1) The document proposes that general relativity can emerge from quantum mechanics in the limit of many degrees of freedom, similar to how thermodynamics emerges from classical mechanics with many particles.
2) It suggests defining statistical ensembles over wave functions using an "infoton field" to obtain a spatially covariant description of quantum information, represented by an information tensor.
3) A dual theory description of computational complexity is developed using the infoton field, arriving at a Klein-Gordon theory with an inverse metric related to computational parameters like the number of qubits. This provides a space-time covariant description of quantum computation.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces.
Key words: Metric space, Contraction Mapping, Fixed point Theorem, Quasi Metric Space, p-Convergent, p-orbit ally continuous.
Kriging and spatial design accelerated by orders of magnitudeAlexander Litvinenko
We suggest new method, which combine a low-rank tensor technique and Fast Fourier Transformation to speed up kriging, estimation of the variance and geostatistical optimal design. We do 3D Kriging with O(10e+13) estimation points, A-criterion for optimal design with O(10e+12) points in 30 sec.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces.
Key words: Metric space, Contraction Mapping, Fixed point Theorem, Quasi Metric Space, p-Convergent, p-orbit ally continuous.
Kriging and spatial design accelerated by orders of magnitudeAlexander Litvinenko
We suggest new method, which combine a low-rank tensor technique and Fast Fourier Transformation to speed up kriging, estimation of the variance and geostatistical optimal design. We do 3D Kriging with O(10e+13) estimation points, A-criterion for optimal design with O(10e+12) points in 30 sec.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
A Note on “ Geraghty contraction type mappings”IOSRJM
In this paper, a fixed point result for Geraghty contraction type mappings has been proved. Karapiner [2] assumes to be continuous. In this paper, the continuity condition of has been replaced by a weaker condition and fixed point result has been proved. Thus the result proved generalizes many known results in the literature [2-7].
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
On Clustering Financial Time Series - Beyond CorrelationGautier Marti
Financial correlation matrices are noisy. Most of their coefficients are meaningless. RMT advocates that the intrinsic dimension is much lower than O(N^2). Clustering can help to reduce the dimension. But, it can also work on other information than mere correlation...
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Similar to Vitaly Vanchurin "General relativity from non-equilibrium thermodynamics of quantum information" (20)
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Vitaly Vanchurin "General relativity from non-equilibrium thermodynamics of quantum information"
1. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
GENERAL RELATIVITY FROM
NON-EQUILIBRIUM THERMODYNAMICS OF
QUANTUM INFORMATION
VITALY VANCHURIN
UNIVERSITY OF MINNESOTA, DULUTH
based on arxiv:1603.07982, 1706.02229, 1707.05004 and “shoulders of giants”
2. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
MAIN IDEA:
General Relativity emerges from Quantum Mechanics with many d.o.f.
GR = lim
N→∞
QM
(just like Thermodynamics emerges from Classical Mechanics with many d.o.f.)
OUTLINE:
I. SPATIAL METRIC from QUANTUM INFORMATION
define statistical ensembles using information as constraint
derive a spatially covariant description of quantum information
II. SPACE-TIME METRIC from QUANTUM COMPUTATION
define a dual theory description of computational complexities
derive a space-time covariant description of quantum comp.
III. GRAVITY from NON-EQUILIBRIUM THERMODYNAMICS
define thermodynamic variables in the limit of local equilibrium
derive an equation for a non-equilibrium entropy production
3. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
WAVE FUNCTION FOR QUBITS, QUTRITS AND QUDITS
Consider a vector in Hilbert space with preferred t.p. factorization
|ψ =
2D
−1
X=0
ψX
|X ≡
2D
−1
X=0
ψX
D
i=1
|Xi
where Xi
∈ {0, 1} for qubits, Xi
∈ {0, 1, 2} for qutrits, etc.
Then components ψX
’s define a wave-function representation of |ψ
For qubits it is useful to think of ψX
as a function on D dim. lattice
For qutrits the periodicity of the lattice is 3 (or in general k for qudits).
