# Quantum Theory, Namely the Pure and Reversible Theory of Information

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Complete Set of Information-Theoretic Principles for Quantum Theory

**Figure 1.**Alice’s laboratory. Alice has at disposal many devices, each of them having an input system and an output system (represented by different wires) and possibly a set of outcomes labelling different processes that can take place. The devices can be connected in series and in parallel to form circuits. A circuit with no input and no output wires represents an experiment starting from the preparation of a state with a given source and ending with some measurement(s). Specifying a theory for Alice’s laboratory means specifying which are the allowed devices and specifying a rule to predict the probability of outcomes in such experiments.

**Principle 1**

**(Causality)**

**Principle 2**

**(Fine-Grained Composition)**

**Principle 3**

**(Perfect Distinguishability)**

**Principle 4**

**(Ideal Compression)**

**Figure 2.**Compressing information. Alice encodes information (here represented by a pile of books) in a suitable system carrying the smallest possible amount of data (here a USB stick). The most advantageous situation is when the compression is lossless (after the encoding Bob is able to perfectly retrieve the information) and maximally efficient (the encoding system contains only the pure states needed to convey the information compatible with ρ).

**Principle 5**

**(Local tomography)**

**Figure 3.**Local Tomography. Alice can reconstruct the state of compound systems using only local measurements on the components. A world where this property did not hold would contain global information that cannot be accessed with local experiments.

**Principle 6**

**(Purity and Reversibility of Physical Processes)**

- physical systems are associated to complex Hilbert spaces;
- the maximum number of perfectly distinguishable states of the system is equal to the dimension of the corresponding Hilbert space;
- the pure states of a system are described by the unit vectors in the corresponding Hilbert space (up to a global phase);
- the reversible processes on a system are described by the unitary operators on the corresponding Hilbert space (up to a global phase);
- the measurements on a system are described by resolutions of the identity in terms of positive operators ${\left\{{P}_{i}\right\}}_{i\in X}$ on the corresponding Hilbert space (aka POVMs, see, e.g., [30] for a didactical presentation);
- the mixed states of a system are described by density matrices on the corresponding Hilbert space;
- the probabilities of outcomes in a measurement are given by the Born rule ${p}_{i}=\mathrm{Tr}\left[{P}_{i}\rho \right]$, where ρ is the density matrix representing the system’s state and $\mathrm{Tr}$ denotes the trace of a matrix;
- the Hilbert space associated to a composite system is the tensor product of the Hilbert spaces associated to the components;
- random processes are described by completely positive trace-preserving maps.

