Category: Quantum Mechanics

The Quantum Physics includes the description of all quantum phenomena on wide and resized scale… From entaglements, passing through nuclear scattering and teleportation, until to the quantum effects on which the electronics and the informatics are based

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A Portrait of Space – Time in the Vision of the EPR Paradox Investigation


The Quantum Main Feature


Quantum theory (further information in here), renowned for its perplexing and counterintuitive features, is notably characterized by the phenomenon of Non-Locality a [1]. This aspect of quantum mechanics involves the influence of the measurement of one part of a quantum system on another part, regardless of the distance separating them, provided the system is described by an entangled, non-separable wavefunction b. This phenomenon, dramatically illustrated by the Einstein - Podolsky - Rosen (EPR) Paradox, raises profound questions about the nature of reality and Locality.


It’s Time of Paradoxes…


The EPR paradox highlights two distinct puzzles in quantum mechanics. The first, termed the EPR Locality Paradox, emerges in non-relativistic quantum mechanics and pertains to the non-local correlations enforced by entanglement b (further information in here, Ref. 2). The second, the EPR Influence Paradox, arises in the context of Special Relativity, where the order of measurements on spacelike separated entangled particles can differ between reference frames.


When considering the EPR paradox under special relativity, a deeper mystery arises. For spacelike separated measurements on entangled particles, there exist reference frames where the temporal order of these measurements is reversed. This relativity of Simultaneity [2] introduces additional complexity into understanding the non-local a influences described by quantum mechanics.






The non-local influence between entangled particles appears unattenuated by distance, discriminates specifically between entangled components, and can operate at speeds exceeding that of light. These properties starkly conflict with special relativity, which posits that the speed of light is the ultimate speed limit for causal influences.


By acknowledging that quantum measurements actualize particles into Space - Time, the paradoxical nature of influence between measurements is mitigated. The influence does not propagate instantaneously between pre-existing Space - Time events but rather emerges as measurements occur, respecting the relativistic constraints on Causality [3].


A Quantum and Conceptual Framework relating to Space - Time


A recently proposed conceptual framework suggests that quantum objects do not exist within Space - Time until measured. Before measurement, the Metric (further information in here, Section 4) relations these objects obey are independent of the spacetime metric. This implies that the EPR paradox involves influences operating “outside” Space - Time.


To address the EPR influence paradox, the framework proposes a concept of “Areatime” [4], a reduced-dimensionality representation of Space - Time. Entities existing in areatime do not age relative to those in Space - Time due to orthogonal proper times. Interactions within areatime, when exceeding a certain limit, cause the “Actualization” of one possible state into Space - Time, corresponding to the collapse of the wave function (further information in here) in quantum mechanics.






According to this framework, non-local quantum correlations are a manifestation of events in areatime, independent of Space - Time metric relations. The framework posits that Space - Time itself may be emergent, a result of processes occurring in a higher-dimensional areatime.


This conceptual framework reinterprets the standard path integral formulation (further information in here, Ref. 3) of quantum mechanics, distinguishing between actualizable and actual paths. Before measurement, a quantum particle does not exist in Space - Time, only actualizable paths exist. Measurement collapses these paths into an actual path, rendering the particle real in Space - Time.


Figure 1.   A Pictorial Representation of a Wormhole Interior


Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group


Is There an Evidence of Non – Locality?


Bell’s Theorem provided a method to empirically test the predictions of quantum mechanics against those of local hidden variable theories. Experiments confirmed the violation of Bell’s Inequalities (Equation 1) [5], supporting the non-local predictions of quantum mechanics and challenging the notion of local realism.



[math]\LARGE{|\langle A_{0}B_{0} \rangle + \langle A_{0}B_{1} \rangle + \langle A_{1}B_{0} \rangle - \langle A_{1}B_{1} \rangle| \leq 2}[/math]


[math]\normalsize{|\langle A_{0}B_{0} \rangle + \langle A_{0}B_{1} \rangle + \langle A_{1}B_{0} \rangle - \langle A_{1}B_{1} \rangle| \leq 2}[/math]

Equation 1.   A Bell's Inequality: the Linear Combination of Measurements Averages


This inequality is derived from an imaginary experiment, in which two characters, Alice and Bob, having a particle a person, perform, separately, two different measurements. So [math]\small{A_{0,1}}[/math] and [math]\small{B_{0,1}}[/math] are, respectively, Alice and Bob Binary Measurements; while the symbol [math]\small{\langle \rangle}[/math] stands for the Average of binary measurements.


