## Quantum algorithmic information theory (1996)

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Venue: | Journal of Universal Computer Science |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Svozil96quantumalgorithmic,

author = {K. Svozil},

title = {Quantum algorithmic information theory},

journal = {Journal of Universal Computer Science},

year = {1996},

volume = {2},

pages = {311--346}

}

### OpenURL

### Abstract

### Citations

1694 | An Introduction to Kolmogorov Complexity and its Applications
- Li, Vitanyi
- 1993
(Show Context)
Citation Context ...anishing phases ��;l in the generalized diagonalization process. 5 Quantum algorithmic information Quantum algorithmic information theory can be developed in analogy to algorithmic information the=-=ory [29, 28, 23, 68]-=-. Before proceeding, though, one decisive strategic decision concerning the physical character of the program has to be made. This amounts to a restriction to purely classical prefix-free programs. Th... |

1175 |
On computable numbers, with an application to the Entscheidungs problem
- Turing
- 1936
(Show Context)
Citation Context ...s apparently first been applied by Cantor to demonstrate the non-enumerability of real numbers [25]. It has also been used by Turing for a proof of the recursive undecidability of the halting problem =-=[97]-=-. A brief review of the classical algorithmic argument will be given first. Consider a universal computer C. For the sake of contradiction, consider an arbitrary algorithm B(X) whose input is a string... |

978 |
Quantum Field Theory
- Itzykson, Zuber
- 1980
(Show Context)
Citation Context ...at such multi-particle, multiquanta systems appropriately? The answer is yes. Experiment and theoretical reasoning (the representation theory of the Lorentz group [86] and the spin-statistics theorem =-=[56, 71, 17, 54]) indicate-=- that there are (at least) two basic types of states (quanta, particles): bosonic and fermionic states. Bosonic states have what is called "integer spin;" i.e., s b = 0;~;2~;3~; : : :, where... |

838 |
Theory of Recursive Functions and Effective Computability
- ROGERS
- 1967
(Show Context)
Citation Context ...ithout loss of generality, we consider only empty input strings. jpj stands for the length of p. 4.6 Diagonalization This is neither the place for a comprehensive review of the diagonalization method =-=[83, 77]-=-, nor suffices the author's competence for such an endeavor. Therefore, only a few hallmarks are stated. As already Godel pointed out in his classical paper on the incompleteness of arithmetic [47], t... |

809 | Algorithms for quantum computation: discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...nomial time complexity class with respect to variations of "reasonable" models of computation. In particular, factoring may require polynomial time on quantum computers within "reasonab=-=le statistics" [87]-=-. I would suspect that it is wise of mathematicians and computer scientists to keep an eye on new developments in physics, just as we physicists are required to be open for the great advances in the f... |

732 |
The principles of quantum mechanics
- Dirac
- 1958
(Show Context)
Citation Context ...the volume of phase space has the dimension h f . 3 Introductions to quantum mechanics can be found in Feynman, Leighton & M. Sands [44], Harris [52], Lipkin [69], Ballentine [3], Messiah [74], Dirac =-=[38]-=-, Peres [78], von Neumann [99], and Bell [5], among many other expositions. The history of quantum mechanics is reviewed by Jammer [55]. Wheeler & Zurek [100] published a helpful resource book. 4 The ... |

690 |
Quantum Theory: Concepts and Methods
- Peres
- 1995
(Show Context)
Citation Context ...f phase space has the dimension h f . 3 Introductions to quantum mechanics can be found in Feynman, Leighton & M. Sands [44], Harris [52], Lipkin [69], Ballentine [3], Messiah [74], Dirac [38], Peres =-=[78]-=-, von Neumann [99], and Bell [5], among many other expositions. The history of quantum mechanics is reviewed by Jammer [55]. Wheeler & Zurek [100] published a helpful resource book. 4 The bra-ket nota... |

658 | Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer
- Deutsch
- 1985
(Show Context)
Citation Context ...and p f q = 0 (psq stands for the code of s). The states can, for instance, be realized by some condenser who is discharged (j cbit state 0) or charged (j cbit state 1). In quantum information theory =-=[1, 34, 43, 6, 7, 35, 36]-=-, the most elementary unit of information is the quantum bit, henceforth called qbit. Qbits can be physically represented by a coherent superposition of the two orthonormal 8 states t and f . The qbit... |

