scholarly journals Limitations on quantum dimensionality reduction

2015 ◽  
Vol 13 (04) ◽  
pp. 1440001 ◽  
Author(s):  
Aram W. Harrow ◽  
Ashley Montanaro ◽  
Anthony J. Short
Keyword(s):  
Dimensionality Reduction ◽  
Figure Of Merit ◽  
Classical Case ◽  
Constant Factor ◽  
Quantum States ◽  
Low Rank ◽  
Mixed States ◽  
Correct Figure ◽  
Classic Result ◽  
Euclidean Distances

The Johnson–Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O( log n) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over quantum channels that significantly reduces the dimension of quantum states while preserving the 2-norm distance with high probability. We discuss two tasks for which the 2-norm distance is indeed the correct figure of merit. In the case of the trace norm, we show that the dimension of low-rank mixed states can be reduced by up to a square root, but that essentially no dimensionality reduction is possible for highly mixed states.

scholarly journals Supervised dimensionality reduction for big data

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Joshua T. Vogelstein ◽  
Eric W. Bridgeford ◽  
Minh Tang ◽  
Da Zheng ◽  
Christopher Douville ◽  
...  
Keyword(s):  
Dimensionality Reduction ◽  
Data Science ◽  
Real Data ◽  
Low Rank ◽  
Conditional Moment ◽  
Desktop Computer ◽  
Reduction Techniques ◽  
Reduction Methods ◽  
The Individual ◽  
Low Dimensional

AbstractTo solve key biomedical problems, experimentalists now routinely measure millions or billions of features (dimensions) per sample, with the hope that data science techniques will be able to build accurate data-driven inferences. Because sample sizes are typically orders of magnitude smaller than the dimensionality of these data, valid inferences require finding a low-dimensional representation that preserves the discriminating information (e.g., whether the individual suffers from a particular disease). There is a lack of interpretable supervised dimensionality reduction methods that scale to millions of dimensions with strong statistical theoretical guarantees. We introduce an approach to extending principal components analysis by incorporating class-conditional moment estimates into the low-dimensional projection. The simplest version, Linear Optimal Low-rank projection, incorporates the class-conditional means. We prove, and substantiate with both synthetic and real data benchmarks, that Linear Optimal Low-Rank Projection and its generalizations lead to improved data representations for subsequent classification, while maintaining computational efficiency and scalability. Using multiple brain imaging datasets consisting of more than 150 million features, and several genomics datasets with more than 500,000 features, Linear Optimal Low-Rank Projection outperforms other scalable linear dimensionality reduction techniques in terms of accuracy, while only requiring a few minutes on a standard desktop computer.

scholarly journals SPIN–MOMENTUM CORRELATION IN RELATIVISTIC SINGLE-PARTICLE QUANTUM STATES

2010 ◽  
Vol 08 (03) ◽  
pp. 517-528 ◽  
Author(s):  
M. A. JAFARIZADEH ◽  
M. MAHDIAN
Keyword(s):  
Single Particle ◽  
Inertial Frame ◽  
Quantum States ◽  
Moving Frame ◽  
Lorentz Transformations ◽  
Mixed States ◽  
Wigner Rotation ◽  
Lorentz Boost ◽  
Momentum Correlation ◽  
Spin Momentum

This paper is concerned with the spin–momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigenstates (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin and momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not an observer-dependent quantity in the inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and have shown that entanglement decreases with respect to the increase in observer velocity. Finally, we argue that Wigner rotation is induced by Lorentz transformations and can be realized as a controlling operator.

