Approximate and ensemble local entanglement transformations for multipartite states

David Gunn, Martin Hebenstreit, Cornelia Spee, Julio I. de Vicente, and Barbara Kraus

Phys. Rev. A 108, 052401 – Published 1 November 2023

Abstract

Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n \geq 5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.

Received 21 July 2023
Accepted 9 October 2023

DOI:https://doi.org/10.1103/PhysRevA.108.052401

Physics Subject Headings (PhySH)

Quantum entanglement Quantum information theory

Quantum Information, Science & Technology

Authors & Affiliations

David Gunn¹, Martin Hebenstreit¹, Cornelia Spee¹, Julio I. de Vicente ^2,3, and Barbara Kraus^1,4

¹Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
²Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, E-28911, Leganés (Madrid), Spain
³Instituto de Ciencias Matemáticas (ICMAT), E-28049 Madrid, Spain
⁴Department of Physics, QAA, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany

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Vol. 108, Iss. 5 — November 2023

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Images

Figure 1
Approximate transformations: Blue dashed lines represent branches of an ensemble LOCC transformation. Black solid lines represent deterministic LOCC transformations. Figure 1 concerns general faithful transformations. The orange (green) disk represents the $δ$ ( $ε)$ vicinity around the initial (final) state. $Λ_{1} \in T_{ens}^{δ, ϕ} (ψ, ϕ)$ maps a state $| {\tilde{ψ}}_{1} 〉$ (which is $δ$ -near $| ψ 〉$ ) to an ensemble of states, ${(p_{i}, | {\tilde{ϕ}}_{i} 〉)}$ , such that the average fidelity with $| ϕ 〉$ (represented by the gray dot) is greater than or equal to $1 - ε$ . $Λ_{2} \in T_{det}^{δ, ε} (ψ, ϕ)$ is a map that corresponds to a protocol which deterministically transforms $| {\tilde{ψ}}_{2} 〉$ to $| {\tilde{ϕ}}_{4} 〉$ . Transformation in $T_{p max}^{δ, ε} (ψ, ϕ) \subseteq T_{ens}^{δ, ε}$ corresponds to transformations with the structure of $Λ_{1}$ but with the additional constraint that $p_{i} = p_{max} (| \tilde{ψ} 〉 \to | {\tilde{ϕ}}_{i} 〉)$ for one of the outputs in the $ε$ vicinity (i.e., in the above example, either $| {\tilde{ϕ}}_{1} 〉$ or $| {\tilde{ϕ}}_{3} 〉$ ). Figure 1 considers transformations within SLOCC classes. In this case, we consider only output states that are SLOCC equivalent to a state in the initial $δ$ vicinity (indicated by the incomplete orange fill in the final vicinity). $Λ_{3} \in T_{ens-SLOCC}^{δ, ε} (ψ, ϕ)$ transforms a state $| \tilde{ψ_{1}} 〉$ to an ensemble of states SLOCC equivalent to $| \tilde{ψ_{1}} 〉$ . $Λ_{4} \in T_{det-SLOCC}^{δ, ε} (ψ, ϕ)$ deterministically transforms $| {\tilde{ψ}}_{2} 〉$ to an SLOCC equivalent state $| {\tilde{ϕ}}_{3} 〉$ .
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Figure 2
Approximate transformations within an SLOCC class are more powerful than deterministic transformations within an SLOCC class.
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Figure 3
In the multipartite case, the optimal transformation is not always deterministic. There are choices of states $| ψ 〉$ and $| ϕ 〉$ and $ε > 0$ such that a faithful ensemble transformation of $| ψ 〉$ is possible (blue dashed lines), but there is no deterministic transformation of $| ψ 〉$ into the $ε$ vicinity around $| ϕ 〉$ .
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Figure 4
Strong numerical evidence for the existence of a faithful transformation for which no deterministic transformations between the $ε$ vicinities of the initial and final states exist. Thus, approximate transformations are more powerful than deterministic transformations.
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Figure 5
The protocol proposed in this section. Instead of a standard OSBP, the parties sequentially measure. In the event one fails, yielding a four-qubit state, the parties transform the state probabilistically with a standard OSBP to a four-qubit state such that the output fidelity is optimized. In case of an unsuccessful measurement, yielding a three-qubit state, the state is transformed deterministically to a product state.
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Figure 6
Plot of the fidelities relevant to the proof that approximate transformations are more than deterministic transformations between $ε$ vicinities. The red line corresponds to $F_{prot} (λ)$ , the blue line corresponds to ${| 〈 ψ | ϕ (λ) 〉 |}^{2}$ , the yellow line corresponds to $F_{triv}^{UB} (λ)$ , and the green line corresponds to $F_{1 | 2345}^{max} (λ) .$ As $λ$ approaches 1/2, if one chooses $ε = 1 - F_{prot} (λ)$ , then $ε$ becomes sufficiently small that all states in the initial vicinity are isolated. Moreover, we see that, in this limit, $F_{prot} (λ)$ (the red line) is greater than the fidelity achievable with deterministic transformations. Therefore, there is a $λ < 1 / 2$ and $ε = 1 - F_{prot} (λ)$ such that an approximate transformation is possible, but there are no deterministic transformations between the $ε$ vicinities.
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