Figure 2.   Illustration of the Hologram for a Sphere


A New Perspective for Universe Structure


The EPR paradoxes challenge the understanding of locality and causality. By proposing that quantum objects only actualize within Space - Time upon measurement, and suggesting that non-local influences originate outside Space - Time, a coherent conceptual framework can be constructed. This framework not only aligns with experimental observations but also provides a novel perspective on the fundamental nature of reality and the structure of Space - Time. If accurate, it implies that the fabric of Space - Time and the phenomena we observe are emergent properties of deeper, underlying processes in areatime.




(Figure 1)

  1. Nature. "Steering is an essential feature of non-locality in quantum theory" https://www.nature.com/articles/s41467-018-06255-5

  2. Semantic Scholar. "Derivation of Bell’s locality condition from the relativity of simultaneity" https://www.semanticscholar.org/paper/Derivation-of-Bell's-locality-condition-from-the-of-Blood/39b33473ea59b73f71acc38e10a752a832bf8cf3

  3. National Institute of Health. "On Explaining Quantum Correlations: Causal vs. Non-Causal" https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8151236/

  4. arXiv.org. "Time in Quantum Theory" https://arxiv.org/pdf/0705.4638

  5. ResearchGate. "Can violations of Bell’s inequalities be considered as the final proof of quantum physics ?" https://www.researchgate.net/publication/257528290_Can_violations_of_Bell's_inequalities_be_considere_as_the_final_proof_of_quantum_physics


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How Can Teleportation involve Time Travels?


Albert Einstein's Theory of General Relativity permits the existence of Closed-Timelike Curves (CTCs) [1], which are paths within Space - Time that, if traversed, would enable a traveler to interact with their own Past self, whether that traveler be human or elemental particle. Kurt Gödel was among the first to highlight the possibility of CTCs, and subsequent research has proposed various Space - Time configurations accommodating these curves.


However, such scenarios of Time Travel inevitably introduce paradoxes, such as the infamous Grandfather Paradox [2], wherein the time traveler inadvertently alters the past in a way that prevents their own existence. This concept troubled even Einstein, who was close friends with Gödel. The reconciliation of CTCs with quantum mechanics poses a formidable challenge, tackled through various approaches, including Path-Integral Techniques [3].


The Removed “Memories” Approach...


Any theory aiming to unify quantum mechanics and gravity must address the complexities inherent in closed timelike curves, which introduce nonlinearities that challenge the Linearity [4] of conventional quantum mechanics. Deutsch proposed a resolution in his influential work, suggesting a Self-Consistency Condition (Equation 1) concerning the states within CTCs. This condition demands equivalence between measurements at the CTC's entrance and exit. However, this formulation necessitates the assumption of Factorization, implying invalidation of Future [5] "memories". However, Deutsch's theory has faced criticism for apparent inconsistencies.



[math]\LARGE{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]




[math]\large{\rho_{CTC} = Tr_{A}[U(\rho_{A} \otimes \rho_{CTC})U^{\dagger}}][/math]



Equation 1.   Deutsch Self-Consistency Condition form



where [math]\small{\rho_{CTC}}[/math] is the density matrix (further information in here, Section 1) of the system state, [math]\small{A}[/math] inside the CTC; [math]\small{Tr_{A}}[/math] is the Trace of [math]\small{A}[/math]; [math]\small{U}[/math] is the Unitary Matrix; [math]\small{\rho_{A}}[/math] is the density matrix of [math]\small{A}[/math] and [math]\small{U^{\dagger}}[/math] is the transpose unitary matrix.


