Generation of minimum-energy entangled states

Nicolò Piccione, Benedetto Militello, Anna Napoli, and Bruno Bellomo

Phys. Rev. A 103, 062402 – Published 1 June 2021

Abstract

Quantum technologies exploiting bipartite entanglement could be made more efficient by using states having the minimum amount of energy for a given entanglement degree. Here, we study how to generate these states in the case of a bipartite system of arbitrary finite dimension either by applying a unitary transformation to its ground state or through a zero-temperature thermalization protocol based on turning on and off a suitable interaction term between the subsystems. In particular, we explicitly identify three possible unitary operators and five possible interaction terms. On one hand, two of the three unitary transformations turn out to be easily decomposable in terms of local elementary operations and a single nonlocal one, making their implementation easier. On the other hand, since the thermalization procedures can be easily adapted to generate many different states, we numerically show that, for each degree of entanglement, generating minimum-energy entangled states costs, in general, less than generating the vast majority of the other states.

Received 26 October 2020
Revised 12 February 2021
Accepted 23 March 2021

DOI:https://doi.org/10.1103/PhysRevA.103.062402

Physics Subject Headings (PhySH)

Entanglement production Quantum entanglement Quantum gates

Quantum Information, Science & Technology

Authors & Affiliations

Nicolò Piccione ^1,*, Benedetto Militello ^2,3, Anna Napoli ^2,3, and Bruno Bellomo ¹

¹Institut UTINAM, CNRS UMR 6213, Université Bourgogne Franche-Comté, Observatoire des Sciences de l'Univers THETA, 41 bis avenue de l'Observatoire, F-25010 Besançon, France
²Università degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segrè, via Archirafi 36, I-90123 Palermo, Italy
³INFN Sezione di Catania, via Santa Sofia 64, I-95123 Catania, Italy

^*nicolo.piccione@univ-fcomte.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 103, Iss. 6 — June 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Distribution of randomly generated pure states with respect to the entropy of entanglement $E$ , and the efficiency of the protocol (a) or the amount of energy used to create them (b). For these plots, the thermalization protocol is based on the simple approach. The relevant Hamiltonians have spectra $σ (H_{A}) = \{0, 2, 4\}$ and $σ (H_{B}) = \{0, 1, 6, 9\}$ in arbitrary units. Both the entanglement and the expended energy are normalized to 1, respectively, with respect to $ln (3)$ and $2 max σ (H_{0})$ . Each graph is the result of an interpolation of $10^{9}$ random states distributed in a $1000 \times 1000$ grid, which covers the whole range of values (even those not shown on the graph; e.g., the grid of efficiency goes from zero to one). The colors correspond to ${log}_{10} (1 + c)$ , where $c$ is the number of states in each grid element. In the bar legend, we report the value of $1 + c$ , so that, e.g., 1 corresponds to the case of counting equal to zero. The blue lines give the position of the MEESs. Regarding the efficiency, we note that it is quite high for every state and that the MEESs are among the worst ones. However, regarding the expended energy, they are much cheaper than the vast majority of all the possible pure states that can be generated.
Reuse & Permissions
Figure 2
The same plots of Fig. 1 but with the thermalization protocol based on the modified simple approach. Here, we have used a different set of states with respect to the case of Fig. 1 since the modified simple approach cannot generate all the states but only those having the form of Eq. (6). In particular, we have randomly generated $10^{9}$ states of this latter kind. As in Fig. 1, the efficiency is quite high for every state with the MEESs being among the worst ones but also costing less than the vast majority of all the possible pure states that can be obtained.
Reuse & Permissions
Figure 3
The same plots of Fig. 1 but with the thermalization protocol based on the transformation of $H_{0}$ through $U_{S}$ . Here, the same states of Fig. 1 have been used. Compared to Fig. 1, the efficiency is remarkably lower, but in this case the MEESs are among the best ones. As in all the other figures, they are cheaper than the vast majority of all the possible pure states that can be generated.
Reuse & Permissions
Figure 4
The same plots of Fig. 1 but with the thermalization protocol based on the transformation of $H_{0}$ through ${\tilde{U}}_{A}$ . Here, the same states of Fig. 2 have been used. With respect to the unitary global approach of Fig. 3, the efficiency is generally higher. The MEESs result to be cheaper to generate than the vast majority of all the possible pure states that can be obtained. The striking difference in the distribution comes from the difference of the sample set of states.
Reuse & Permissions
Figure 5
The same plots of Fig. 1 but with the thermalization protocol based on the transformation of $H_{0}$ through ${\tilde{U}}_{B}$ . Here, the same states of Fig. 2 have been used. This is the only case in which MEESs practically attain the maximum efficiency, even if only for the low entanglement zone. Concerning the expended energy, the MEESs are cheaper to generate than the vast majority of the all the possible pure states that can be generated.
Reuse & Permissions
Figure 6
Comparison of the five approaches analyzed in Sec. 5. For both plots, entanglement, quantified by the entropy of entanglement normalized to 1, is reported on the $x$ axis, whereas on the $y$ axis are shown, respectively, efficiency (a) or expended energy (b). Depending on how much entanglement is required, the most efficient states to generate can be obtained through the MSSG unitary approach based on ${\tilde{U}}_{B}$ or the modified simple one. We also observe that, as predicted analytically, the modified simple approach always performs better than the nonmodified one and that the unitary approach based on $U_{S}$ is worse than the other two for any value of the entanglement.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information