While in relativity theory space evolves over time into a single entity known as spacetime, quantum theory lacks a standard notion of how to encapsulate the dynamical evolution of a quantum state into a single "state over time". Recently it was emphasized in the work of Fitzsimons, Jones and Vedral that if such a state over time is to encode not only spatial but also temporal correlations which exist within a quantum dynamical process, then it should be represented not by a density matrix, but rather, by a pseudo-density matrix . A pseudo-density matrix is a hermitian matrix of unit trace whose marginals are density matrices, and in this work, we make use a factorization system for quantum channels to associate a pseudo-density matrix with a quantum system which is to evolve according to a finite sequence of quantum channels. We then view such a pseudo-density matrix as the quantum analog of a local patch of spacetime, and we make an in-depth mathematical analysis of such quantum dynamical pseudo-density matrices and the properties they satisfy. We also show how to explicitly extract quantum dynamics from a given pseudo-density matrix, thus solving an open problem posed in the literature.
While in relativity theory space evolves over time into a single entity known as spacetime, quantum theory lacks a standard notion of how to encapsulate the dynamical evolution of a quantum state into a single "state over time". Recently it was emphasized in the work of Fitzsimons, Jones and Vedral that if such a state over time is to encode not only spatial but also temporal correlations which exist within a quantum dynamical process, then it should be represented not by a density matrix, but rather, by a pseudo-density matrix . A pseudo-density matrix is a hermitian matrix of unit trace whose marginals are density matrices, and in this work, we make use a factorization system for quantum channels to associate a pseudo-density matrix with a quantum system which is to evolve according to a finite sequence of quantum channels. We then view such a pseudo-density matrix as the quantum analog of a local patch of spacetime, and we make an in-depth mathematical analysis of such quantum dynamical pseudo-density matrices and the properties they satisfy. We also show how to explicitly extract quantum dynamics from a given pseudo-density matrix, thus solving an open problem posed in the literature.
In ordinary, non-relativistic, quantum physics, time enters only as a parameter and not as an observable: a state of a physical system is specified at a given time and then evolved according to the prescribed dynamics. While the state can, and usually does, extend across all space, it is only defined at one instant of time, in conflict with special relativity where space and time are treated on an equal footing. Here we ask what would happen if we defined the notion of the quantum density matrix for multiple spatial and temporal measurements. We introduce the concept of a pseudo-density matrix which treats space and time indiscriminately. This matrix in general fails to be positive for timelike separated measurements, motivating us to define a measure of causality that discriminates between spacelike and timelike correlations. Important properties of this measure, such as monotonicity under local operations, are proved. Two qubit NMR experiments are presented that illustrate how a temporal pseudo-density matrix approaches a genuinely allowed density matrix as the amount of decoherence is increased between two consecutive measurements.
In ordinary, non-relativistic, quantum physics, time enters only as a parameter and not as an observable: a state of a physical system is specified at a given time and then evolved according to the prescribed dynamics. While the state can, and usually does, extend across all space, it is only defined at one instant of time. Here we ask what would happen if we defined the notion of the quantum density matrix for multiple spatial and temporal measurements. We introduce the concept of a pseudo-density matrix (PDM) which treats space and time indiscriminately. This matrix in general fails to be positive for measurement events which do not occur simultaneously, motivating us to define a measure of causality that discriminates between spatial and temporal correlations. Important properties of this measure, such as monotonicity under local operations, are proved. Two qubit NMR experiments are presented that illustrate how a temporal pseudo-density matrix approaches a genuinely allowed density matrix as the amount of decoherence is increased between two consecutive measurements.
Quantum correlations have a very fundamental significance in the field of quantum
computation and quantum information theory. Though it is very well defined in the
study of spatially separated systems, quantum mechanics allows for the full range of
correlation between temporally separated systems too. In this project, our aim is to
study it's notion in the time domain. Recently Joseph F. Fitzsimon et al. has given a
new approach to the study of quantum correlation in the time domain by introducing
the concept of Pseudo-Density Matrix (PDM) whereas, a very fundamental development
on the temporal correlation has been done by Legget- Garg in 1985( which can't be
accounted by any macro-realist theory). Here, we set out to understand the PDM
notion for different cases and the significance of it to hold the correlations in time
domain using a quantifying tool. This study is extended to find out the correspondences
or incompatibility of it with Legget-Garg inequality and Concurrence - a measure of
quantum entanglement.
Space-time is one of the most essential, yet most mysterious concepts in physics. In quantum mechanics it is common to understand time as a marker of instances of evolution and define states around all the space but at one time; while in general relativity space-time is taken as a combinator, curved around mass. Here we present a unified approach on both space and time in quantum theory, and build quantum states across spacetime instead of only on spatial slices. We no longer distinguish measurements on the same system at different times with measurements on different systems at one time and construct spacetime states upon these measurement statistics. As a first step towards non-relativistic quantum field theory, we consider how to approach this in the continuous-variable multi-mode regime. We propose six possible definitions for spacetime states in continuous variables, based on four different measurement processes: quadratures, displaced parity operators, position measurements and weak measurements. They are motivated by the pseudo-density matrix formulation among indefinite causal structures and the path integral formalism. We show that these definitions lead to desirable properties, and raise the differences and similarities between spatial and temporal correlations. An experimental proposal for tomography is presented, construing the operational meaning of the spacetime states.
Recent efforts to formulate quantum mechanics in a way that treats space and time on a more equal footing have led to a large variety of spacetime-oriented approaches. In this work we present a detailed study of spacetime states, the objects that play the role of quantum states in the recently introduced framework of spacetime quantum mechanics, and show that the main proposals in the literature are different manifestations of the same underlying object. Path integrals, quantum states over time, pseudo-density matrices, the Page and Wootters mechanism, superdensity operators, and timelike-entanglement proposals all arise from spacetime states through particular evaluations, reduced information, linear maps, or quantum channels. This unification provides explicit mathematical representations of these formalisms, reveals relations among them, and clarifies the spacetime information each one captures. We also study the broader relevance of the spacetime-state point of view for Leggett-Garg inequalities, OTOCs, temporal tensor networks, fermionic systems, relativistic QFTs, quantum reference frames, and classical physics, together with additional insights and perspectives revealed by the common unifying framework.
The standard formalism of quantum theory treats space and time in fundamentally different ways. In particular, a composite system at a given time is represented by a joint state, but the formalism does not prescribe a joint state for a composite of systems at different times. If there were a way of defining such a joint state, this would potentially permit a more even-handed treatment of space and time, and would strengthen the existing analogy between quantum states and classical probability distributions. Under the assumption that the joint state over time is an operator on the tensor product of single-time Hilbert spaces, we analyse various proposals for such a joint state, including one due to Leifer and Spekkens, one due to Fitzsimons, Jones and Vedral, and another based on discrete Wigner functions. Finding various problems with each, we identify five criteria for a quantum joint state over time to satisfy if it is to play a role similar to the standard joint state for a composite system: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time steps is associative. We show that no construction satisfies all these requirements. If Hermiticity is dropped, then there is an essentially unique construction that satisfies the remaining four criteria.