Generalized probabilistic theory

<h2 id="definition">Definition</h2>
A GPT is specified by a number of mathematical structures, namely:

<ul><li>a family of state spaces, each of which represents a physical system;</li>
<li>a composition rule (usually corresponds to a <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a>), which specifies how joint state spaces are formed;</li>
<li>a set of measurements, which map states to probabilities and are usually described by an <a href="/facts/Effect_algebra/lf3JEbCb">effect algebra</a>;</li>
<li>a set of possible physical operations, i.e., transformations that map state spaces to state spaces.</li></ul>
It can be argued that if one can prepare a state 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and a different state 
 
 
 
 y
 
 
 {\displaystyle y}
 
, then one can also toss a (possibly biased) coin which lands on one side with probability 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and on the other with probability 
 
 
 
 1
 −
 p
 
 
 {\displaystyle 1-p}
 
 and prepare either 
 
 
 
 x
 
 
 {\displaystyle x}
 
 or 
 
 
 
 y
 
 
 {\displaystyle y}
 
, depending on the side the coin lands on. The resulting state is a statistical mixture of the states 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 and in GPTs such statistical mixtures are described by convex combinations, in this case 
 
 
 
 p
 x
 +
 (
 1
 −
 p
 )
 y
 
 
 {\displaystyle px+(1-p)y}
 
. For this reason all state spaces are assumed to be <a href="/facts/Convex_set/vdAuJRJl">convex sets</a>. Following a similar reasoning, one can argue that also the set of measurement outcomes and set of physical operations must be convex.
Additionally it is always assumed that measurement outcomes and physical operations are affine maps, i.e. that if 
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
 is a physical transformation, then we must have 
 
 
 
 Φ
 (
 p
 x
 +
 (
 1
 −
 p
 )
 y
 )
 =
 p
 Φ
 (
 x
 )
 +
 (
 1
 −
 p
 )
 Φ
 (
 y
 )
 
 
 {\displaystyle \Phi (px+(1-p)y)=p\Phi (x)+(1-p)\Phi (y)}
 
and similarly for measurement outcomes. This follows from the argument that we should obtain the same outcome if we first prepare a statistical mixture and then apply the physical operation, or if we prepare a statistical mixture of the outcomes of the physical operations.
Note that physical operations are a subset of all affine maps which transform states into states as we must require that a physical operation yields a valid state even when it is applied to a part of a system (the notion of "part" is subtle: it is specified by explaining how different system types compose and how the global parameters of the composite system are affected by local operations).
For practical reasons it is often assumed that a general GPT is embedded in a finite-dimensional vector space, although infinite-dimensional formulations exist.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<h2 id="classical-quantum-and-beyond">Classical, quantum, and beyond</h2>
Classical theory is a GPT where states correspond to probability distributions and both measurements and physical operations are stochastic maps. One can see that in this case all state spaces are <a href="/facts/Simplex/0FCfNRN8">simplexes</a>.
Standard <a href="/facts/Quantum_information_theory/Bl9QIvHj">quantum information theory</a> is a GPT where system types are described by a natural number 
 
 
 
 D
 
 
 {\displaystyle D}
 
 which corresponds to the complex Hilbert space dimension. States of the systems of Hilbert space dimension 
 
 
 
 D
 
 
 {\displaystyle D}
 
 are described by the normalized positive semidefinite matrices, i.e. by the <a href="/facts/Density_matrix/A1XdylJa">density matrices</a>. Measurements are identified with <a href="/facts/POVM/ouBuwcck">Positive Operator valued Measures (POVMs)</a>, and the physical operations are <a href="/facts/Completely_positive_map/GJGjwqgh">completely positive maps</a>. Systems compose via the tensor product of the underlying complex Hilbert spaces.
Real quantum theory is the GPT which is obtained from standard quantum information theory by restricting the theory to real Hilbert spaces. It does not satisfy the axiom of local tomography.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> 
The framework of GPTs has provided examples of consistent physical theories which cannot be embedded in quantum theory and indeed exhibit very non-quantum features. One of the first ones was Box-world, the theory with maximal non-local correlations.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> Other examples are theories with third-order interference<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> and the family of GPTs known as generalized bits.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>
Many features that were considered purely quantum are actually present in all non-classical GPTs. These include the impossibility of universal broadcasting, i.e., the <a href="/facts/No-cloning_theorem/cLYH4pQE">no-cloning theorem</a>;<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> the existence of incompatible measurements;<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> and the existence of entangled states or entangled measurements.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a><a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quantum_foundations/ALdNv2sn">Quantum foundations</a></li></ul>