A New View: Overcoming the CTCs


In contrast, while acknowledging the strangeness of time travel quantum mechanics, P-CTCs, based on the Novikov Principle, appear to offer a less problematic framework. The concept of Probabilistic Closed Timelike Curves (P-CTCs) [6] was initially conceived to tackle the enigma posed by the integration of quantum mechanics into the framework of General Relativity, particularly concerning closed-timelike curves. However, its implications extend beyond this specific domain, offering insights into the potentiality of time travel in alternative scenarios.



N   [math]\LARGE{[\rho]\propto {Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]




N  [math]\large{[\rho]\propto{Tr_{E}[U_{AE}] = C_{A}\rho C_{A}^{\dagger}}}[/math]



Equation 2.   P-CTC External System Time Evolution Equation


N [math]\small{[\rho]}[/math] is the Time Evolution of External System; [math]\small{\rho}[/math] is the density matrix of external system; [math]\small{Tr_{E}}[/math] is the trace of the Hilbert space (further information in here, Section 2), [math]\small{E}[/math] over the system into the CTC; [math]\small{C_{A}}[/math] is the partial trace of [math]\small{A}[/math]; [math]\small{U_{AE}}[/math] is the unitary matrix coupling the internal and external systems; [math]\small{{C_{A}}^{\dagger}}[/math] is the transpose partial trace of [math]\small{A}[/math].


Fundamentally, any quantum theory that permits non-linear processes like the Projection onto specific states, such as the Entangled States (further information in here, Ref. 2) associated with P-CTCs, inherently allows for the prospect of time travel, even in the absence of Space - Time configurations supporting closed-timelike curves. The P-CTS mechanism translates mathematically to the time evolution of the external system being , with the absence of evolution enforced if certain conditions (Equation 2) are met. The paradigm of non-general relativistic P-CTCs can be instantiated through the generation and projection onto entangled pairs of particle-antiparticle. This method mirrors renowned Wheeler's Thought Experiment [7] of a telephone call through time.


craiyon_170508_a_small_ball_passing_through_a_white_vertical_wall

Figure 1.   A typical Macroscopic Example of Quantum Tunneling: a Ball (Subatomic Particle) which overcomes a Wall (Potential Energy Barrier)


Although the process of projection is inherently nonlinear, defying deterministic implementation within conventional quantum mechanics, it can be executed in a probabilistic manner. Consequently, experimental validation of P-CTCs is achievable through Quantum Teleportation experiments, where outcomes corresponding to the desired entangled-state output are selectively post-processed. Should it transpire that the linearity of quantum mechanics is merely an approximation, and projection onto specific states indeed manifests, such occurrences could potentially be witnessed at the singularities (further information in here) of black holes.


In such a scenario, even in the absence of general relativistic closed-timelike curves, the realization of time travel might still be feasible. The theoretical framework of P-CTCs elucidates that quantum time travel can be conceived as a form of retrograde Quantum Tunneling (Figure 1), permitting temporal traversal devoid of a classical trajectory from future to past. P-CTCs rely on Destructive Interference (Figure 2) to prevent self-contradictory events, emphasizing a different self-consistency condition from Deutsch's approach.


Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group


Two Different Perspectives


Illustrating the link between P-CTCs and teleportation provides further insights, showcasing their behavior through Qubits. This demonstration underscores the compatibility of P-CTCs with Higher-Dimensional Systems and the extension to infinite-dimensional scenarios. Presently, no definitive conclusion favors either approach (Deutsch or P-CTCs), given their respective foundations and consistency with different theoretical frameworks. The aspiration in elaborating on the theory of P-CTCs is that it may furnish valuable insights for formulating a Quantum Theory of Gravity. By shedding light on one of the most enigmatic ramifications of general relativity—the prospect of time travel—this theory may contribute significantly to our understanding of Gravity at the quantum level.


craiyon_174102_mystical_blue_and_gold_water_ripples

Figure 2.   A classic Experiment of Waves Interference: Ripples in the Water






  1. arXiv. "Can we travel to the past? Irreversible physics along closed timelike curves "https://arxiv.org/pdf/1912.04702.pdf

