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arXiv:1306.0430 · Physical Review A

Genuine fidelity gaps associated with a sequential decomposition of genuinely entangling isometry and unitary operations

Hamed Saberi

Published Dec 12, 2013• 0 citations
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Abstract

We draw attention to the existence of ``genuine'' fidelity gaps in an ancilla-assisted sequential decomposition of genuinely entangling isometry and unitary operations of quantum computing. The gaps arise upon a bipartite decomposition of a multiqubit operation within a one-way sequential recipe in which an ancillary system interacts locally and only once with each qubit in a row. Given the known ``no-go'' associated with such a theoretically and experimentally desirable decomposition, various figures of merit are introduced to analyze the optimal ``fidelity'' with which an arbitrary genuinely entangling operation may admit such a sequential decomposition. An efficient variational matrix-product-operator protocol is invoked in order to obtain numerically the minimal values of the fidelity gaps incurred upon the sequential decomposition of genuine entanglers. We call the values of the gaps so obtained ``genuine'' in light of possible connections to the concept of the genuine multipartite entanglement and since they turn out to be independent of the ancilla dimension and the initial states the associated isometries or unitaries act upon.

DOI
10.1103/physreva.88.062315

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We draw attention to the existence of genuine'' fidelity gaps in an ancilla-assisted sequential decomposition of genuinely entangling isometry and unitary operations of quantum computing. The gaps arise upon a bipartite decomposition of a multiqubit operation within a one-way sequential recipe in which an ancillary system interacts locally and only once with each qubit in a row. Given the known no-go'' associated with such a theoretically and experimentally desirable decomposition, various figures of merit are introduced to analyze the optimal fidelity'' with which an arbitrary genuinely entangling operation may admit such a sequential decomposition. An efficient variational matrix-product-operator protocol is invoked in order to obtain numerically the minimal values of the fidelity gaps incurred upon the sequential decomposition of genuine entanglers. We call the values of the gaps so obtained genuine'' in light of possible connections to the concept of the genuine multipartite entanglement and since they turn out to be independent of the ancilla dimension and the initial states the associated isometries or unitaries act upon.
We consider the recently proposed ``no-go'' theorem of Lamata et al. [Phys. Rev. Lett. 101, 180506 (2008)] on the impossibility of sequential implementation of global unitary operations with the aid of an itinerant ancillary system and view the claim within the language of Kraus representation. By virtue of an extremely useful tool for analyzing entanglement properties of quantum operations, namely, operator-Schmidt decomposition, we provide alternative proof to the no-go theorem and also study the role of initial correlations between the qubits and ancilla in sequential preparation of unitary entanglers. Despite the negative response from the no-go theorem, we demonstrate explicitly how the matrix-product operator (MPO) formalism provides a flexible structure to develop protocols for sequential implementation of such entanglers with an optimal fidelity. The proposed numerical technique, which we call variational matrix-product operator (VMPO), offers a computationally efficient tool for characterizing the ``globalness'' and entangling capabilities of nonlocal unitary operations.
In quantum information, sequential protocols are of utmost importance which involve a very demanding task, especially in the implementation of a global unitary acting on N qubits. In principle, this can be simplified by decomposing the unitary in sequential two-body interactions, between each qubit and an ancillary system, in a manner where each qubit is acted upon only once by the ancilla. This study shows that the sequential decomposition of a global unitary that entangles two or more qubits is impossible in general, even in the case of an infinite-dimensional ancilla.
We establish operational limits on the broadcasting of genuine multipartite entanglement (GME) in quantum networks. Using a distributed protocol in which each of N parties locally implements optimal 1 →\to→ 2 cloning via beam-splitter interactions, we derive exact expressions for the broadcast fidelity of Greenberger-Horne-Zeilinger (GHZ), W states, and Cluster states in a limited setting. We show that the fidelity decays exponentially with system size as [c(R)]N^NN, providing a quantitative expression of multipartite entanglement monogamy in the broadcasting setting, and that both state families share a universal normalisation factor arising from independent post-selection probabilities. Most significantly, we prove a no-go result for simultaneous GME certification: for all reflectivities and all system sizes, the two broadcast copies cannot be simultaneously certified as genuinely multipartite entangled within the standard framework of fidelity-based witnesses. We further check this behaviour for three- and four-party cluster states, finding consistent results that support the generality of the no-go beyond the GHZ and W families. This obstruction arises from the redistribution of multipartite coherence, which both reduces the achievable fidelity and increases the corresponding certification threshold. Our results reveal a fundamental trade-off between the broadcastability of multipartite entanglement and its operational certifiability, and delineate intrinsic limits on entanglement distribution in quantum networks.
We study the possibility for a global unitary applied on an arbitrary number of qubits to be decomposed in a sequential unitary procedure, where an ancillary system is allowed to interact only once with each qubit. We prove that sequential unitary decompositions are in general impossible for genuine entangling operations, even with an infinite-dimensional ancilla, being the controlled-NOT gate a paradigmatic example. Nevertheless, we find particular nontrivial operations in quantum information that can be performed in a sequential unitary manner, as is the case of quantum error correction and quantum cloning.
Not all unitary operations upon a set of qubits can be implemented by sequential interactions between each qubit and an ancillary system. We analyze the specific case of sequential quantum cloning, 1-->M, and prove that the minimal dimension D of the ancilla grows linearly with the number of clones M. In particular, we obtain D=2M for symmetric universal quantum cloning and D=M+1 for symmetric phase-covariant cloning. Furthermore, we provide a recipe for the required ancilla-qubit interactions in each step of the sequential procedure for both cases.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.