<h2 id="references">References</h2>

<ol>
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<li id="fn:2">Aubrun, Guillaume; Lami, Ludovico; Palazuelos, Carlos; Plávala, Martin (2022). "Entanglement and Superposition Are Equivalent Concepts in Any Physical Theory". Physical Review Letters. 128 (16): 160402. arXiv:2109.04446. Bibcode:2022PhRvL.128p0402A. doi:10.1103/PhysRevLett.128.160402. ISSN 0031-9007. PMID 35522482. S2CID 237453629. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Barnum, H. and Barrett, J. and Leifer, M. and Wilce, A. (2012). "Teleportation in general probabilistic theories". Proceedings of Symposia in Applied Mathematics. 71: 25–48.{{cite journal}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Template:Cite_journal" target="_blank">/wiki/Template:Cite_journal</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Ludwig, Günther (2012-12-06). An Axiomatic Basis for Quantum Mechanics: Volume 1 Derivation of Hilbert Space Structure. Springer Science & Business Media. ISBN 978-3-642-70029-3. <a href="978-3-642-70029-3" target="_blank">978-3-642-70029-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hardy, L. (2001). "Quantum Theory From Five Reasonable Axioms". arXiv:quant-ph/0101012. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
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<li id="fn:7">Wetering, John van de (2019-12-18). "An effect-theoretic reconstruction of quantum theory". Compositionality. 1: 1. arXiv:1801.05798. doi:10.32408/compositionality-1-1. ISSN 2631-4444. <a href="https://compositionality-journal.org/papers/compositionality-1-1/" target="_blank">https://compositionality-journal.org/papers/compositionality-1-1/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
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<li id="fn:16">Ludwig, Günther (1972). "An improved formulation of some theorems and axioms in the axiomatic foundation of the Hilbert space structure of quantum mechanics". Communications in Mathematical Physics. 26 (1): 78–86. Bibcode:1972CMaPh..26...78L. doi:10.1007/BF01877548. ISSN 0010-3616. S2CID 120013555. <a href="http://projecteuclid.org/euclid.cmp/1103858004" target="_blank">http://projecteuclid.org/euclid.cmp/1103858004</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
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<li id="fn:18">Hartkämper, A; Neumann, H (1974). Foundations of Quantum Mechanics and Ordered Linear Spaces: Advanced Study Institute Marburg 1973. Berlin, Heidelberg: Springer-Verlag : Springer e-books. ISBN 978-3-540-38650-6. <a href="978-3-540-38650-6" target="_blank">978-3-540-38650-6</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Ludwig, Günther (2012-12-06). An Axiomatic Basis for Quantum Mechanics: Volume 1 Derivation of Hilbert Space Structure. Springer Science & Business Media. ISBN 978-3-642-70029-3. <a href="978-3-642-70029-3" target="_blank">978-3-642-70029-3</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Barrett, J. (2007). "Information processing in generalized probabilistic theories". Phys. Rev. A. 75 (3): 032304. arXiv:quant-ph/0508211. Bibcode:2007PhRvA..75c2304B. doi:10.1103/PhysRevA.75.032304. S2CID 119504263. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Hardy, L. (2001). "Quantum Theory From Five Reasonable Axioms". arXiv:quant-ph/0101012. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Chiribella, Giulio; D’Ariano, Giacomo Mauro; Perinotti, Paolo (2011-07-11). "Informational derivation of quantum theory". Physical Review A. 84 (1): 012311. arXiv:1011.6451. Bibcode:2011PhRvA..84a2311C. doi:10.1103/PhysRevA.84.012311. ISSN 1050-2947. S2CID 15364117. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Chiribella, Giulio; D’Ariano, Giacomo Mauro; Perinotti, Paolo (2010-06-30). "Probabilistic theories with purification". Physical Review A. 81 (6): 062348. arXiv:0908.1583. Bibcode:2010PhRvA..81f2348C. doi:10.1103/PhysRevA.81.062348. S2CID 35205469. <a href="https://link.aps.org/doi/10.1103/PhysRevA.81.062348" target="_blank">https://link.aps.org/doi/10.1103/PhysRevA.81.062348</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
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<li id="fn:25">Kuramochi, Yui (2020-02-17). "Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex". Positivity. 24 (5): 1479–1486. arXiv:1912.00563. doi:10.1007/s11117-020-00742-0. ISSN 1385-1292. S2CID 208527451. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
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<li id="fn:27">Barrett, J. (2007). "Information processing in generalized probabilistic theories". Phys. Rev. A. 75 (3): 032304. arXiv:quant-ph/0508211. Bibcode:2007PhRvA..75c2304B. doi:10.1103/PhysRevA.75.032304. S2CID 119504263. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
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<li id="fn:31">Kuramochi, Yui (2020-02-17). "Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex". Positivity. 24 (5): 1479–1486. arXiv:1912.00563. doi:10.1007/s11117-020-00742-0. ISSN 1385-1292. S2CID 208527451. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Plávala, Martin (2016-10-12). "All measurements in a probabilistic theory are compatible if and only if the state space is a simplex". Physical Review A. 94 (4): 042108. arXiv:1608.05614. Bibcode:2016PhRvA..94d2108P. doi:10.1103/PhysRevA.94.042108. ISSN 2469-9926. S2CID 119115973. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Aubrun, Guillaume; Lami, Ludovico; Palazuelos, Carlos; Plávala, Martin (2021). "Entangleability of cones". Geometric and Functional Analysis. 31 (2): 181–205. arXiv:1911.09663. doi:10.1007/s00039-021-00565-5. S2CID 208202463. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Aubrun, Guillaume; Lami, Ludovico; Palazuelos, Carlos; Plávala, Martin (2022). "Entanglement and Superposition Are Equivalent Concepts in Any Physical Theory". Physical Review Letters. 128 (16): 160402. arXiv:2109.04446. Bibcode:2022PhRvL.128p0402A. doi:10.1103/PhysRevLett.128.160402. ISSN 0031-9007. PMID 35522482. S2CID 237453629. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
</ol>

Generalized probabilistic theory open-in-new

Generalized probabilistic theory