In all cases it is convenient to replace the discrete Xi
with continuous xi
,
differences ∆ with differentiations ∂i, sums with integrals , etc.
4. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
STATISTICAL DEPENDENCE OR ENTANGLEMENT
Question: What is a good measure of entanglement of variables i and j?
Related Question: What is a good measure of statistical dependence
between i and j described by distribution P(x) ≡ ψ∗
(x)ψ(x)?
For statistically dependent random variables we know that
P(x) = P(x1
)P(x2
)...P(xD
) (1)
or
log(P(x)) = log(P(x1
)) + log(P(x2
)) + ... + log(P(xD
)).
Then if we expand the left hand side around a global maxima
log(P(x)) ≈ log(P(y)) −
1
2
(xi
− yi
)Σij(xi
− yi
) + ... (2)
then a good measure of statistical dependence is
Σij ≡ −2
∂2
∂xi∂xj
log(P(x))
x=y
(3)
5. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
FISHER INFORMATION MATRIX
More generally the Hessian matrix (which is a local quantity)
Σij(x) ≡ −2
∂2
∂xi∂xj
log(P(x)) (4)
allows us to approximate the distribution as a sum of Gaussians
P(x) ∝
m
exp −
1
2
xi
− yi
m Σij(ym) xj
− y
j
m (5)
To obtain a measure of statistical dependence between i’s and j’s qubits
(or subsystems) the quantity must be summed (or integrated) over
different values with perhaps different weights. One useful choice is
Aij ≡
1
4
dN
x P(x)Σij(x) = −
1
4
dN
x P(x)
∂2
∂xi∂xj
log(P(x))
where the factor of 1/4 is introduced for future convenience.
It can be shown that Aij is the so-called Fisher information matrix
obtained from shifts of coordinates x.
6. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
FUBINI-STUDY METRIC
For periodic/vanishing boundary conditions the matrix reduces to
Aij = dN
x
∂ P(x)
∂xi
∂ P(x)
∂xj
(6)
Then one can try to define information matrix
Aij = dN
x
∂|ψ(x)|
∂xi
∂|ψ(x)|
∂xj
(7)
but it does not measure well certain quantum entanglements.
A better object is a straightforward generalization, i.e.
Aij = dN
x
∂ψ∗
(x)
∂xi
∂ψ(x)
∂xj
. (8)
which is closely related to the so-called Fubini-Study metric.
We will refer to Aij (for both statistical and quantum systems) as
information matrix.
7. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
INFOTON FIELD OR UNNORMALIZED WAVE FUNCTION
Consider a dual field ϕ(x) (we shall call infoton) in the
sample/configuration space defined (for now) as
ϕ(x) ∝ ψ(x) (9)
and then the information matrix is
Aij ∝ dN
x
∂ϕ∗
(x)
∂xi
∂ϕ(x)
∂xj
. (10)
Next step is to define distributions over |ψ and so one can think of this
as “2nd
quantization”, i.e. prob. distribution over prob. amplitudes.
More precisely, we shall construct statistical ensembles P[ϕ] that would
define probabilities of pure states
P[|ψ ] =
ϕ∝ψ
DϕDϕ∗
P[ϕ] (11)
So we are now dealing with mixed states, but instead of density
matrices we will work with statistical ensembles described by P[ϕ].
8. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
STATISTICAL ENSEMBLE OVER WAVE FUNCTIONS
What we really want is machinery to define distributions over
“microscopic” quantum states subject to “macroscopic” constraints.
For example, we might want to define a statistical ensemble over
infoton ϕ such that the (expected) information matrix is
Aij = ¯Aij (12)
for a given Hermitian matrix ¯Aij.
Statistical ensembles are usually defined using partition functions
Z = DϕDϕ∗
exp (−S[ϕ]) (13)
If the theory is local then the (Euclidean) action S is given by an integral
over a local function L of fields and its derivatives, e.g.
S[ϕ] = dN
x gij ∂ϕ∗
(x)
∂xi
∂ϕ(x)
∂xj
+ λϕ∗
(x)ϕ(x) (14)
where the values of gij
do not depend on x and the “mass-squared”
constant λ must be chosen so that the infoton field ϕ (which is
proportional to wave-functions ψ) is on average normalized.
9. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
INFORMATION TENSOR
For more general ensembles (e.g. over sums of Gaussians) gij can
depend on coordinates x and thus to play the role of a metric tensor.
To make the expression covariant we will also add |g| to the volume
integral and replace partial derivatives with covariant derivatives, i.e.
S = dD
x |g| gij
(x) iϕ∗
(x) jϕ(x) + λ(x)ϕ∗
(x)ϕ(x) (15)
Then, we can define a covariant information tensor as
Aij(x) ≡ iϕ∗
(x) jϕ(x). (16)
and a covariant probability scalar
N(x) ≡ ϕ∗
(x)ϕ(x). (17)
Note that both Aij(x) and N(x) are local quantities in configuration
space.
10. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
STRESS TENSOR
These two quantities can be used to express the stress tensor
Tij = (iϕ∗
j)ϕ + gij gkl
kϕ∗
lϕ + λϕ∗
ϕ (18)
= 2A(ij) + gij gkl
Akl + λN (19)
where A(µν) ≡ 1
2
(Aµν + Aµν ).
Then for a given expected information tensor ¯A(ij) and probability
density ¯N, the macroscopic parameters gij(x) and λ(x) are to be chosen
such that
N = ¯N (20)
and
Tij = 2 ¯A(ij) + gij gkl ¯Akl + λ ¯N . (21)
Note that the corresponding free energy depends on only
“macroscopic” parameters gij(x) and λ(x) (as it should) , i.e.
F[gij, λ] ≡ − log(Z[gij, λ]) = − log DϕDϕ∗
exp −S[ϕ, gij, λ]
11. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
ACTION-COMPLEXITY CONJECTURE
Once again, consider a quantum system of D qubits.
All states are points on 2D
dim. unit sphere separated by distance O(1)
if you were allowed to move along geodesics.
Now imagine that you are only allowed to move in O(D2
) orthogonal
directions out of O(2D
).
More precisely, at any point you are allowed to only apply O(D) of one-
qubit gates or O(D2
) of two- qubit gates.
This is like playing a very high-dimensional maze with many walls and
very few pathways.
Question: What is the shortest distance (also known as computational
complexity) connecting an arbitrary pair of points on the unit sphere?
Action-complexity conjecture: There exist a dual field theory whose action
equals to computational complexity of the shortest quantum circuit connecting
any pair of states,
C(|ψout , |ψin ) = S[ϕ] (22)
where ϕ is a collective notation for all degrees of freedom.
12. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
DUAL THEORIES
Consider dual theories with d.o.f. represented by infoton field, i.e.
C(|ψout , |ψin ) =
T
0
dt L ϕX
(t),
dϕX
(t)
dt
. (23)
We set initial/final conditions
|ψin =
X
ψX
in|X ∝
X
ϕX
(0)|X (24)
|ψout =
X
ψX
out|X ∝
X
ϕX
(T)|X ,
and demand that the (yet to be discovered) dual theory satisfies the
following symmetries/constrains:
States remain (approximately) normalized, i.e.
X
ϕX(t)ϕX
(t) ≈ 1 (25)
Theory is invariant under permutations of bits, i.e. interactions
depend only on Hamming distance h(I, J) between strings of bits I
and J, e.g. h(0, 7) = 3, h(2, 6) = 1.
13. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
DUAL LAGRANGIAN
Then the leading terms of the Lagrangian can be written as
L(ϕX, ˙ϕX) = α
X
˙ϕX ˙ϕX
+ λ
X
ϕXϕX
+
X,Y
f(h(X, Y))ϕXϕY
+ ... (26)
where f(h(X, Y)) is some function of Hamming distance h(X, Y).
And we arrive at a path integral expression
Z(|ψout , |ψin ) =
|ψout =ϕX
(T)|X
|ψin =ϕX(0)|X
d2D
ϕ∗
d2D
ϕ ei T
0 dt(α ˙ϕX ˙ϕX
+λϕXϕX
+fX
YϕXϕY
)
where the Einstein summation convention is assumed.
