. We characterise the best ...achievable performance of the dPBT when both the resource state and the measurement is optimised. Surprisingly, the best possible fidelity for an arbitrary number of ports and dimension of the teleported state is given by the largest eigenvalue of a particular matrix-Teleportation Matrix. It encodes the relationship between a certain set of Young diagrams and emerges as the optimal solution to the relevant semidefinite programme.
In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic ...case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports
N
, the number of teleported systems
k
, and local dimension
d
. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number
N
of ports increases as
o
(
N
)
still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.
We analyse the problem of transmitting a number of unknown quantum states or one composite system in one go. We derive a lower bound on the performance of such process, measured in the entanglement ...fidelity. The obtained bound is effectively computable and outperforms the explicit values of the entanglement fidelity calculated for the pre-existing variants of the port-based protocols, allowing for teleportation of a much larger amount of quantum information. The comparison with the exact formulas and similar analysis for the probabilistic scheme is also discussed. In particular, we present the closed-form expressions for the entanglement fidelity and for the probability of success in the probabilistic scheme in the qubit case in the picture of the spin angular momentum.
Efficient Multi Port-Based Teleportation Schemes Studzinski, Michal; Mozrzymas, Marek; Kopszak, Piotr ...
IEEE transactions on information theory,
2022-Dec., 2022-12-00, Volume:
68, Issue:
12
Journal Article
Peer reviewed
Open access
In this manuscript we analyse generalised port-based teleportation (PBT) schemes, allowing for transmitting more than one unknown quantum state (or a composite quantum state) in one go, where the ...state ends up in several ports at Bob's side. We investigate the efficiency of our scheme discussing both deterministic and probabilistic case, where parties share maximally entangled states. It turns out that the new scheme gives better performance than various variants of the optimal PBT protocol used for the same task. All the results are presented in group-theoretic manner depending on such quantities like dimensions and multiplicities of irreducible representations in the Schur-Weyl duality. The presented analysis was possible by considering the algebra of permutation operators acting on <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the <inline-formula> <tex-math notation="LaTeX">n- </tex-math></inline-formula>fold tensor product of the Hilbert space with finite dimension, we present construction of the respective irreducible matrix representations, which are in fact matrix irreducible representations of the Walled Brauer Algebra. I turns out that the introduced formalism, and symmetries beneath it, appears in many aspects of theoretical physics and mathematics - theory of anti ferromagnetism, aspects of gravity theory or in the problem of designing quantum circuits for special task like for example inverting an unknown unitary.
Abstract
Port-based teleportation (PBT) is a protocol of quantum teleportation in which a receiver does not have to apply correction to the transmitted state. In this protocol two spatially separated ...parties can teleport an unknown quantum state only by exploiting joint measurements on number of shared
d
-dimensional maximally entangled states (resource state) together with a state to be teleported and one way classical communication. In this paper we analyse for the first time the recycling protocol for the deterministic PBT beyond the qubit case. In the recycling protocol the main idea is to re-use the remaining resource state after one or many rounds of PBT for further processes of teleportation. The key property is to learn how much the underlying resource state degrades after every round of the teleportation process. We measure this by evaluating quantum fidelity between respective resource states. To do so we first present analysis of the square-root measurements used by the sender in PBT by exploiting the symmetries of the system. In particular, we show how to effectively evaluate their square-roots and composition. These findings allow us to present the explicit formula for the recycling fidelity involving only group-theoretic parameters describing irreducible representations in the Schur–Weyl duality. For the first time, we also analyse the degradation of the resource state for the optimal PBT scheme and show its degradation for all
d
⩾ 2. In the both versions, the qubit case is discussed separately resulting in compact expression for fidelity, depending only on the number of shared entangled pairs.
Positive maps from irreducibly covariant operators Kopszak, Piotr; Mozrzymas, Marek; Studziński, Michał
Journal of physics. A, Mathematical and theoretical,
10/2020, Volume:
53, Issue:
39
Journal Article
Peer reviewed
Open access
In this paper, we discuss positive maps induced by (irreducibly) covariant linear operators for finite groups. The application of group theory methods allows deriving some new results of a different ...kind. In particular, a family of necessary conditions for positivity, for such objects is derived, stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map. In the low-dimensional cases, for the permutation group S(3) and the quaternion group Q, the necessary and sufficient conditions are given, together with the discussion on their decomposability. In higher dimensions, we present positive maps induced by a three-dimensional irreducible representation of the permutation group S(4) and d-dimensional representation of the monomial unitary group MU(d). In the latter case, we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction. We show that the generalised Choi map can be obtained by considered covariant maps induced by the monomial unitary group. As an additional result, a novel interpretation of the Fujiwara-Algolet conditions for positivity and complete positivity is presented. Finally, in the end, a new form of an irreducible representation of the symmetric group S(n) is constructed, allowing us to simplify the form of certain Choi-Jamiołkowski images derived for irreducible representations of the symmetric group.
In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation ...protocols. Namely, for a given finite group
G
with arbitrary subgroup
H
, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup
H
, as a matrix representation of
H
, is completely reduced to diagonal block form with an irreducible representation of
H
in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on
n
systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group
S
(
n
-
1
)
⊂
S
(
n
)
. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.
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