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Optimality of the pretty good measurement for port-based teleportation

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Abstract

Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communication. In this paper, we give an explicit proof that the so-called pretty good measurement, or square-root measurement, is optimal for the PBT protocol with independent copies of maximally entangled states as the port state. We then show that the very same measurement remains optimal even when the port state is optimized to yield the best possible PBT protocol. Hence, there is one particular pretty good measurement achieving the optimal performance in both cases. The following well-known facts are key ingredients in the proofs of these results: (i) the natural symmetries of PBT, leading to a description in terms of representation-theoretic data; (ii) the operational equivalence of PBT with certain state discrimination problems, which allows us to employ duality of the associated semidefinite programs. Along the way, we rederive the representation-theoretic formulas for the performance of PBT protocols proved in Studziński et al. (Sci Rep 7(1):1–11, 2017) and Mozrzymas et al. (N J Phys 20(5):053006, 2018) using only standard techniques from the representation theory of the unitary and symmetric groups. Providing a simplified derivation of these beautiful formulas is one of the main goals of this paper.

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Notes

  1. In this paper, we are only concerned with so-called deterministic PBT as described above. There is another variant of the protocol called probabilistic PBT, in which the protocol teleports the target state perfectly, but may abort with a certain probability. Both variants were introduced in the original papers [15, 16], and we refer to [8] for a more detailed comparison of the two variants.

  2. Here, \({\bar{X}}\) denotes complex conjugation with respect to the basis used to define \(|\varPhi ^+\rangle \).

  3. Note that we set \(\mu _{d+1} = -\infty \) in (4.8), and hence the summand for \(i=d\) always appears in the sum. For a Young diagram \(\mu \vdash _d N\) with \(\mu _d = 0\), the resulting \(\mu -\varepsilon _d\) is not a Young diagram anymore.

  4. Any two purifications of \(\phi _{A^N}\) on \(A^NB^N\) are related by an isometry acting on \(B^N\), so we may assume that Bob applies a suitable isometry on \(B^N\) to obtain the state in (5.1) before starting the protocol. The entanglement fidelity of the resulting protocol will be no worse than the original one.

  5. In analogy to the discussion about the operator X in Sect. 4, one can show that the coefficients \(y_{\mu ,i}\) defined with respect to the decomposition (4.8) vanish whenever \(\mu -\varepsilon _i\) does not correspond to a Young diagram \(\alpha \vdash _d N-1\).

  6. In [8], the PBT protocol based on N maximally entangled states and the associated pretty good measurement is called the standard protocol.

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Acknowledgements

I would like to thank Christian Majenz, Connor Paul-Paddock, and Michael Walter for valuable discussions and helpful feedback. I am also grateful to the anonymous referee for useful comments on an earlier version of this manuscript, and permission to reproduce their proof of Lemma 1. This research was partially funded through the Army Research Lab CDQI program.

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Proof of Lemma 1

Proof of Lemma 1

In this appendix, we give a proof of Lemma 1, which is restated below for convenience. For a positive semidefinite operator X, the generalized inverse \(X^{-1}\) and the orthogonal projection \(\varPi _X\) onto \({{\,\mathrm{supp}\,}}X\) are defined as in Sect. 2. Note that \({{\,\mathrm{supp}\,}}X = (\ker X)^\perp = {{\,\mathrm{im}\,}}X\) for positive semidefinite X.

Lemma 1

(restated) Let X be a positive semidefinite operator on a Hilbert space \(\mathcal {H}\). For some \(K\in {\mathbb {N}}\) and \(c\in {\mathbb {R}}\) let \(\lbrace |\xi _k\rangle \rbrace _{k=1}^K\subset {{\,\mathrm{im}\,}}(X)\) be a collection of nonzero vectors such that \(\langle \xi _j|X^{-1}|\xi _k\rangle = \delta _{j,k} c\) for \(1\le j, k\le K\). Then, \(c>0\) and

$$\begin{aligned} X \ge \frac{1}{c} \sum _{k=1}^K |\xi _k\rangle \langle \xi _k|. \end{aligned}$$
(A.1)

Proof

We prove this lemma by induction on K.