609 |
Quantum Cryptography: Public Key Distribution and Coin Tossing
- Bennett, Brassard
- 1984
(Show Context)
Citation Context ...y adds noise at the same time. This noise can be interpreted as spontaneous emission of field quanta (photons) in the process of amplification. One application of this feature is quantum cryptography =-=[13, 12, 11]-=-. Thereby, the impossibility to copy qbits is used for a cryptographic communication via quantum channels. 4.2.2 Context dependence of qbits This section could be skipped at first reading. Assume that... |

590 |
Can Quantum-Mechanical Description of Physical Reality
- Einstein, Podolsky, et al.
- 1935
(Show Context)
Citation Context ... to copy qbits is used for a cryptographic communication via quantum channels. 4.2.2 Context dependence of qbits This section could be skipped at first reading. Assume that in an EPR-type arrangement =-=[41]-=- one wants to measure the product P =m 1 x m 2 x m 1 y m 2 y m 1 z m 2 z of the direction of the spin components of each one of the two associated particles 1 and 2 along the x, y and z-axes. Assume t... |

552 |
Speakable and Unspeakable in Quantum Mechanics
- Bell
- 1987
(Show Context)
Citation Context ... f . 3 Introductions to quantum mechanics can be found in Feynman, Leighton & M. Sands [44], Harris [52], Lipkin [69], Ballentine [3], Messiah [74], Dirac [38], Peres [78], von Neumann [99], and Bell =-=[5]-=-, among many other expositions. The history of quantum mechanics is reviewed by Jammer [55]. Wheeler & Zurek [100] published a helpful resource book. 4 The bra-ket notation introduced by Dirac is wide... |

482 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
(Show Context)
Citation Context ...l ones [59]. Furthermore, in order to obtain sufficient statistical data, a "great" (non-polynomially bounded) number of single particles may be needed [91]. We shall not pursue these matter=-=s further [36, 14, 15, 6, 27, 87]-=-. 4.5 Nomenclature Consider a (not necessarily universal) quantum computer C and its ith program p i , which, at time t 2 Z, can be described by a quantum state C(t; p i ). Let C(p) = s stand for a co... |

462 |
Logical Reversability of Computation
- Bennett
- 1973
(Show Context)
Citation Context ...reby violating quantum mechanics. 4 Quantum recursion theory 4.1 Reversible computation and deletion of (q)bits As a prelude to quantum computation, we briefly review classical reversible computation =-=[61, 8, 45, 9, 66]-=-. This type of computation is characterized by a single-valued inverse transition function. That is, logical functions are performed which do not have a single-valued inverse, such as AND or OR; i.e.,... |

436 |
Mathematische Grundlagen der Quantenmechanik
- Neumann
- 1932
(Show Context)
Citation Context ...exotic' nomenclature prevalent in physics--the Dirac bra-ket notation--the notation of Dunford-Schwartz [39] is adopted. 4 All quantum mechanical entities are represented by objects of Hilbert spaces =-=[99]-=-. A Hilbert space is a linear vector space H over the field F of complex numbers (with vector addition and scalar multiplication), together with a complex function (\Delta; \Delta), the scalar or inne... |

407 | Foundations of Constructive Analysis
- Bishop
- 1967
(Show Context)
Citation Context ...s. Basic to all these considerations is quantum theory, which is most conveniently expressible in Hilbert space. 1 Information is physical, so is computation The reasoning in constructive mathematics =-=[16, 19, 20]-=- and recursion theory, at least insofar as their applicability to worldly things is concerned, makes implicit assumptions about the operationalizability of the entities of discourse. It is this postul... |

397 | Simulating physics with computers
- Feynman, Shor
- 1982
(Show Context)
Citation Context ...ntained. After the processing, the result is obtained by an irreversible measurement. The processing of qbits, however, would require an exponential space overhead on classical computers in cbit base =-=[42]. Thus, in-=- order to remain tractable, the corresponding qbits should be implemented on truly quantum universal computers. It should be noted, however, that the fixed point qbit "solution" to the above... |

391 | Uber formal unentscheidbare Satze der Principia Mathematica und verwadter Systeme I”, Monatshefte fur Mathematik und Physik 38: 173-198. For the English version: “On formally undecidable propositions - Godel - 1931 |

344 |
Irreversibility and heat generation in the computing process
- Landauer
- 2000
(Show Context)
Citation Context ...reby violating quantum mechanics. 4 Quantum recursion theory 4.1 Reversible computation and deletion of (q)bits As a prelude to quantum computation, we briefly review classical reversible computation =-=[61, 8, 45, 9, 66]-=-. This type of computation is characterized by a single-valued inverse transition function. That is, logical functions are performed which do not have a single-valued inverse, such as AND or OR; i.e.,... |

