High-dimensional crossover of optimized port-based teleportation
- Field
- Topics
Problem
What is the high-dimensional crossover fidelity of optimized deterministic port-based teleportation?
Let \(F_d^*(N)\) be the maximum entanglement fidelity of teleporting an unknown \(d\)-dimensional state with \(N\) ports, optimizing both the resource state and Alice’s measurement while Bob only selects a port. Use
For each fixed \(c>0\), define
Does the limit in Eq. (2) exist for every \(c>0\), and if so, what is \(\varphi(c)\)? Equation (1) fixes the fidelity convention.
Source
The question is explicitly posed or retained as open in the cited primary literature [Christandl21][Yoshida26]. The statement is rewritten here to make its hypotheses and success criterion self-contained.
Progress
Finite-resource optimization is already characterized by a teleportation-matrix eigenvalue. In particular,
\begin{equation} F_d^*(N)=\frac{N}{d^2}\qquad(1\leq N\leq d). \tag{3} \end{equation}This exact regime does not reach \(N\sim c d^2\) for fixed positive \(c\). [Mozrzymas18]
The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
The optimized exact probabilistic protocol has success probability
\begin{equation} p_d^*(N)=\frac{N}{N+d^2-1}. \tag{4} \end{equation}Converting failure into an arbitrary port output gives a deterministic protocol with fidelity at least \(p_d^*(N)\). Consequently,
\begin{equation} \liminf_{d\to\infty}F_d^*(N_d)\geq\frac{c}{1+c}. \tag{5} \end{equation}Section 8 explicitly identifies the fixed-ratio limit as an open direction. [Christandl21]
The displayed definitions, constraints, and target bounds are recorded in Eqs. (4), (5).
An elementary no-signalling consequence of the protocol model gives
\begin{equation} F_d^*(N)\leq\min\left\{1,\frac{N}{d^2}\right\}. \tag{6} \end{equation}Indeed, for a fixed port \(B_i\), its subnormalized reference–output state in branch \(i\) is bounded above by the unconditional state \(\mathbb I_R/d\otimes\rho_{B_i}\). Its overlap with \(\Phi_d\) is therefore at most \(1/d^2\); summing over \(N\) selected-port branches proves the bound. Combining this derivation with the preceding lower bound yields the useful, not claimed sharp, bracket
\begin{equation} \frac{c}{1+c} \leq\liminf_{d\to\infty}F_d^*(N_d) \leq\limsup_{d\to\infty}F_d^*(N_d) \leq\min\{1,c\}. \tag{7} \end{equation}This paragraph supplies a direct derivation rather than attributing the bracket to a new theorem of the cited paper. [Christandl21]
The displayed definitions, constraints, and target bounds are recorded in Eqs. (6), (7).
The 2026 correspondence relates the problem to unitary estimation with \(N_d-1\) queries, but its order estimate \(1-F_d^*(N)=\Theta(d^4N^{-2})\) does not determine \(\varphi(c)\). In particular, a fixed-\(d\) expansion cannot be substituted into this joint limit without uniform error control. [Yoshida26]
Comment
A literature check through 15 September 2026 located no determination of this crossover function or general proof of the displayed limit. This is distinct from the fixed-dimensional asymptotic-coefficient problem: the input dimension grows together with the resource, so dimension-dependent remainders matter. The result would quantify high-dimensional port requirements rather than extrapolating a fixed-dimensional approximation.
References
- [Mozrzymas18]
- M. Mozrzymas, M. Studziński, S. Strelchuk, and M. Horodecki, "Optimal Port-based Teleportation," New Journal of Physics 20, 053006 (2018).DOIarXiv
- [Christandl21]
- M. Christandl, F. Leditzky, C. Majenz, G. Smith, F. Speelman, and M. Walter, "Asymptotic performance of port-based teleportation," Communications in Mathematical Physics 381, 379–451 (2021).DOIarXiv
- [Yoshida26]
- S. Yoshida, Y. Koizumi, M. Studziński, M. T. Quintino, and M. Murao, "One-to-One Correspondence between Deterministic Port-Based Teleportation and Unitary Estimation," IEEE Transactions on Information Theory 72, 2358–2377 (2026).DOIarXiv