Note that:
fX
Y ≡ f(h(X, Y)) in computational basis and to transform to other
basis it must be treated as a rank (1, 1) tensor under U 2D
.
Roughly speaking, we expect the function f(h) to quickly vanish
for h > 2, i.e. penalizing more than two q-bit gates.
It will be convenient to denote the three relevant constants as
β ≡ f(0), γ ≡ f(1) and δ ≡ f(2) (in addition to α defined above).
14. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
LATTICE FIELD THEORY
The path integral can also be written as a quantum field theory path integral
on D dimensional torus with only 2D
lattice points
Then in a continuum limit the path integral would be given by
Z(|ψout , |ψin ) = Dϕ∗
Dϕ exp i
T
0
dt dD
x ˜L(ϕ(x), ∂µϕ(x)) (27)
where tildes denote spacetime quantities and µ labels D + 1 dimentions.
15. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
LAGRANGIAN DENSITY
After some math we arrive at Klein-Gordon theory
˜L(ϕ(x), ∂µϕ(x) = ˜gµν
∂µϕ∗
(x)∂ν ϕ(x) − m2
ϕ∗
(x)ϕ(x) (28)
where the “mass’-squared’
m2
≡ − β + Dγ +
D(D − 1)
2
δ l−D−1
(29)
and the inverse “metric” is
˜g00
≡ αl1−D
(30)
˜gii
≡ −
1
2
(γ + (D − 1)δ) l1−D
(31)
˜gij
≡
1
2
δl1−D
, (32)
where i, j ∈ {1, ..., D} and i = j.
For the path integral to be finite we need the mass squared to be
positive and all but one eigenvalues of the metric to be negative, e.g.
α > 0; γ > 0; δ > −
γ
D
; β < −Dγ −
D(D − 1)
2
δ.
16. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
LOCAL COMPUTATIONS
More generally the infoton field theories defined by Lagrangian
˜L(ϕ(x), ∂µϕ(x) = ˜gµν
(x)∂µϕ∗
(x)∂ν ϕ(x) − λ(x)ϕ∗
(x)ϕ(x) (33)
can give a dual description to the theories of computation of qudits
Computations in each hypercube are described by one/two qubit gates
but these computations share each other’s memory on boundaries.
The qubit computers associated with each hypercube run separately,
but exchange information and thus the results of computations.
17. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
INFORMATION-COMPUTATION TENSOR
Now that we have a fully covariant action we can look at a covariant
generalization of the information tensor, i.e.
Aµν ≡ ν ϕ∗
µϕ. (34)
The tensor Aµν is related to the the energy momentum tensor
Tµν = −2A(µν) + ˜gµν ˜gαβ
Aαβ (35)
which implies that it should satisfy the following equation
ν
A(µν) −
1
2
A˜gµν = 0 (36)
Space-space components, i.e. ij, provide a good measure of
informational dependence between (k-local and x-local) subsystems.
Space-time component, i.e. 0i, measures the amount that a given qubit i
is contributing to the computations (zero if iϕ or 0ϕ vanishes)
Thus it is useful to think of 00 as a “density of computations” and of 0i
as a “flux of computations”, which together with ij form a generally
covariant information-computation tensor Aµν (defined for a network
of parallel computers or on a D dimensional dual space-time).
18. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
EMERGENT GRAVITY
Let us go back to “spatial” partition function
Z = DϕDϕ∗
exp (−S[ϕ]) (37)
described by the action with only spatial covariance
S = dD
x |g| gij
(x) iϕ∗
(x) jϕ(x) + λ(x)ϕ∗
(x)ϕ(x) (38)
The corresponding free energy can be expanded as
F[gij, λ, ] ≡ − log(Z[gij, λ, ]) ≈ (39)
≈ dD
x |g| gij
Aij + λ N − S.
If we turn on a random, but unitary dynamics of wave functions then
the infoton field should also evolve accordingly.
But if we want to keep the form of the ensemble to remain the same,
then the macroscopic parameters gij(x) and λ(x) must evolve as well.