342 |
Rapid solution of problems by quantum computation
- Deutsch, Jozsa
- 1992
(Show Context)
Citation Context ...and p f q = 0 (psq stands for the code of s). The states can, for instance, be realized by some condenser who is discharged (j cbit state 0) or charged (j cbit state 1). In quantum information theory =-=[1, 34, 43, 6, 7, 35, 36]-=-, the most elementary unit of information is the quantum bit, henceforth called qbit. Qbits can be physically represented by a coherent superposition of the two orthonormal 8 states t and f . The qbit... |

323 | Algorithmic Information Theory
- Chaitin
- 1987
(Show Context)
Citation Context ...gs over A. Any program p i is coded as a classical sequence pp i q = s 1i s 2i \Delta \Delta \Delta s ni 2 A , s ji 2 A. Whenever possible, pp i q will be abbreviated by p i . We assume prefix coding =-=[51, 29, 28, 92, 23]-=-; i.e., the domain of C is prefix-free such that no admissible program is the prefix of another admissible program. Furthermore, without loss of generality, we consider only empty input strings. jpj s... |

312 |
Classical Recursion Theory I
- Odifreddi
- 1989
(Show Context)
Citation Context ...ithout loss of generality, we consider only empty input strings. jpj stands for the length of p. 4.6 Diagonalization This is neither the place for a comprehensive review of the diagonalization method =-=[83, 77]-=-, nor suffices the author's competence for such an endeavor. Therefore, only a few hallmarks are stated. As already Godel pointed out in his classical paper on the incompleteness of arithmetic [47], t... |

258 |
D.V.Shirkov, Introduction to the Theory of Quantized Fields
- Bogoliubov
- 1959
(Show Context)
Citation Context ...at such multi-particle, multiquanta systems appropriately? The answer is yes. Experiment and theoretical reasoning (the representation theory of the Lorentz group [86] and the spin-statistics theorem =-=[56, 71, 17, 54]) indicate-=- that there are (at least) two basic types of states (quanta, particles): bosonic and fermionic states. Bosonic states have what is called "integer spin;" i.e., s b = 0;~;2~;3~; : : :, where... |

246 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
Citation Context ...and p f q = 0 (psq stands for the code of s). The states can, for instance, be realized by some condenser who is discharged (j cbit state 0) or charged (j cbit state 1). In quantum information theory =-=[1, 34, 43, 6, 7, 35, 36]-=-, the most elementary unit of information is the quantum bit, henceforth called qbit. Qbits can be physically represented by a coherent superposition of the two orthonormal 8 states t and f . The qbit... |

223 |
The problem of hidden variables in quantum mechanics
- Kochen, Specker
- 1967
(Show Context)
Citation Context ...present. A classical bit remains unchanged, no matter by what methods it is inferred. It obeys classical logic. It can be copied. No doubts can be left. By contrast, quantum information is contextual =-=[57, 58]. A quantu-=-m bit may appear different, depending on the method by which it is inferred. Quantum bits cannot be copied or "cloned" [102, 37, 72, 75, 46, 26]. Classical tautologies are not necessarily sa... |

222 |
Quantum mechanical computers
- Feynman
- 1985
(Show Context)
Citation Context |

219 |
Conservative Logic
- Fredkin, Toffoli
- 1982
(Show Context)
Citation Context ...reby violating quantum mechanics. 4 Quantum recursion theory 4.1 Reversible computation and deletion of (q)bits As a prelude to quantum computation, we briefly review classical reversible computation =-=[61, 8, 45, 9, 66]-=-. This type of computation is characterized by a single-valued inverse transition function. That is, logical functions are performed which do not have a single-valued inverse, such as AND or OR; i.e.,... |

201 | Experimental quantum cryptography
- Bennett, Bessette, et al.
- 1992
(Show Context)
Citation Context ...y adds noise at the same time. This noise can be interpreted as spontaneous emission of field quanta (photons) in the process of amplification. One application of this feature is quantum cryptography =-=[13, 12, 11]-=-. Thereby, the impossibility to copy qbits is used for a cryptographic communication via quantum channels. 4.2.2 Context dependence of qbits This section could be skipped at first reading. Assume that... |

151 |
Quantum Theory and Measurement
- Wheeler, Zurek
- 1983
(Show Context)
Citation Context ... [69], Ballentine [3], Messiah [74], Dirac [38], Peres [78], von Neumann [99], and Bell [5], among many other expositions. The history of quantum mechanics is reviewed by Jammer [55]. Wheeler & Zurek =-=[100]-=- published a helpful resource book. 4 The bra-ket notation introduced by Dirac is widely used in physics. To translate expressions into the bra-ket notation, the following identifications work for mos... |

