Port-based teleportation is a variant of quantum teleportation, where the receiver can choose one of the ports in his part of the entangled state shared with the sender, but cannot apply other recovery operations.We show that the optimal fidelity of deterministic port-based teleportation (dPBT) using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> + 1 ports to teleport a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>-dimensional state is equivalent to the optimal fidelity of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>-dimensional unitary estimation using n calls of the input unitary operation. From any given dPBT, we can explicitly construct the corresponding unitary estimation protocol achieving the same optimal fidelity, and vice versa. Using the obtained one-to-one correspondence between dPBT and unitary estimation, we derive the asymptotic optimal fidelity of port-based teleportation given by 1 − <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(d⁴)N−2</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> ≤ 1−Ω(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>4)<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>−2, which improves the previously known result given by 1 − <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(d⁵)N−2</i> ≤ F ≤ 1 − Ω(d2)N−2. We also show that the optimal fidelity of unitary estimation for the case n ≤ d − 1 is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> = n+1/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d²</i> , and this fidelity is equal to the optimal fidelity of unitary inversion with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> − 1 calls of the input unitary operation even if we allow indefinite causal order among the calls.

In this paper, we present new families of quantum channels for which corresponding minimum output Rényi p-entropy is not additive. Our manuscript is motivated by the results of Grudka et al. and we focus on channels characterized by both extensions and subspaces of the antisymmetric subspace in$\mathbb {C}^{d} \otimes \mathbb {C} ^{d}$, which exhibit additivity breaking for$p\gt 2$.

Let$U_{d}$be a unitary operator representing an arbitrary$d$-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number$k$of calls of$U_{d}$into its complex conjugate$\overline {U_{d}}$. Our circuits admit a parallel implementation and are proven to be optimal for any$k$and$d$with an average fidelity of$\left \langle{ {F}}\right \rangle =\frac {k+1}{d(d-k)}$. Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call ($k=1$) of the operation$U_{d}$, and the special case of$k=d-1$calls. We then show that our results encompass optimal transformations from$k$calls of$U_{d}$to$f(U_{d})$for any arbitrary homomorphism$f$from the group of$d$-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.

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$n$systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the$n-$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.