Let first \(K=1\). Recall that \(\varPi _X |\xi \rangle = |\xi \rangle \) by assumption, and \(\varPi _X = \sqrt{X}\sqrt{X^{-1}}\) by the definitions of the square root and generalized inverse of X. For any \(|\psi \rangle \in \mathcal {H}\),

$$\begin{aligned} \langle \psi | \xi \rangle \langle \xi |\psi \rangle = |\langle \psi |\xi \rangle |^2 = |\langle \psi | \varPi _X|\xi \rangle |^2&= |\langle \psi |\sqrt{X}\sqrt{X^{-1}}|\xi \rangle |^2 \end{aligned}$$
(A.2)
$$\begin{aligned}&\le \langle \psi | X |\psi \rangle \langle \xi | X^{-1} | \xi \rangle \end{aligned}$$
(A.3)
$$\begin{aligned}&= c \langle \psi | X |\psi \rangle , \end{aligned}$$
(A.4)

where (A.3) follows from the Cauchy-Schwarz inequality. Then \(|\xi \rangle \langle \xi | \le cX\) holds since \(|\psi \rangle \in \mathcal {H}\) was arbitrary. Taking traces on both sides of this operator inequality and using \(|\xi \rangle \ne 0\) and \(X\ge 0\) shows \(c>0\), from which the induction base case \(\frac{1}{c}|\xi \rangle \langle \xi | \le X\) follows.

Let now \(K>1\). Applying the argument above to \(|\xi _K\rangle \) shows that \(Y{:}{=}X - c^{-1}|\xi _K\rangle \langle \xi _K|\) is positive semidefinite. In order to use the induction hypothesis, we need to verify that (a) \(|\xi _j\rangle \in {{\,\mathrm{im}\,}}(Y)\) for \(1\le j\le K-1\) and (b) \(\langle \xi _j|Y^{-1}|\xi _k\rangle = \delta _{j,k} c\) for \(1\le j,k\le K-1\).

To show (a), observe that for any \(1\le j\le K-1\),

$$\begin{aligned} YX^{-1} |\xi _j\rangle&= \left( X-\frac{1}{c}|\xi _K\rangle \langle \xi _K|\right) X^{-1} |\xi _j\rangle = XX^{-1} |\xi _j\rangle - \frac{1}{c} |\xi _K\rangle \langle \xi _K|X^{-1}|\xi _j\rangle \nonumber \\&= \varPi _X |\xi _j\rangle = |\xi _j\rangle , \end{aligned}$$
(A.5)

since \(\langle \xi _K|X^{-1}|\xi _j\rangle = 0\) and \(|\xi _j\rangle \in {{\,\mathrm{im}\,}}(X)\) for \(1\le j\le K-1\) by assumption.

To show (b), we apply \(Y^{-1}\) to both sides of (A.5), giving

$$\begin{aligned} Y^{-1}|\xi _j\rangle = Y^{-1}YX^{-1}|\xi _j\rangle = \varPi _YX^{-1} |\xi _j\rangle . \end{aligned}$$
(A.6)

Taking the inner product with any \(|\xi _k\rangle \) for \(1\le k\le K-1\) and using (a) then shows that

$$\begin{aligned} \langle \xi _k | Y^{-1} |\xi _j\rangle = \langle \xi _k| \varPi _Y X^{-1} |\xi _j\rangle = \langle \xi _k | X^{-1} |\xi _j\rangle = \delta _{j,k} c. \end{aligned}$$
(A.7)

We may therefore apply the induction hypothesis to Y and the vectors \(\lbrace |\xi _j\rangle \rbrace _{j=1}^{K-1}\), giving

$$\begin{aligned} Y = X - \frac{1}{c} |\xi _K\rangle \langle \xi _K| \ge \frac{1}{c} \sum _{j=1}^{K-1} |\xi _j\rangle \langle \xi _j|. \end{aligned}$$
(A.8)

Rearranging this inequality yields the assertion of Lemma 1. \(\square \)

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Leditzky, F. Optimality of the pretty good measurement for port-based teleportation. Lett Math Phys 112, 98 (2022). https://doi.org/10.1007/s11005-022-01592-5

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