Periodic Reporting for period 4 - TempoQ (Temporal Quantum Correlations)
Reporting period: 2020-11-01 to 2021-10-31
Quantum information processing combines quantum theory and information theory in a way that both subjects benefit from each other. While the real-world and commercial implementation of quantum information processing may take some more time, the methods of quantum information theory have already led to new insights into quantum theory. Moreover, the quest for quantum information processing has triggered experimental progress on the manipulation and control of quantum systems with trapped ions, polarized photons, atoms in optical lattices and solid state systems.
Entanglement between two or more particles is often believed to be the central phenomenon in quantum information theory. Consequently, many works have been devoted to the study of entanglement theory. There are, however, other correlations in quantum theory, which are not captured by the framework of entanglement theory. Such correlations appear if sequences of measurements on a single quantum system are made, and not measurements on two distant particles. These temporal quantum correlations have recently moved into the centre of attention for several reasons. First, the improvement of coherence times in quantum optics experiments made it possible to observe temporal quantum correlations with trapped ions or polarized photons. Second, certain theoretical models of quantum computation are naturally formulated in terms of sequential measurements on a quantum system. An example is the paradigm of “measurement-based quantum computing”, where first a complex state is prepared, and then the computation is performed by making measurements on that state only. Finally, there has been a revived interest in foundational issues of quantum mechanics, such as the Kochen-Specker theorem or Leggett-Garg inequalities. Both results use temporal quantum correlations as a central ingredient.
It is the central aim of this proposal to reach a full understanding of temporal quantum correlations. For that, we will study the different notions of temporal correlations and their connection to information theory. Furthermore, we will study the relation to hidden variable theories and whether temporal quantum correlations can be used to prove the quantumness of a system. Finally, we will study implementations using nanomechanical oscillators and applications in measurement-based quantum computation and quantum device testing.
Entanglement between two or more particles is often believed to be the central phenomenon in quantum information theory. Consequently, many works have been devoted to the study of entanglement theory. There are, however, other correlations in quantum theory, which are not captured by the framework of entanglement theory. Such correlations appear if sequences of measurements on a single quantum system are made, and not measurements on two distant particles. These temporal quantum correlations have recently moved into the centre of attention for several reasons. First, the improvement of coherence times in quantum optics experiments made it possible to observe temporal quantum correlations with trapped ions or polarized photons. Second, certain theoretical models of quantum computation are naturally formulated in terms of sequential measurements on a quantum system. An example is the paradigm of “measurement-based quantum computing”, where first a complex state is prepared, and then the computation is performed by making measurements on that state only. Finally, there has been a revived interest in foundational issues of quantum mechanics, such as the Kochen-Specker theorem or Leggett-Garg inequalities. Both results use temporal quantum correlations as a central ingredient.
It is the central aim of this proposal to reach a full understanding of temporal quantum correlations. For that, we will study the different notions of temporal correlations and their connection to information theory. Furthermore, we will study the relation to hidden variable theories and whether temporal quantum correlations can be used to prove the quantumness of a system. Finally, we will study implementations using nanomechanical oscillators and applications in measurement-based quantum computation and quantum device testing.
The work can be divided into three fields of research. The first field of research is the theory of temporal quantum correlations. These are the probabilities that occur if a sequence of measurements is made on a single system, see also Fig. 1. An important task is to study the correlations that can arise in quantum mechanics. In [J. Hoffmann et al., New J. Phys. 20, 102001 (2018)] we first determined the polytope of temporal quantum correlations coming from the most general measurements. We then showed that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identified the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test. We worked with an experimental group on the implementation using trapped ions [C. Spee et al., New J. Phys. 22, 023028 (2020)]. Finally, we discussed in detail the dimension dependence of the temporal correlations, giving also constructions of nonlinear witnesses for the quantum dimension [Y. Mao et al., arXiv:2005.13964] and introduced the notion of genuine multipartite entanglement in time [S. Milz et al., SciPost Phys. 10, 141 (2021)].
The second field of research concerns the connection between temporal correlations and the quantumness of a system. One point is here the derivation of noncontextuality inequalities. We provided a general method to derive Bell inequalities (which are also noncontextuality inequalities) for hypergraph states, which can be described by a nonlocal stabilizer formalism [M. Gachechiladze et al., Phys. Rev. Lett. 116, 070401 (2016)]. The violation of these inequalities increases exponentially with the particle number, but is robust against particle loss. More recently, we provided a systematic method to generalize Bell inequalities to more parties [F. Bernards et al., Phys. Rev. Lett. 125, 200401 (2020)] and this method can be used to study temporal Leggett-Garg or contextuality inequalities. A second major result concerns the nature of contextuality. We were able to show that the Kochen-Specker set with 18 vectors from Cabello et al. [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] is the minimal set for any dimension, verifying a longstanding conjecture by Peres [Z.P. Xu et al., Phys. Rev. Lett. 124, 230401 (2020)].
