Probing quantum entanglement from magnetic-sublevels populations: beyond spin squeezing inequalities
By: Guillem Müller-Rigat, Maciej Lewenstein, and Irénée Frérot,
2022–12–29
The paper presents a systematic approach to constructing novel entanglement criteria for quantum systems based on Zeeman-sublevel population measurements. This approach goes beyond spin squeezing inequalities (SSI), which are a major tool for probing quantum entanglement in many-body systems. The authors illustrate their approach on ground states of spin-1 and spin-2 Bose-Einstein condensates (BECs), and show that it can be used to reveal finer features of quantum entanglement not captured by SSI.
Introduction
The paper begins by reviewing the concept of spin squeezing, which is a measure of the degree of quantum entanglement in a many-body system. SSI are a set of inequalities that can be used to detect spin squeezing, and they have been used to study a variety of quantum systems, including BECs. However, the authors note that SSI cannot detect all forms of quantum entanglement, and they propose a new approach to constructing entanglement criteria that is based on Zeeman-sublevel population measurements.
SSI are a set of inequalities that can be used to detect spin squeezing, which is a measure of the degree of quantum entanglement in a many-body system. However, SSI cannot detect all forms of quantum entanglement. This is because SSI are based on the assumption that the system is in a pure state. However, there are many forms of quantum entanglement that can exist in mixed states.
For example, consider a system of two spins that are in a mixed state. This state can be represented by a density matrix, which is a mathematical object that describes the probabilities of finding the spins in different states. If the two spins are perfectly entangled, then the density matrix will have the form of a Werner state. However, if the two spins are only partially entangled, then the density matrix will have a more general form.
SSI cannot detect entanglement in mixed states that are not Werner states. This is because SSI are only sensitive to the degree of spin squeezing, which is a measure of the degree of entanglement in pure states. However, there are other measures of entanglement that can be used to detect entanglement in mixed states.
Applications to a Spin-1 BEC
This section applies the authors’ approach to constructing entanglement criteria to a spin-1 Bose-Einstein condensate (BEC).
A spin-1 BEC is a system of atoms that have spin 1. The atoms can be in one of three Zeeman sublevels, which have spins of +1, 0, or -1. The authors show that the Zeeman-sublevel population measurements can be used to construct a new entanglement criterion that is more sensitive to quantum entanglement than SSI.
The authors begin by reviewing the concept of spin squeezing, which is a measure of the degree of quantum entanglement in a many-body system. SSI are a set of inequalities that can be used to detect spin squeezing, and they have been used to study a variety of quantum systems, including BECs. However, the authors note that SSI cannot detect all forms of quantum entanglement, and they propose a new approach to constructing entanglement criteria that is based on Zeeman-sublevel population measurements.
The authors then show how this new approach can be used to construct a new entanglement criterion for spin-1 BECs. The criterion is based on the idea that the Zeeman-sublevel population measurements can be used to reconstruct the density matrix of the BEC. If the density matrix is entangled, then the criterion will be violated.
The authors then apply the new criterion to a variety of spin-1 BEC states, including the ground state and several excited states. They show that the criterion can be used to detect entanglement in all of these states.
The second section of the paper concludes by discussing the implications of the results for the study of quantum entanglement in spin-1 BECs. The authors argue that their approach provides a more powerful tool for probing quantum entanglement in these systems than SSI.
Applications to a Spin-2 BEC
A spin-2 BEC is a system of atoms that have spin 2. The atoms can be in one of five Zeeman sublevels, which have spins of +2, +1, 0, -1, or -2. The authors show that the Zeeman-sublevel population measurements can be used to construct a new entanglement criterion that is more sensitive to quantum entanglement than SSI.
The authors begin by reviewing the concept of spin squeezing, which is a measure of the degree of quantum entanglement in a many-body system. SSI are a set of inequalities that can be used to detect spin squeezing, and they have been used to study a variety of quantum systems, including BECs. However, the authors note that SSI cannot detect all forms of quantum entanglement, and they propose a new approach to constructing entanglement criteria that is based on Zeeman-sublevel population measurements.
The authors then show how this new approach can be used to construct a new entanglement criterion for spin-2 BECs. The criterion is based on the idea that the Zeeman-sublevel population measurements can be used to reconstruct the density matrix of the BEC. If the density matrix is entangled, then the criterion will be violated.
The authors then apply the new criterion to a variety of spin-2 BEC states, including the ground state and several excited states. They show that the criterion can be used to detect entanglement in all of these states.
The third section of the paper concludes by discussing the implications of the results for the study of quantum entanglement in spin-2 BECs. The authors argue that their approach provides a more powerful tool for probing quantum entanglement in these systems than SSI.
Conclusions
The conclusions section of the paper discusses the implications of the results for the study of quantum entanglement in many-body systems. The authors argue that their approach provides a more powerful tool for probing quantum entanglement in these systems than SSI.
The authors begin by discussing the limitations of SSI. They note that SSI can only detect entanglement in pure states, and they cannot detect entanglement in mixed states that are not Werner states. The authors also note that SSI are not always sensitive to entanglement, even in pure states.
The authors then discuss the advantages of their approach. They note that their approach can be used to detect entanglement in both pure and mixed states, and it is more sensitive to entanglement than SSI. They also note that their approach can be used to detect entanglement in a wider range of states than SSI.
The authors conclude by suggesting that their approach could be used to study other forms of quantum entanglement, such as entanglement between different components of a BEC. They also suggest that their approach could be used to develop new quantum technologies, such as quantum sensors and quantum computers.
Originally published at http://quantumsumm.wordpress.com on July 18, 2023.