  2. ResearchGate. "Grandfather paradox from a new perspective" https://www.researchgate.net/publication/361446083_Grandfather_paradox_from_a_new_perspective#fullTextFileContent

  3. Galileo.phys. "Path Integrals in Quantum Mechanics" https://galileo.phys.virginia.edu/classes/751.mf1i.fall02/PathIntegrals.htm

  4. ScienceDirect. "Consistency and linearity in quantum theory" https://pdf.sciencedirectassets.com/271541/1-s2.0-S0375960100X01843/1-s2.0-S0375960198002898/main.pdf?X

  5. LinkedIn. "Time Travel is Real: Unraveling the Wonders of Traveling to the Future" https://www.linkedin.com/pulse/time-travel-real-unraveling-wonders-traveling-future-manjunath-m-r/

  6. American Physical Society. "Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency" https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.106.040403

  7. Horizon IIT. "The Delayed Choice Quantum Eraser – does the future affect the past?" https://horizoniitm.github.io/dcqe/

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The Implications of Quantum Entanglement on Space-Time: A Focus on the Time Direction


About the "entangled" Microscopic and Macroscopic Realities


Through a detailed investigations on both the Conformal Field Theory (CFT) and Gravity (Space - Time) sides, it’s possible to find the profound connections between initial correlations in quantum systems and the geometric structure of Dual Space-Times. Recent developments in Anti - de - Sitter (AdS)/CFT correspondence [1] have uncovered intriguing links between Quantum Information Theory and Gravity, specifically focusing on the structure of Quantum Entanglement [2] (Figure 1) in conformal field theories (CFTs) and its impact on the dual spacetime. The entanglement structure of quantum subsystems is argued to be a key determinant of classically connected spacetimes.


Let's start with the Quantum Entanglement!

In the standard approach, we commence by examining two independent conformal field theories (CFTs) on the sphere [math]\normalsize{S^d}[/math] (x time) (Figure 2). These CFTs correspond to subsystems, Left (L) and Right (R), with their Hilbert Spaces decomposed as:


[math]\LARGE{{H_{LR}} = {H_{L}} \otimes {H_{R}}}[/math]



[math]\Large{H_{LR} = H_{L} \otimes H_{R}}[/math]

Equation 1.   Decomposed Hilbert Space of LR Entangled State



[math]\large{H_{LR}}[/math] is the Hilbert Space for LR Entangled State; [math]\large{H_{L}}[/math] is the Hilbert Space for L Subsystem and [math]\large{H_{R}}[/math] is the Hilbert Space for R Subsystem ([math]\large{\otimes}[/math] is the Product Operator, further information in here, Section 2).


Initially uncorrelated, the joint state is a product state, [math]\normalsize{\rho_{LR}}[/math]


[math]\LARGE{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/math]



[math]\Large{\rho_{LR} = \rho_{L} \otimes \rho_{R} = |\Psi_{\beta}\Psi_{\beta}|} [/math]


Equation 2.   LR Entangled State Density Matrix


where [math]\normalsize{\rho_{L}}[/math] and [math]\normalsize{\rho_{R}}[/math] are the Density Matrix(ces) for the left and right subsystems, representing Thermal States [3]; [math]\normalsize{\Psi_{\beta}}[/math] is the LR Entangled State Wavefunction. For initially entangled states, such as the thermofield double state, the joint state [math]\normalsize{\rho_{LR}}[/math] involves entangled pure states for subsystems L and R.


Due particelle interagenti con un fascio di energia

Figure 1.   An abstract illustration of the two-Particles Quantum Entanglement


In the AdS/CFT framework, this uncorrelated state corresponds to disconnected AdS spacetimes. We quantify correlations using mutual information as follows.


[math]\LARGE{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]




[math]\large{{I({\rho_{LR}})} = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR}})}}[/math]



Equation 3.   The Quantum Information equation for the LR State


[math]\large{S(\rho_{LR})}[/math], [math]\large{S(\rho_{L})}[/math] and [math]\large{S(\rho_{R})}[/math] are the LR State, L and R Subsystems Entropies, respectively.