145 |
A single quantum cannot be cloned, Nature 299
- Wootters, Zurek
- 1982
(Show Context)
Citation Context ...ts can be left. By contrast, quantum information is contextual [57, 58]. A quantum bit may appear different, depending on the method by which it is inferred. Quantum bits cannot be copied or "clo=-=ned" [102, 37, 72, 75, 46, 26]-=-. Classical tautologies are not necessarily satisfied in quantum information theory. Quantum bits obey quantum logic. And, as has been argued before, they are coherent superpositions of classical info... |

136 |
The Philosophy of Quantum Mechanics
- Jammer
- 1974
(Show Context)
Citation Context ...], Harris [52], Lipkin [69], Ballentine [3], Messiah [74], Dirac [38], Peres [78], von Neumann [99], and Bell [5], among many other expositions. The history of quantum mechanics is reviewed by Jammer =-=[55]-=-. Wheeler & Zurek [100] published a helpful resource book. 4 The bra-ket notation introduced by Dirac is widely used in physics. To translate expressions into the bra-ket notation, the following ident... |

128 | Die gegenwärtige situation in der quantenmechanik - Schrödinger - 1935 |

116 |
Computability & unsolvability
- Davis
- 1958
(Show Context)
Citation Context ...such an endeavor. Therefore, only a few hallmarks are stated. As already Godel pointed out in his classical paper on the incompleteness of arithmetic [47], the undecidability theorems of formal logic =-=[31]-=- (and the theory of recursive functions [83, 77]) are based on semantical paradoxes such as the liar [2] or Richard's paradox. A proper translation of the semantic paradoxes results in the diagonaliza... |

107 |
The unreasonable effectiveness of Mathematics in the Natural Sciences
- Wigner
- 1960
(Show Context)
Citation Context ...es this irreversible observation take place? Since the physical laws (with the possible exception of the weak force) are time-reversible, the act of irreversible observation must, according to Wigner =-=[101]-=-, occur in the consciousness, thereby violating quantum mechanics. 4 Quantum recursion theory 4.1 Reversible computation and deletion of (q)bits As a prelude to quantum computation, we briefly review ... |

85 |
Theory of Photons and Electrons
- Rohrlich
(Show Context)
Citation Context ...at such multi-particle, multiquanta systems appropriately? The answer is yes. Experiment and theoretical reasoning (the representation theory of the Lorentz group [86] and the spin-statistics theorem =-=[56, 71, 17, 54]) indicate-=- that there are (at least) two basic types of states (quanta, particles): bosonic and fermionic states. Bosonic states have what is called "integer spin;" i.e., s b = 0;~;2~;3~; : : :, where... |

82 |
The thermodynamics of computation--a review
- Bennett
- 1982
(Show Context)
Citation Context |

79 |
Information is Physical
- Landauer
- 1991
(Show Context)
Citation Context ...tructive [16, 19, 96]) formalism should correspond to some physically operationalizable [21] property. Hence, any theory of information, if applicable, has to deal with entities which are operational =-=[21, 64, 62, 60, 65]. In -=-Bridgman's words [22, p. V], "the meaning of one's terms are to be found by an analysis of the operations which one performs in applying the term in concrete situations or in verifying the truth ... |

77 |
Quantum mechanical Hamiltonian models of Turing machines
- Benioff
- 1982
(Show Context)
Citation Context |

73 |
Experimental realization of any discrete unitary operator
- Reck, Zeilinger, et al.
- 1994
(Show Context)
Citation Context ...ned in (V) below. It has recently been demonstrated that (by an analog embodiment using particle beams) every selfadjoint operator in a finite dimensional Hilbert space can be experimentally realized =-=[82]-=-. (III) The result of any single measurement of the observable A on a state x 2 H can only be one of the real eigenvalues of the corresponding Hermitean operator A. If x is in a coherent superposition... |

65 |
Quantum Mechanics
- Ballentine
- 1989
(Show Context)
Citation Context ...cording to the fact that the volume of phase space has the dimension h f . 3 Introductions to quantum mechanics can be found in Feynman, Leighton & M. Sands [44], Harris [52], Lipkin [69], Ballentine =-=[3]-=-, Messiah [74], Dirac [38], Peres [78], von Neumann [99], and Bell [5], among many other expositions. The history of quantum mechanics is reviewed by Jammer [55]. Wheeler & Zurek [100] published a hel... |