The third research area concerns the applications of temporal correlations. In [T. Simnacher et al., Phys. Rev. A 99, 062319 (2019)] we presented a general method to characterize and test quantum memories based on their ability to preserve coherence. We introduced a quality measure for quantum memories and characterize it in detail for the qubit case. The measure can be estimated from sparse experimental data by looking at few temporal correlations only. In [M. Gachechiladze et al., Phys. Rev. A 99, 052304 (2019)] we introduced a deterministic scheme of universal measurement-based computation, using only Pauli measurements a hypergraph state. This scheme leads to a different notion of computational depth, in contrast to the usual cluster-state scheme.
In summary, the project was extremely successful, not only in the sense of reaching its original ambitions goals, but also in opening novel research directions. The results have been published in international high-impact journals (such as Nature Communications, Physical Review Letters, and Review of Modern Physics) and attracted the attention of the international scientific community. In addition, four workshops with totally about 190 participants have been organized. Finally, several of the team members received prizes, e.g. or for the best paper on quantum foundations.
The second field of research concerns the connection between temporal correlations and the quantumness of a system. One point is here the derivation of noncontextuality inequalities. We provided a general method to derive Bell inequalities (which are also noncontextuality inequalities) for hypergraph states, which can be described by a nonlocal stabilizer formalism [M. Gachechiladze et al., Phys. Rev. Lett. 116, 070401 (2016)]. The violation of these inequalities increases exponentially with the particle number, but is robust against particle loss. More recently, we provided a systematic method to generalize Bell inequalities to more parties [F. Bernards et al., Phys. Rev. Lett. 125, 200401 (2020)] and this method can be used to study temporal Leggett-Garg or contextuality inequalities. A second major result concerns the nature of contextuality. We were able to show that the Kochen-Specker set with 18 vectors from Cabello et al. [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] is the minimal set for any dimension, verifying a longstanding conjecture by Peres [Z.P. Xu et al., Phys. Rev. Lett. 124, 230401 (2020)].
The third research area concerns the applications of temporal correlations. In [T. Simnacher et al., Phys. Rev. A 99, 062319 (2019)] we presented a general method to characterize and test quantum memories based on their ability to preserve coherence. We introduced a quality measure for quantum memories and characterize it in detail for the qubit case. The measure can be estimated from sparse experimental data by looking at few temporal correlations only. In [M. Gachechiladze et al., Phys. Rev. A 99, 052304 (2019)] we introduced a deterministic scheme of universal measurement-based computation, using only Pauli measurements a hypergraph state. This scheme leads to a different notion of computational depth, in contrast to the usual cluster-state scheme.
In summary, the project was extremely successful, not only in the sense of reaching its original ambitions goals, but also in opening novel research directions. The results have been published in international high-impact journals (such as Nature Communications, Physical Review Letters, and Review of Modern Physics) and attracted the attention of the international scientific community. In addition, four workshops with totally about 190 participants have been organized. Finally, several of the team members received prizes, e.g. or for the best paper on quantum foundations.
This project has gone beyond the state of the art in several research directions. First, we developed a general theory of temporal quantum correlations. This led to methods to characterize the dimensionality of quantum devices which have already been implemented in experiments. Second, we developed a theory of coherence estimation, which can be used to certify the proper functioning of quantum memories. Third, we developed the theory of hypergraph states, which are an interesting family of multiparticle quantum states.
Two more results should be mentioned as unexpected, but opening new research directions for the entire field. First, we established connection between the marginal problem (asking whether there is a global pure state compatible with some given marginals) and the problem of characterizing quantum correlations, especially entanglement [X.D. Yu et al., Nature Communications 12, 1012 (2021)]. This allows to transfer results from one field to the other and back. Second, combining modern methods of graph theory and convex optimization, we solved the Peres conjecture for contextuality [Z.P. Xu et al., Phys. Rev. Lett. 124, 230401 (2020)]. This result allows to identify the minimal and cleanest scenarios where quantum contextuality and temporal quantum correlations can lead to an advantage in information processing.
Two more results should be mentioned as unexpected, but opening new research directions for the entire field. First, we established connection between the marginal problem (asking whether there is a global pure state compatible with some given marginals) and the problem of characterizing quantum correlations, especially entanglement [X.D. Yu et al., Nature Communications 12, 1012 (2021)]. This allows to transfer results from one field to the other and back. Second, combining modern methods of graph theory and convex optimization, we solved the Peres conjecture for contextuality [Z.P. Xu et al., Phys. Rev. Lett. 124, 230401 (2020)]. This result allows to identify the minimal and cleanest scenarios where quantum contextuality and temporal quantum correlations can lead to an advantage in information processing.