In a low - Entropy (Figure 3) environment,


[math]\boxed{\LARGE{{I({\rho_{LR}})}} = 0} \hspace{0.5cm} \LARGE{\Rightarrow}[/math]     [math]\LARGE{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];


[math]\LARGE{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]


[math]\boxed{\large{{I({\rho_{LR}})} = 0}}[/math]

[math]\LARGE{\Downarrow}[/math]

[math]\large{0 = {S({\rho_{L}})} + {S({\rho_{R}})} - {S{(\rho_{LR})}}}[/math];


[math]\large{{S({\rho_{LR}})} = {S({\rho_{L}})} + {S{(\rho_{R})}}}[/math]


Equation 4.   Low-Entropy Conditions Quantum Information


According to the Second Law of Thermodinamics (Equation 5), the Entropy of the composite system must increase as individual entropies evolve, giving:



[math]\LARGE{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]



[math]\Large{{\Delta{S}({\rho_{L}})} + {\Delta{S}({\rho_{R}})} \geq 0}[/math]


Equation 5.   The Second Law of Thermodynamics for the LR State


[math]\large{\Delta{S}(\rho)}[/math] is the Entropy Variation for each subsystem and state involved.


In the absence of initial correlations, the dual Space - Time is composed of disconnected AdS regions, while initial entanglement leads to classical connectivity. The degree of entanglement is shown to dynamically influence the connectivity of the dual spacetime. Disentangling Degrees of Freedom decreases mutual information and Entropy.


Traveling the Space-Time aboard the Thermodynamic Arrow of Time


Building upon recent debates on the Thermodynamic Arrow of Time [4], it has been established a connection between the initial conditions of quantum correlations and the emergence of a preferred direction for the arrow of time. If there are no initial correlations, the arrow of time is directed toward increasing Entropy. However, in contrast to the uncorrelated case, initial correlations alter the entropy evolution. The thermodynamic arrow can now reverse, allowing for both orientations.


CRAIYO~1

Figure 2.   A 3D Representation of a [math]\small{S^d}[/math] Sphere


And on Gravity … Side?

Furthermore, the concepts of Space - Time Sidedness [5] and Time - Orientability have to be discussed. Initial entanglement in the composite quantum system is argued to lead to a time-unoriented, one-sided Space - Time, while decreasing entanglement results in a time-oriented, two-sided Space - Time. In the latter condition, the dual spacetime features disconnected components with opposing time orientations, reflecting the reversed arrows of time in the individual CFTs.


The Fluctuations between Entanglement States


The effects of varying the degree of entanglement between the dual CFTs affect the Space - Time. High correlations are associated with a connected one-sided spacetime, while disentangling the degrees of freedom leads to a disconnected two-sided Space - Time. The maximal entanglement is interpreted as building a connection between the two sides of Space - Time.


craiyon_163844_red_and_blue_balls_mixed

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group


Just a Multi-Effect Dynamics


As shown, the insights into the relationship between quantum entanglement, Space - Time sidedness, and the thermodynamic arrow of time, within the AdS/CFT correspondence framework, highlight the crucial play of dynamic between initial correlations and the geometric dual structure, in understanding the emergence and orientation of the thermodynamic arrow of time.






  1. nLab.org. "AdS-CFT correspondence in nLab"https://ncatlab.org/nlab/show/AdS-CFT+correspondence

  2. IOPscience. "Quantum Entanglement and Its Application in Quantum Communication" https://iopscience.iop.org/article/10.1088/1742-6596/1827/1/012120

  3. Astronomy & Astrophysics. "The thermal state of molecular clouds in the Galactic center: evidence for non-photon-driven heating" https://www.aanda.org/articles/aa/full_html/2013/02/aa20096-12/aa20096-12.html

  4. Forbes. "No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time" https://www.forbes.com/sites/startswithabang/2019/11/22/no-thermodynamics-does-not-explain-our-perceived-arrow-of-time/?sh=4694b68c3109

  5. vXra.org. "The Placement of Two-sided Time in Physics" https://vixra.org/pdf/1906.0353v2.pdf



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