64 | The Undecidable - Davis - 1965 |

53 | The Quantum Challenge to Structural Complexity Theory
- Berthiaume, Brassard
(Show Context)
Citation Context ...l ones [59]. Furthermore, in order to obtain sufficient statistical data, a "great" (non-polynomially bounded) number of single particles may be needed [91]. We shall not pursue these matter=-=s further [36, 14, 15, 6, 27, 87]-=-. 4.5 Nomenclature Consider a (not necessarily universal) quantum computer C and its ith program p i , which, at time t 2 Z, can be described by a quantum state C(t; p i ). Let C(p) = s stand for a co... |

53 |
Constructive Mathematics
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...s. Basic to all these considerations is quantum theory, which is most conveniently expressible in Hilbert space. 1 Information is physical, so is computation The reasoning in constructive mathematics =-=[16, 19, 20]-=- and recursion theory, at least insofar as their applicability to worldly things is concerned, makes implicit assumptions about the operationalizability of the entities of discourse. It is this postul... |

53 |
Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt
- Einstein
- 1905
(Show Context)
Citation Context ... fundamental constant h; i.e., e = hn. That was a bold step in a time of the predominant continuum models of classical mechanics. In extension of Planck's discretized resonator energy model, Einstein =-=[40]-=- proposed a quantization of the electromagnetic field. Every field mode of frequency n could carry a discrete number of light quanta of energy hn per quantum. The present quantum theory is still a con... |

51 |
Communication by EPR devices
- Dieks
(Show Context)
Citation Context ...ts can be left. By contrast, quantum information is contextual [57, 58]. A quantum bit may appear different, depending on the method by which it is inferred. Quantum bits cannot be copied or "clo=-=ned" [102, 37, 72, 75, 46, 26]-=-. Classical tautologies are not necessarily satisfied in quantum information theory. Quantum bits obey quantum logic. And, as has been argued before, they are coherent superpositions of classical info... |

48 |
Quantum cryptography, or unforgeable subway tokens. CRYPTO
- Bennett, Brassard, et al.
- 1982
(Show Context)
Citation Context ...y adds noise at the same time. This noise can be interpreted as spontaneous emission of field quanta (photons) in the process of amplification. One application of this feature is quantum cryptography =-=[13, 12, 11]-=-. Thereby, the impossibility to copy qbits is used for a cryptographic communication via quantum channels. 4.2.2 Context dependence of qbits This section could be skipped at first reading. Assume that... |

42 |
Quantum Mechanics
- Davydov
- 1965
(Show Context)
Citation Context ...ic oscillator potential, which can be solved wonderfully by these methods (and they are presented over and over again in standard courses of quantum mechanics), but not many more. (See, for instance, =-=[33]-=- for a detailed treatment of various Hamiltonians H.) For a quantum mechanical treatment of a two-state system, see appendix A. For a review of the quantum theory of multiple particles, see appendix B... |

37 | Theory of Recursive Functions and E ective Computability - Rogers - 1967 |

34 |
What’s wrong with these elements of reality
- Mermin
- 1990
(Show Context)
Citation Context ...and z-axes. Assume that the operators are normalized such that jm j i j = 1, i 2 fx;y;zg, j 2 f1;2g. One way to determine P is measuring and, based on these measurements, "counterfactually inferr=-=ing" [78, 73]-=- the 11 Copying of qbits would allow circumvention of the Heisenberg uncertainty relation by measuring two incompatible observables on two identical qbit copies. It would also allow faster-than-light ... |

31 |
Information and Randomness:An Algorithmic perspective, 2 nd Ed.; Springer-Verlag
- Calude
- 2002
(Show Context)
Citation Context ...gs over A. Any program p i is coded as a classical sequence pp i q = s 1i s 2i \Delta \Delta \Delta s ni 2 A , s ji 2 A. Whenever possible, pp i q will be abbreviated by p i . We assume prefix coding =-=[51, 29, 28, 92, 23]-=-; i.e., the domain of C is prefix-free such that no admissible program is the prefix of another admissible program. Furthermore, without loss of generality, we consider only empty input strings. jpj s... |

31 | A formal theory of inductive inference: Part
- Solomonoff, J
- 1964
(Show Context)
Citation Context ...y, the computer could be initially prepared in the non-halting state f . After completion of the task, the halting bit is again switched to the halting state t. In analogy to the fully classical case =-=[29, 28, 88, 23], th-=-e quantum halting amplitude 15 W can be defined as a weighted expectation over all computations of C with classical input p i (jp i j stands for the length of p i ) Wj �� C(p )2H 2 \Gammaj p j=2 (... |