Quantum Physics and Quantum Computation
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Entropy Gain and Information Loss by Measurements
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Preprint: Current (08/22/2019)
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Abstract
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When the von Neumann entropy of a quantum system is increased by measurements, part of its information is lost.
To faithfully reflect such a gain-loss relation, we propose the Information Retrievability (IR) and Information Loss (IL),
which depend only on the density matrix of the system before and after measurements. We explain that,
after a pure quantum state collapses to a maximal mixed state, it gets the maximal entropy together with the maximal info loss,
and its discrete uniform distribution contains only classical info. Then we compute the entropy, IR and IL for systems of single-qubit,
entangled qubits (like Bell, GHZ, W state) and the 2-qubit Werner (mixed) state with their dependence on various parameters.
We notice that, since the data-exchange between the two observers in Bell tests can recover certain critical quantum info,
the related quantum entropy should be removable (a possible dilemma linked to quantum non-locality).
We show that, measuring the Bell, GHZ and the marginally entangled Werner state will produce the same minimal entropy gain,
accompanied by equal minimal info loss.
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Reduced Matrix, Operator Chain and IO Kets: New Formats to Demystify Quantum Logic Circuits
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Preprint: Current (12/2019)
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Abstract
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Given the diagram of a 4-qubit circuit with 5 controlled gates, like a quantum full adder,
how do you derive its input-output (IO) logic and its 16 x 16 matrix representation?
Here we first propose the Reduced Matrix Format, representing most m-qubit gates by m x m diagonal matrices.
Next we introduce the Operator Chain Format, which represents any n-gate circuit by a chain of n-operators,
making it easy to derive its IO logic. Moreover, with the help of the decimal state notation,
the unitary matrix of the circuit now can be simply written in the one row IO Kets Format.
Our new formats are then applied to various popular multi-qubit gates and to several important multi-gate circuits.
Finally, as a demonstration, we demystify the quantum full adder by using our formats
to deriving its quantum logic and unitary matrix.
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Applied Mathematics and Mathematical Physics:
From Dirac Notation to Probability Bracket Notation
Author: Dr. Xing M (Sherman) Wang
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Dirac notation (or Bra-ket notation) is a very powerful and indispensable tool for modern physicists.
Unfortunately it is only taught in
Quantum Mechanics . I believe it would be great to introduce it in Applied Mathematics (like Linear Algebra).
On the other hand, while studying probability theories, I felt that it would be very helpful if we had a similar notation to represent or derive probabilistic formulas.
That was why I posted following articles online, in which Dirac Notation was introduced to IR, and Probability Bracket Notation was proposed and applied to IR, too.
No mater you agree or disagree with my work, I welcome and appreciate your opinions.
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| How to email to the author? |
| Subject line: | About the articles on your web site |
| Email address: | swang (at) shermanlab (dot) com,
or from arxiv.org if you are a member |
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Probability Bracket Notation, Wick-Matsubara Relation, Density Operators,
and Microscopic Probability Modeling
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arXiv. 09/112024, : Current (08/20/2024)
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Abstract
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Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN)
in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary
Schrodinger equation in the Hilbert space transforms into the master equation of a microscopic probabilistic
process (MPP) in the probability space. In this article, we first study the MPP of the system of a single particle,
we show that the energy expectation of the MPP eventually approaches the lowest energy level in its initial
condition and its von Neumann entropy finally vanishes. Then we explore the MPP for the quantum system of
identical particles in the Fock space, we recover the expected occupation number of particles and the
grand partition function in quantum statistics by connecting time with temperature (the Wick-Matsubara relation).
We also reproduce the internal energy of an ideal gas in thermodynamics by using the relation.
To address the entropy issue and relate the PBN with research topics of statistics in the literature,
we express the density operators and the von Neumann entropy in the probability space by using the PBN.
The Wick-Matsubara relation plus the PBN might provide a new way of microscopic probability modeling.
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Probability Brackets Notation for Probability Modeling
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Axioms 2024, 13(8), 564PDF: Current (08/20/2024)
(Published in a Special Issue:
"Stochastic Processes in Quantum Mechanics and Classical Physics")
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Abstract
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Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN),
by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae,
such as normalizations and expectations in systems of one or more random variables, can now be written
in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity.