XII.—On the Theory of Unimolecular Gas Reactions: A Quantum Harmonic Oscillator Model

1954 ◽  
Vol 64 (2) ◽  
pp. 161-174 ◽  
Author(s):  
N. B. Slater
Keyword(s):  
High Pressure ◽  
Classical Mechanics ◽  
Classical Case ◽  
Constant Factor ◽  
Quantum Case ◽  
Fundamental Vibration ◽  
Fixed Temperature ◽  
Dissociation Probability ◽  
Pressure Scale ◽  
Continuum Of States

SynopsisThe writer's theory of unimolecular dissociation rates, based on the treatment of the molecule as a harmonically vibrating system, is put in a form which covers quantum as well as classical mechanics. The classical rate formulæ are as before, and are also the high-temperature limits of the new quantum formulæ. The high-pressure first-order rate k∞ is found first from the Gaussian distribution of co-ordinates and momenta of harmonic systems, and is justified for the quantum-mechanical case by Bartlett and Moyal's phase-space distributions. This leads to a re-formulation of k∞ as a molecular dissociation probability averaged over a continuum of states, and to a general rate for any pressure of the gas.The high-pressure rate k∞ is of the form ve-F/kT, where v and F depend, in the quantum case, on the temperature T; but v is always between the highest and lowest fundamental vibration frequencies of the molecule. Concerning the decline of the general rate k with pressure at fixed temperature, k/k∞ is to a certain approximation the same function of as was tabulated earlier for the classical case, apart from a constant factor changing the pressure scale in the quantum case.

A note on the degree conjecture for separability of multipartite quantum states

2021 ◽  
pp. 2050048
Author(s):  
Zhen Wang ◽  
Ming-Jing Zhao ◽  
Zhi-Xi Wang
Keyword(s):  
Quantum States ◽  
Point Graph ◽  
Mixed States ◽  
Degree Condition ◽  
Graph Laplacians ◽  
Nearest Point ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Math Phys ◽  
Degree Criterion

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

Low-rank tensors applications for dimensionality reduction of complex hydrocarbon reservoirs

Energy ◽  
2021 ◽  
pp. 122680
Author(s):  
Fahad Iqbal Syed ◽  
Temoor Muther ◽  
Amirmasoud Kalantari Dahaghi ◽  
Shahin Neghabhan
Keyword(s):  
Dimensionality Reduction ◽  
Hydrocarbon Reservoirs ◽  
Low Rank

scholarly journals Generalized notions of sparsity and restricted isometry property. Part I: a unified framework

2019 ◽  
Vol 9 (1) ◽  
pp. 157-193 ◽  
Author(s):  
Marius Junge ◽  
Kiryung Lee
Keyword(s):  
Banach Space ◽  
Inverse Problems ◽  
Compressed Sensing ◽  
Dimensionality Reduction ◽  
Group Actions ◽  
Function Spaces ◽  
Restricted Isometry Property ◽  
Low Rank ◽  
Unified Framework ◽  
Order Tensor

Abstract The restricted isometry property (RIP) is an integral tool in the analysis of various inverse problems with sparsity models. Motivated by the applications of compressed sensing and dimensionality reduction of low-rank tensors, we propose generalized notions of sparsity and provide a unified framework for the corresponding RIP, in particular when combined with isotropic group actions. Our results extend an approach by Rudelson and Vershynin to a much broader context including commutative and non-commutative function spaces. Moreover, our Banach space notion of sparsity applies to affine group actions. The generalized approach in particular applies to high-order tensor products.

scholarly journals CUMULATIVE MEASURE OF CORRELATION FOR MULTIPARTITE QUANTUM STATES

2014 ◽  
Vol 28 (07) ◽  
pp. 1450050 ◽  
Author(s):  
ANDRÉ L. FONSECA DE OLIVEIRA ◽  
EFRAIN BUKSMAN ◽  
JESÚS GARCÍA LÓPEZ DE LACALLE
Keyword(s):  
Phase Transitions ◽  
State Space ◽  
Present Article ◽  
Critical Points ◽  
Quantum Phase Transitions ◽  
The State ◽  
Quantum States ◽  
Quantum Phase ◽  
Mixed States ◽  
Different Dimensions

The present article proposes a measure of correlation for multiqubit mixed states. The measure is defined recursively, accumulating the correlation of the subspaces, making it simple to calculate without the use of regression. Unlike usual measures, the proposed measure is continuous additive and reflects the dimensionality of the state space, allowing to compare states with different dimensions. Examples show that the measure can signal critical points (CPs) in the analysis of Quantum Phase Transitions (QPTs) in Heisenberg models.

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