Quantum computing theory is at the intersection of math, physics and computer science. Experiment also can involve electrical engineering. Eventually you will want to learn aspects of all of these fields, but when starting you can use any for an entry into the field.

Quantum logic In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac[25] David Hilbert[26] John von Neumann[27] and Hermann Weyl[28] the possible states of a quantum mechanical system are symbolized [29] as unit vectors called state vectors.

Formally, these reside in a complex separable Hilbert space —variously called the state space or the associated Hilbert space of the system—that is well defined up to a complex number of norm 1 the phase factor.

In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes.

Each observable is represented by a maximally Hermitian precisely: Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the Quantum cryptography research papers spectrum is discrete, the observable can attain only those discrete eigenvalues.

In the formalism of quantum Quantum cryptography research papers, the state of a system at a given time is described by a complex wave functionalso referred to as state vector in a complex vector space.

For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variablessuch as position and momentum, to arbitrary precision.

For instance, electrons may be considered to a certain probability to be located somewhere within a given region of space, but with their exact positions unknown.

Contours of constant probability density, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.

The possible results of a measurement are the eigenvalues of the operator representing the observable—which explains the choice of Hermitian operators, for which all the eigenvalues are real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.

Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. The probabilistic nature of quantum mechanics thus stems from the act of measurement.

This is one of the most difficult aspects of quantum systems to understand.

It was the central topic in the famous Bohr—Einstein debatesin which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied.

Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangledso that the original quantum system ceases to exist as an independent entity.

For details, see the article on measurement in quantum mechanics. Instead, it makes a prediction using a probability distribution ; that is, it describes the probability of obtaining the possible outcomes from measuring an observable.

Often these results are skewed by many causes, such as dense probability clouds.

Probability clouds are approximate but better than the Bohr model whereby electron location is given by a probability functionthe wave function eigenvaluesuch that the probability is the squared modulus of the complex amplitudeor quantum state nuclear attraction.

Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as eigenstates of the observable "eigen" can be translated from German as meaning "inherent" or "characteristic".

Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence.

However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum since they are conjugate pairs or its energy and time since they too are conjugate pairs.Providing researchers with access to millions of scientific documents from journals, books, series, protocols and reference works.

International Journal of Scientific and Research Publications, Volume 2, Issue 6, June 1 ISSN benjaminpohle.com Research Directions in Quantum Cryptography and Quantum Key Distribution Ms. Deepa Harihar Kulkarni Assistant Professor, SKN College of Engineering Research Directions in Quantum Cryptography and Quantum Key.

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Support every statement you make with evidence. Because this is a research paper, there shouldn’t be any remarks that you make that cannot be supported by facts directly from your research.

"For the beginning student in quantum computation, the text is particularly useful for its breadth, background, and computations. The book covers the basic topics in enough depth and detail for students to continue on to research papers.

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