## 3. Conservation of Information and the Purification Principle

## 4. Discussion and Conclusions

## Acknowledgments

## References

- Wheeler, J.A. ‘A Practical Tool’, but puzzling too. New York Times. 12 December 2000. Available online: http://www.nytimes.com/2000/12/12/science/12ESSA.html (accessed on 1 September 2012).
- Redei, M. Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud. Hist. Phil. Mod. Phys.
**1997**, 27, 493–510. [Google Scholar] [CrossRef] - von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1932. [Google Scholar]
- Birkhoff, G.; von Neumann, J. The logics of quantum mechanics. Ann. Math.
**1936**, 37, 823–843. [Google Scholar] [CrossRef] - Wheeler, J.A. Information, physics, quantum: The search for links. In Complexity, Entropy, and the Physics of Information; Zurek, W., Ed.; Addison-Wesley: Redwood City, CA, USA, 1990; p. 5. [Google Scholar]
- Wootters, W.K. The acquisition of information from quantum measurements. Ph.D. thesis, University of Texas at Austin, Austin, TX, USA, 1980. [Google Scholar]
- von Weizsacker, C.F. The Structure of Physics; Görnitz, T., Lyre, H., Eds.; Springer: Dodrecht, The Netherlands, 2006. [Google Scholar]
- Zeilinger, A. A foundational principle for quantum mechanics. Found. Phys.
**1999**, 29, 631–643. [Google Scholar] [CrossRef] - Brukner, Č.; Zeilinger, A. Information and fundamental elements of the structure of quantum theory. In Time, Quantum, Information; Castell, L., Ischebeck, O., Eds.; Springer: Berlin, Heidelberg, Germany, 2003; pp. 323–354. [Google Scholar]
- Wootters, W.K.; Zurek, W.H. A single quantum cannot be cloned. Nature
**1982**, 299, 802–803. [Google Scholar] [CrossRef] - Dieks, D. Communication by EPR devices. Phys. Lett. A
**1982**, 92, 271–272. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Peres, A.; Wootters, W.K. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett.
**1993**, 70, 1895–1899. [Google Scholar] [CrossRef] [PubMed] - Bennett, C.H.; Brassard, G. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India, 9–12 December 1984; pp. 175–179.
- Ekert, A.K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.
**1991**, 67, 661–663. [Google Scholar] [CrossRef] [PubMed] - Grover, L.K. A fast quantum mechanical algorithm for database search. In Proceedings of 28th Annual ACM Symposium on the Theory of Computing (STOC), Philadelphia, PA, USA, 22–24 May 1996; pp. 212–219.
- Shor, P.W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput.
**1997**, 26, 1484–1509. [Google Scholar] [CrossRef] - Fuchs, C.A. Quantum mechanics as quantum information, mostly. J. Mod. Opt.
**2003**, 50, 987–1023. [Google Scholar] [CrossRef] - Brassard, G. Is information the key? Nat. Phys.
**2005**, 1, 2–4. [Google Scholar] [CrossRef] - Pawlowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Zukowski, M. Information causality as a physical principle. Nature
**2009**, 461, 1101–1104. [Google Scholar] [CrossRef] [PubMed] - Hardy, L. Quantum theory from five reasonable axioms. arXiv
**2001**. [Google Scholar] - D’Ariano, G.M. Probabilistic theories: What is special about quantum mechanics? In Philosophy of Quantum Information and Entanglement; Bokulich, A., Jaeger, G., Eds.; Cambridge University Press: Cambridge, UK, 2010; pp. 85–126. [Google Scholar]
- Goyal, P.; Knuth, K.H.; Skilling, J. Origin of complex quantum amplitudes and Feynman’s rules. Phys. Rev. A
**2010**, 81, 022109. [Google Scholar] [CrossRef] - Dakic, B.; Bruckner, Č. Quantum theory and beyond: Is entanglement special? In Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed.; Cambridge University Press: Cambridge, UK, 2011; pp. 365–392. [Google Scholar]
- Masanes, L.; Müller, M. A derivation of quantum theory from physical requirements. New J. Phys.
**2011**, 13, 063001. [Google Scholar] [CrossRef] - Hardy, L. Reformulating and reconstructing quantum theory. arXiv
**2011**. [Google Scholar] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Informational derivation of quantum theory. Phys. Rev. A
**2011**, 84, 012311. [Google Scholar] [CrossRef] - Brukner, Č. Questioning the rules of the game. Physics
**2011**, 4, 55. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Probabilistic theories with purification. Phys. Rev. A
**2010**, 81, 062348. [Google Scholar] [CrossRef] - Coecke, B. Quantum picturalism. Contemp. Phys.
**2010**, 51, 59–83. [Google Scholar] [CrossRef] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Hardy, L. Foliable operational structures for general probabilistic theories. In Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed.; Cambridge University Press: Cambridge, UK, 2011; p. 409. [Google Scholar]
- Popescu, S.; Rohrlich, D. Quantum nonlocality as an axiom. Found. Phys.