And if so, can one describe the emergent dynamics of gij(x) and λ(x)
using dynamical equations, e.g. Einstein equations, corrections?
19. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
THERMODYNAMIC VARIABLES
We define (local) thermodynamic variables
information tensor aij ≡ Aij
metric tensor gij
≡ gij
particle number scalar n ≡ N
chemical potential scalar m ≡ λ
entropy scalar s ≡
S
dDx |g|
free energy scalar f ≡ gij
aij + mn − s (40)
Equation (40) together with the First Law of thermodynamics
0 = mdn + gij
daij − ds (41)
gives us the Gibbs-Duhem Equation
df = ndm + aijdgij
. (42)
20. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
ONSAGER TENSOR
Non-equilibrium entropy production (which is to be extremized)
S[g, ϕ] ≡ dD+1
x |g| L(ϕ, g) −
1
2κ
R(g) + Λ (43)
By following the standard prescription we expand entropy production
1
2κ
R = gαβ,µJµαβ
(44)
where the generalized forces are taken to be
gαβ,µ ≡
∂gαβ
∂xν
(45)
and fluxes are expanded to the linear order in generalized forces
Jµαβ
= Lµν αβ γδ
gγδ,ν . (46)
and thus
1
2κ
R = Lµν αβ γδ
gαβ,µgγδ,ν . (47)
where Lµν αβ γδ
is the Onsager tensor.
21. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
ONSAGER RECIPROCITY RELATIONS
Onsager relations force us to only consider Onsager tensors that are
symmetric under exchange (µ, α, β) ↔ (ν, γ, δ), i.e.
Lµν αβ γδ
= Lνµ βα δγ
(48)
To illustrate the procedure, let us first consider a tensor
Lµν αβ γδ
=
1
2κ
gαν
gβδ
gµγ
+ gαγ
gβν
gµδ
− gαγ
gβδ
gµν
(49)
for which the flux can be rewritten as
Jµαβ
=
1
κ
gαγ
gβδ
Γµ
γδ (50)
where Γµ
γδ are Christoffel symbols and κ is some constant.
If we inset it back into the entropy functional we get
dD+1
x |g|
1
2κ
R =
1
κ
dD+1
x |g|gµν
Γα
µν,α + Γβ
µν Γα
αβ (51)
Note quite what one needs for GR to emerge.
22. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
GENERAL RELATIVITY
The overall space of Onsager tensors is pretty large, but it turns out that
a very simple choice leads to general relativity, i.e.
Lµν αβ γδ
=
1
8κ
gαν
gβδ
gµγ
+ gαγ
gβν
gµδ
− gαγ
gβδ
gµν
− gαβ
gγδ
gµν
.
It has a lot more symmetries and as a result of these symmetries we are
led to a fully covariant theory of general relativity
dD+1
x |g|
1
2κ
R =
1
κ
dD+1
x |g|gµν
Γα
ν[µ,α] + Γβ
ν[µΓα
α]β
By varying the full action with respect to metric (what is equivalent to
minimization of entropy production) we arrive at the Einstein equations
Rµν −
1
2
Rgµν + Λgµν = κ Tµν (52)
where the Ricci tensor is
Rµν ≡ 2 Γα
ν[µ,α] + Γβ
ν[µΓα
α]β (53)
Of course, this result is expected to break down far away from
equilibrium (dark matter?) What about inflation and dark energy?
23. METRIC FROM INFORMATION SPACE-TIME FROM COMPUTATION GRAVITY FROM THERMODYNAMICS
SUMMARY
I. SPATIAL METRIC from QUANTUM INFORMATION
defined statistical ensembles using information as constraint
derived a spatially covariant description of quantum information
II. SPACE-TIME METRIC from QUANTUM COMPUTATION
defined a dual theory description of computational complexities
derived a space-time covariant description of quantum comp.
III. GRAVITY from NON-EQUILIBRIUM THERMODYNAMICS
defined thermodynamic variables in the limit of local equilibrium
derived an equation for a non-equilibrium entropy production