Mixed Schur-Weyl duality in port-based teleportation

Created on November 06, 2026

2026   ·   research

In this post I will give an introduction to \((p,q)\)-mixed Schur-Weyl duality in the simple case of \((p,q)=(n-1,1)\), which is the relevant one for port-based teleportation [1], [2], and has also been used recently to discuss data hiding schemes [3]. The discussion is mainly based on the exposition in [4], [5].

Our starting point is to consider a quantum state on $n$ quantum systems of local dimension $d$ of the form

\[\rho = \sum_{i=1}^{n-1} \Phi^+_{in} \otimes \mathbb{I}_{(in)^c},\]

where \(|\Phi^+\rangle_{(ij)} = \sum_{\alpha=1}^d |\alpha\rangle_i \otimes |\alpha\rangle_j\) denotes an unnormalized state on systems \(i\) and \(j\), and \(\mathbb{I}_{(in)^c}\) denotes the identity on the \(n-2\) systems \(1,\dots,n\) excluding systems \(i,j\). Here we are using the common shorthand notation \(\psi\equiv |\psi\rangle\langle\psi|\) for the rank-1 density matrix associated with a pure state. I will be working with unnormalized states for cleaner exposition throughout, but of course all results hold by including normalization factors to obtain quantum states with unit trace.

The maximally entangled state $$ \Phi^+\rangle$$ has the unitary symmetry
\[(U\otimes \overline{U})|\Phi^+\rangle = |\Phi^+\rangle \quad \text{for all unitaries $U$,}\]
where the bar over an operator denotes entry-wise complex conjugation (defined with respect to the same basis used in the definition of $$ \Phi^+\rangle$$.

References

  1. [1]S. Ishizaka and T. Hiroshima, “Asymptotic teleportation scheme as a universal programmable quantum processor,” Physical Review Letters, vol. 101, no. 24, p. 240501, 2008, doi: 10.1103/PhysRevLett.101.240501. [doi]
  2. [2]S. Ishizaka and T. Hiroshima, “Quantum teleportation scheme by selecting one of multiple output ports,” Physical Review A, vol. 79, no. 4, p. 042306, 2009, doi: 10.1103/PhysRevA.79.042306. [doi]
  3. [3]O. Akresh, J. Beckey, and F. Leditzky, “Tight Universal Bounds on Quantum Data Hiding with Multipartite Werner States,” arXiv preprint, Sept. 2026, Available at: https://arxiv.org/abs/2609.40332 [arXiv]
  4. [4]M. Christandl, F. Leditzky, C. Majenz, G. Smith, F. Speelman, and M. Walter, “Asymptotic Performance of Port-Based Teleportation,” Communications in Mathematical Physics, vol. 381, no. 1, pp. 379–451, Jan. 2021, doi: 10.1007/s00220-020-03884-0. [doi] [arXiv]
  5. [5]J. Fei, S. Timmerman, and P. Hayden, “Efficient Quantum Algorithm for Port-based Teleportation,” arXiv preprint, 2023, Available at: https://arxiv.org/abs/2310.01637 [arXiv]