Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

Lü, Wei; Jia, Zhih-Ahn; Kaszlikowski, Dagomir; Sheng, T. T.

doi:10.48550/arxiv.2202.10989

Cited by 2 publications

(5 citation statements)

References 29 publications

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“…They form a basis for the real vector space of d×d Hermitian operators. An explicit example is generalized Gell-Mann matrices (GGM) [36] (See [37,Sec. 2] for an explicit matrix expression we will use).…”

Section: A Pseudo-density Operatormentioning

confidence: 99%

“…This R A is called a PDO. Notice that when all qudits are measured at the same instant of time, we obtain the normal Bloch representation of a multipartite state [37]. We will denote the set of all PDOs for an event set A as PDO(A).…”

Section: A Pseudo-density Operatormentioning

confidence: 99%

“…Consider a collection of PDOs R = {R A1 , • • • , R An }, a G-symmetry of R is a group G equipped with a representation for each i, g → Φ i g ∈ QPC such that Φ i g (R Ai ) = R Ai for all g ∈ G. Remark 19. In [37] the antilinear quantum channels are investigated, which are crucial for describing the discrete symmetries of an open quantum system and characterizing the quantum entanglement of the mixed quantum state. For the pseudo-density operator, we can also introduce the antilinear quantum pseudo-channel.…”

Section: Space-time Lindbladian and Symmetrymentioning

confidence: 99%

See 2 more Smart Citations

Quantum space-time marginal problem: global causal structure from local causal information

Jia¹,

Song²,

Kaszlikowski³

2023

Preprint

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Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators, and quantum causality between the involved events in an experiment is encoded in the corresponding pseudo-density operator. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the pseudo-density operators; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle, which can be solved effectively by using a neural network. The notion of quantum pseudo-channel is also introduced and we demonstrate that the quantum pseudo-channel marginal problem can be solved by transforming it into a pseudo-density operator marginal problem via the channel-state duality. CONTENTS I. Introduction 1 II. Quantum space-time causality and pseudo-density operator formalism 2 A. Pseudo-density operator 3 B. Quasi-probabilistic mixture of space-time product states 5 C. Space-time purification 6 III. Pseudo density operator marginal problem 6 A. Space-time separable marginal problem 7 B. Space-time symmetric extension 8 C. Polygamy of space-time correlations 8 D. Classical quasi-probability marginal problem 8 IV. Inferring global space-time state from reduced space-time states 9 A. Entropy of space-time states 9 B. Space-time maximum entropy principle 11 C. The neural network approach to inferring the global space-time state 12 V. Conclusion and discussion 12 Acknowledgments 13A. Quantum pseudo-channel 13 1. Quantum pseudo-channel as higher-order maps 13

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Section: A Pseudo-density Operatormentioning

confidence: 99%

Section: A Pseudo-density Operatormentioning

confidence: 99%

Section: Space-time Lindbladian and Symmetrymentioning

confidence: 99%

See 1 more Smart Citation

Quantum space-time marginal problem: global causal structure from local causal information

Jia¹,

Song²,

Kaszlikowski³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…, R An }, a G-symmetry of R is a group G equipped with a representation for each i, g → Φ i g ∈ QPC such that Φ i g (R A i ) = R A i for all g ∈ G. Remark 22. In [67] the antilinear quantum channels are investigated, which are crucial for describing the discrete symmetries of an open quantum system and characterizing the quantum entanglement of the mixed quantum state. For the PDO, we can also introduce the antilinear QPC.…”

Section: B2 Space-time Lindbladian and Symmetrymentioning

confidence: 99%

“…This R A is called a PDO. Notice that when all qudits are measured at the same instant of time, we obtain the normal Bloch representation of a multipartite state [67]. We will denote the set of all PDOs for an event set A as PDO(A).…”

Section: Data Availability Statementmentioning

confidence: 99%

Quantum space-time marginal problem: global causal structure from local causal information

Jia,

Song,

Kaszlikowski

2023

New J. Phys.

View full text Add to dashboard Cite

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators (PDOs), and quantum causality between the involved events in an experiment is encoded in the corresponding PDO. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the reduced PDOs; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle. The notion of quantum pseudo-channel (QPC) is also introduced and we demonstrate that the QPC marginal problem can be solved by transforming it into a PDO marginal problem via the channel-state duality.

show abstract

Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

Cited by 2 publications

References 29 publications

Quantum space-time marginal problem: global causal structure from local causal information

Quantum space-time marginal problem: global causal structure from local causal information

Quantum space-time marginal problem: global causal structure from local causal information

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