The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets
are identified with probability vectors and our P-bra system is comparable with Doi’s state function or
Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process,
and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also,
some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN
by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained
domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.
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Probability Brackets Notation: Probability Spaces, Time evolution of Markov Processes and Transition Probabilities
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PDF: Current (09/20/2021)
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Abstract
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Following Dirac’s notation in quantum mechanics (QM), we proposed Probability Bracket Notation (PBN).
In PBN, probability spaces are associated with random variables as base observables; miscellaneous formulas
(like normalizations and expectations) can be written in compact, basis-independent expressions;
the time evolution of Markov processes can also be structured in a similar way.
Transforming from Schrodinger's picture to Heisenberg's, the base observable becomes time-dependent, representing a stochastic process;
many important relations for transition probabilities of Markov processes, such as Chapman-Kolmogorov equations,
can be obtained by just inserting proper time-dependent P-identities.
We hope that PBN could provide an alternative way to study and explore probability modeling, especially for those familiar with QM.
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Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling
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PDF: Current (11/07/2009)
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Abstract
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After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional
density operator (CDO), we investigate probability spaces associated with various quantum systems: system
with one observable (discrete or continuous), system with two commutative observables (independent or
dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive
unified expressions of conditional expectation (CE), conditional probability (CP), and absolute
probability (AP): they have the same format for discrete or continuous spectrum;
they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN);
and they may be useful to deal with CE of non-commutative observables.
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From Dirac Notation to Probability Bracket Notation: Time Evolution and Path Integral under Wick Rotations
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PDF: Current (01/29/2009)
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Abstract
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In this article, we continue to investigate the application of Probability Bracket Notation (PBN).
We show that, under Special Wick Rotation (caused by imaginary-time rotation), the Schrodinger equation of a conservative system
and its path integral in Dirac rotation are simultaneously shifted to the master equation and its Euclidean path integral of an induced micro diffusion in PBN.
Moreover, by extending to General Wick Rotation and using the anti-Hermitian wave-number operator, we execute the path integral
in Dirac notation side-by-side with the Euclidean path integral in PBN, and derive the Euclidean Lagrangian of induced diffusions
and Smoluchowski equation.
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Probability Bracket Notation and Probability Modeling (a short version of the first article)
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PDF: Current (11/07/2009)
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Abstract
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Inspired by the Dirac notation, a new set of symbols, the Probability Bracket Notation (PBN) is proposed for probability modeling. By applying PBN
to discrete and continuous random variables, we show that PBN could play a similar role in probability spaces as the Dirac notation in Hilbert vector spaces.
The time evolution of homogeneous Markov chains with discrete-time and continuous-time are discussed in PBN. Our system state p-kets are identified with
the probability vectors, while our system state p-bra can be identified with Doi's state function and Peliti's standard bra. We also suggest that,
by transforming from the Schrodinger picture to the Heisenberg picture, the time-dependence of a system p-ket of a homogeneous MC can be shifted to the
observable as a stochastic process.
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Errata (09/06/2012)
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Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales
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PDF: Current (07/16/2007)
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Abstract
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In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles.
After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces
associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of
conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN.
In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory.
Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN.
We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful
tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.
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Probability Bracket Notation, Probability Vectors, Markov Chains and Stochestic Processes
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PDF: Current (09/04/2024)
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Abstract
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Inspired by the Dirac vector probability notation (VPN), we propose the Probability Bracket Notation (PBN),
a new set of symbols defined similarly (but not identically) as in the VPN. Applying the PBN to fundamental
definitions and theorems for discrete and continuous random variables, we show that the PBN could play a
similar role in the probability space as the VBN in the Hilbert vector. Our system P-kets are identified with
the probability vectors in Markov chains (MC). The master equation of homogeneous MC in the Schrodinger pictures
can be basis-independent. Our system P-bra is linked to the Doi state function and the Peliti standard bra.
Transformed from the Schrodinger picture to the Heisenberg picture, the time dependence of the system P-ket
of a homogeneous MC (HMC) is shifted to the observable as a stochastic process. Using the correlations
established by the special Wick rotation (SWR), the microscopic probabilistic processes (MPPs) are
investigated for single and many-particle systems. The expected occupation number of particles
in quantum statistics is reproduced by associating time with temperature (the Wick-Matsubara relation).
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More to come, please visit us again!
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