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.

We give two strengthenings of an inequality for the quantum conditional mutual information of a tripartite quantum state recently proved by Fawzi and Renner, connecting it with the ability to reconstruct the state from its bipartite reductions. Namely, we show that the conditional mutual information is an upper bound on the regularized relative entropy distance between the quantum state and its reconstructed version. It is also an upper bound for the measured relative entropy distance of the state to its reconstructed version. The main ingredient of the proof is the fact that the conditional mutual information is the optimal quantum communication rate in the task of state redistribution. with SðXÞ ρ ≔ −trðρ X log ρ X Þ as the von Neumann entropy. It measures the correlations of subsystems C and R relative to subsystem B. The fact the classical CMI is non-negative is a simple consequence of the properties of the probability distributions; the same fact for the quantum CMI is equivalent to a deep result of quantum information theory-strong subadditivity of the von Neumann entropy [1]. Naturally, this led to a variety of applications in different areas, ranging from quantum information theory [2-4] to condensed matter physics [5][6][7].In the classical case, for every tripartite probability distribution p XYZ , IðX∶ ZjYÞ ¼ min where Sðp∥qÞ ≔ P i p i logðp i =q i Þ is the relative entropy and the minimum is taken over the set MC of all distributions q such that X − Y − Z form a Markov chain. Equivalently, the minimization in the right-hand side of Eq. (2) could be taken over Λ ⊗ id Z ðp YZ Þ, for reconstruction channels Λ∶ Y → YX. In particular, IðX∶ ZjYÞ ¼ 0 if, and only if, X − Y − Z form a Markov chain [which is equivalent to the existence of a channelThe class of tripartite quantum states ρ BCR satisfying IðC∶ RjBÞ ρ ¼ 0 has also been similarly characterized [8]: The B subsystem can be decomposed as B ¼ ⨁ k B L;k ⊗ B R;k (with orthogonal vector spaces B L;k ⊗ B R;k ) and the state written asfor a probability distribution fp k g and states ρ CB L;k ∈ C ⊗ B L;k and ρ B R;k R ∈ B R;k ⊗ R. States of this form are called quantum Markov because in analogy to Markov chains, conditioned on the outcome of the measurement onto fB L;k ⊗ B R;k g, the resulting state on C and R is product. Paralleling the classical case, ρ BCR is a quantum Markov state if, and only if, there exists a reconstruction channel Λ∶ B → BC such that Λ ⊗ id R ðρ BR Þ ¼ ρ BCR [9]. Having generalized the definition of CMI, can we also retain the above equivalence, with the set of quantum Markov states taking the role of Markov chains? Surprisingly, it turns out that this is not the case, [10] and it seems not to be possible to connect states that are close to Markov states with states of small CMI in a meaningful way (see, however, [2,11]). Nonetheless, it might be possible to relate states with small CMI with those that can be approximately reconstructed from their bipartite reductions, i.e., such that Λ ⊗ id R ðρ BR Þ ≈ ρ BCR . Indeed, several conje...

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.

Transmitting data reliably over noisy communication channels is one of the most important applications of information theory, and is well understood for channels modelled by classical physics. However, when quantum effects are involved, we do not know how to compute channel capacities. This is because the formula for the quantum capacity involves maximizing the coherent information over an unbounded number of channel uses. In fact, entanglement across channel uses can even increase the coherent information from zero to non-zero. Here we study the number of channel uses necessary to detect positive coherent information. In all previous known examples, two channel uses already sufficed. It might be that only a finite number of channel uses is always sufficient. We show that this is not the case: for any number of uses, there are channels for which the coherent information is zero, but which nonetheless have capacity.

We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.