Notes
[Notebook: Transfinite Field Theory DB 56]
[Sunday 9 November 2003 - Saturday 15 November 2003]
[page 12]
Sunday 9 November 2003
[page 13]
Reincarnation: periodic function in a constrained space, potential well. Potential is the wheel that keeps incarnating a particle until it finds a way to tunnel out of the potential and become free, acyclic, transfinite.
Truth lies in wilderness. Security lies in wilderness.
Knowledge reduces uncertainty a a point where a knowing system can survive.
Long ago (1980s) the idea that metaphysics is mathematics began to take shape. Now it seems that quantum theory is metaphysics, raising the question: does quantum theory embrace the whole of mathematics. We look for answers in Nielsen and Chuang, Quantum Computation and Quantum information. Nielsen and Chuang Aristotle. Aquinas and Lonergan were all familiar with classical computation and information and formalized many classical theorems of logic. What they missed was the recursive nature of dynamics introduced by Cantor's theorem on the generation of transfinite numbers. Quantum mechanics shows us how these formal systems are actually realized in the physical world. So what we want to say is that quantum mechanics is a realization of mathematics via computational processes.
Nielsen and Chuang page 2: 'Quantum mechanics is a mathematical framework or set of rules for the construction of physical theories/', ie [quantum mechanics] is a metatheory, ie a metaphysics.
page 3: no-cloning relates to non-communication above c [the velocity of light]
developing technology of control of single quantum systems: atoms, electrons, quantum dots, Josephson junctions, etc.
[page 14]
Nielsen and Chuang page 6: 'At the time of writing it is not clear whether Deutsch's notion of a Universal Quantum Computer is sufficient to efficiently simulate an arbitrary physical system.' Deutsch
Deutsch
Shor: factorizing integer and discrete logarithm problem. Shor
Lou Grover: fast search. Grover
page 7: Quantum computers can model quantum mechanical problems.
'What is it that makes quantum computers more powerful than classical computer?' Maybe because quantum or 'natural' using unbounded operators, whereas classical is 'artificial' using bounded operators.
page 8: Shannon: noiseless coding theorem; noisy coding theorem. Shannon
Schumacher: Quantum analogue of noiseless coding theorem (1985)
No analogue yet known for the noisy channel theorem, but we have a theory of quantum error correction.
page 9: 'The theory of quantum error correcting codes was developed to protect quantum states against noise.
superdense coding.
'Recently it has been shown that quantum mechanics can require exponentially less communication to solve certain problems than would be required if the networked computers were classical.
'no unifying theory of networked information theory exists for quantum channels.
One example: reversed channels overcome zero capacity [the foundation of cycles]
['imagine that you are attempting to send quantum information from Alice to Bob through a noisy quantum channel. If that channel has zero capacity for quantum information, then it is impossible to reliable send any information from Alice to Bob. Imagine instead that we consider two copies of the channel, operating in synchrony. Intuitively it is clear (and can be rigorously justified) that such a channel has zero capacity to send quantum information. However, if we instead reverse the direction of one of the channels . . . it turns out that sometimes we can obtain non-zero capacity for the transmission of information from Alice to Bob. Counter-intuitive properties like this illustrate the strange nature of quantum information.']
[page 15]
Quantum cryptography: an eavesdropper disturbs the signal.
Nielsen and Huang page 11: Public key cryptography. Wikipedia
Quantum entanglement: a resource
'In recent years there has been a tremendous effort trying to better understand the properties of entanglement considered as a fundamental resource of Nature of comparable importance to energy, information, entropy or any other fundamental resource' (?).
page 12: '. . . in the broadest terms we have learned that any physical theory, not just quantum mechanics, maybe used as a basis doe a theory of information processing and communication.' (as long as it has a rich enough array of symbols (particles, states) and transformations.
We have learnt to think physically about computation, and computationally about physics.
page 13: 'the beauty of treating qubits as abstract entities is that it gives us the freedom to construct a very general theory of quantum computation and quantum information which does not depend on a specific system for its realization.'
'i. . . in general a qubit's state is a unit vector in a two dimensional complex vector space' (which may be mapped onto a 4D real space).
'The dichotomy between the unobservable state of a qubit and the observations we can make lies at the heart of quantum information and quantum computation. In most of our abstract models of the world there is a direct correspondence between the elements of the abstraction and the real world. . . . The lack
[page 16]
of this direct correspondence in quantum mechanics makes it difficult to intuit the behaviour of quantum systems; however thee us an indirect correspondence, for qubit states can be manipulated and transformed in ways which lead to measurement outcomes which depend distinctly on the different properties of the state. Thus these quantum states have real, experimentally verified consequences. . . '
Nielsen and Huang page 14: 'when a qubit is measures, it only ever give '0' or '1' as the measurement result, probabilistically.'
page 15: 'furthermore, measurement changes the state of a qubit, collapsing it from its superposition of |0> and 1> to a specific state consistent with the measurement result. .. Why does this type of collapse occur? Nobody knows. . . . this behaviour is simply one of the fundamental postulates of quantum mechanics.'
page 16: 'Nature conceals a great deal of hidden information, and even more interestingly, we will see shortly that the potential amount of this hidden information grows exponentially with the number of qubits. Understanding this hidden quantum information is a question that we grapple with for much of this book., and which lies a the heart of what makes quantum mechanics a powerful tool for information processing.'
page 17: The measured correlations in the Bell state are stronger than could ever exist between classical systems.
page 18: Quantum NOT gate: a)> + b|1> goes to a1> + b|0>, ie it is linear. . . .
An example of the permutation of two objects. So we may thing of quantum computation as quantum mechanics in a restricted Hilbert space where ℵ0 = 2. The joining of such spaces to make strings of qubits is artificial, since the natural world (?) operates in the unrestricted space within which are 2D Hilbert spaces such as those representing pin 1/2 states.
[page 17]
We can conduct metaphysics and psychology, like quantum mechanics, in terms of spaces of states, transformations of states, paces of transformations of states, transformations of transformations of states, and so on.
'. . . quantum
Distance - path - algorithm.
Our metaphysical foundation is the proposition 'knowledge exists, so knowledge is possible. What feature of the Universe makes knowledge possible? Recursion.