**1994**, 3, 379–385. [Google Scholar] [CrossRef] - Barrett, J. Information processing in generalized probabilistic theories. Phys. Rev. A
**2007**, 75, 032304. [Google Scholar] [CrossRef] - Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A. A generalized no-broadcasting theorem. Phys. Rev. Lett.
**2007**, 99, 240501. [Google Scholar] [CrossRef] [PubMed] - Barnum, H.; Wilce, A. Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci.
**2011**, 270, 3–15. [Google Scholar] [CrossRef] - Hardy, L. Towards quantum gravity: A framework for probabilistic theories with non-fixed causal structure. J. Phys. A
**2007**, 40, 3081–3099. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Sys. Tech. J.
**1949**, 27, 379–423, 623–656. [Google Scholar] [CrossRef] - Schumacher, B. Quantum coding. Phys. Rev. A
**1995**, 51, 2738–2747. [Google Scholar] [CrossRef] [PubMed] - Wootters, W.K. Local accessibility of quantum states. In Complexity, Entropy and the Physics of Information; Zurek, W.H., Ed.; Addison-Wesley: Boston, MA, USA, 1990; p. 39. [Google Scholar]
- Hawking, S.W. Black hole explosions? Nature
**1974**, 248, 30–31. [Google Scholar] [CrossRef] - Bennet, C.H. Logical reversibility of computation. IBM J. Res. Dev.
**1973**, 17, 525–532. [Google Scholar] [CrossRef] - Fredkin, E.; Toffoli, T. Conservative logic. Int. J. Theor. Phys.
**1982**, 21, 219–253. [Google Scholar] [CrossRef] - Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev.
**1961**, 4, 183. [Google Scholar] [CrossRef] - Preskill, J. Do black holes destroy information? In Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings, Houston Advanced Research Center, Houston, TX, USA, 16–18 January 1992; Kalara, S., Nanopoulos, D.V., Eds.; World Scientific: Singapore, 1993; pp. 22–39. [Google Scholar]
- t’Hooft, G. Dimensional reduction in quantum gravity. arXiv
**2009**. [Google Scholar] - Susskind, L. The world as a hologram. J. Math. Phys.
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef] - Schrödinger, E. Discussion of probability relations between separated systems. Proc. Camb. Phil. Soc.
**1935**, 31, 555–563. [Google Scholar] [CrossRef] - Popescu, S.; Short, A.J.; Winter, A. Entanglement and the foundations of statistical mechanics. Nat. Phys.
**2006**, 2, 754–758. [Google Scholar] [CrossRef] - Preskill, J. Lecture notes on quantum computation. Available online: http://www.theory.caltech.edu/people/preskill/ph229/ (accessed on 1 September 2012).
- Watrous, J. Quantum information and computation lecture notes. Available online: https://cs.uwaterloo.ca/∼watrous/lecture-notes.html (accessed on 1 September 2012).
- Wilde, M. From classical to quantum Shannon theory. arXiv
**2012**. [Google Scholar] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P.; Sacchi, M.F. Efficient use of quantum resources for the transmission of a reference frame. Phys. Rev. Lett.
**2004**, 93, 180503. [Google Scholar] [CrossRef] [PubMed] - Chiribella, G. Group theoretic structures in the estimation of an unknown unitary transformation. J. Phys. Conf. Ser.
**2011**, 284, 012001. [Google Scholar] [CrossRef] - Escher, B.M.; de Matos Filho, R.L.; Davidovich, L. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys.
**2011**, 7, 406. [Google Scholar] [CrossRef] - Wootters, W.K. Entanglement sharing in real-vector-space quantum theory. arXiv
**2010**. [Google Scholar] - Hardy, L. Quantum gravity computers: On the theory of computation with indefinite causal structure. In Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony; Myrvold, W.C., Christian, J., Eds.; Springer: New York, NY, USA, 2009. [Google Scholar]
- Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Beyond causally-ordered quantum computers. arXiv
**2012**. [Google Scholar] - Oreshkov, O.; Costa, F.; Brukner, Č. Quantum correlations with no causal order. arXiv
**2012**. [Google Scholar] - Chiribella, G. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. arXiv
**2011**. [Google Scholar] - Colnaghi, T.; D’Ariano, G.M.; Perinotti, P.; Facchini, S. Quantum computation with programmable connections between gates. arXiv
**2012**. [Google Scholar]

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Quantum Theory, Namely the Pure and Reversible Theory of Information. *Entropy* **2012**, *14*, 1877-1893.
https://doi.org/10.3390/e14101877

**AMA Style**

Chiribella G, D’Ariano GM, Perinotti P. Quantum Theory, Namely the Pure and Reversible Theory of Information. *Entropy*. 2012; 14(10):1877-1893.
https://doi.org/10.3390/e14101877

**Chicago/Turabian Style**

Chiribella, Giulio, Giacomo Mauro D’Ariano, and Paolo Perinotti. 2012. "Quantum Theory, Namely the Pure and Reversible Theory of Information" *Entropy* 14, no. 10: 1877-1893.
https://doi.org/10.3390/e14101877