Reversible dynamics on composite systems

Cite this problem as Problem 42.


Quantum theory allows to create entanglement from product states in a purely reversible way: for example, there is a unitary U (in fact, many) that maps the pure product state |0\rangle\otimes|0\rangle to

 U |0\rangle\otimes |0\rangle = \frac 1 {\sqrt{2}}\left(\strut |0\rangle\otimes |0\rangle+|1\rangle\otimes|1\rangle\right).

Surprisingly, this possibility might be a characteristic feature of quantum theory — which is what we conjecture here. To state the conjecture, we need the basic machinery of generalized probabilistic theories [1]. Consider a d_A-dimensional compact convex set of (normalized) states \Omega_A, affinely embedded into a real vector space A of dimension d_A+1 (in standard complex quantum theory, \Omega_A would be the density matrices over some \C^n, where d_A=n^2-1, and A would be the space of Hermitian n\times n matrices). Similarly, consider another state space \Omega_B embedded analogously into some real vector space B. Now consider a composite state space \Omega_{AB}. If we assume tomographic locality [2], then \Omega_{AB} will be a (d_A+1)(d_B+1)-1-dimensional compact convex set in A\otimes B, containing the analogs of product states, product measurements, and product transformations.

We need one more ingredient to state the open problem: every reversible transformation T on a state space \Omega_A is a linear map such that T\Omega_A=\Omega_A. The reversible transformations form a group \mathcal{G}_A (usually assumed compact), which is a subgroup of all symmetries of that state space. (In quantum theory, this is the group of unitary conjugations \rho\mapsto U\rho U^\dagger, whereas the set of all symmetries would also contain the antiunitary maps.)

Open Problem: Consider two state spaces \Omega_A,\Omega_B such that the reversible transformations \mathcal{G}_A resp.\ \mathcal{G}_B are transitive on the pure states*. Given any (locally-tomographic) composite state space \Omega_{AB} as just described, suppose that there exists any reversible transformation T_{AB} which is not of the form T_A\otimes T_B (that is, a transformation that reversibly creates correlations). Prove, or disprove by counterexample, that this state space must then be embeddable into standard quantum theory.

*This means that every pure state can be mapped to every other by a suitable reversible transformation.


Both a positive and a negative resolution of this open problem would have very interesting consequences. If the conjecture could be proven, then this would characterize quantum theory in an intriguingly simple way, namely as the maximal theory that allows “interesting” reversible computation, while still allowing composite states to be represented tensorially. On the other hand, any non-quantum counterexample would represent a tremendously interesting mathematical discovery, representing a theory whose computational power can be compared against quantum theory. Tony Short was one of the first to spell out the above conjecture at a conference in Barbados in 2012 (among others, including Lluís Masanes).

Partial solutions

With respect to reversible dynamics, there are several interesting examples and partial results:

1. Continuous reversible dynamics. Standard quantum theory over the complex numbers is the only known locally-tomographic theory that admits continuous reversible dynamics.

2. Theories with discrete sets of pure states. If one considers state spaces with a finite number of pure states and discrete transformations, there are two known examples that admit reversible dynamics and local tomography at the same time. Both theories are embeddable in standard complex quantum theory:

a) Classical probability theory, where the states are simply finite probability distributions \Omega_A=\{(p_1,p_2,\ldots,p_{d_A})\,\,|\,\,p_i\geq 0,\sum_i p_i=1\}. On a composite state space AB, one can implement CNOT gates, for example (otherwise interesting classical reversible computation would be impossible).

b) The stabilizer states of quantum theory. Consider the stabilizer states on k qubits, and suppose that these states define the pure states of the composite state space (the Open Problem above would only be concerned with k=2). The Clifford group is transitive on the stabilizer states, see e.g. [13], and it contains non-product transformations like the CNOT gate.

3. Non-locally tomographic theories. An interesting direction to pursue is to give up tomographic locality. A good example is real quantum mechanics (restricting superpositions to have real numbers as amplitudes), where the description of composite systems requires additional global parameters [9]. Another example is given by the composite of quaternionic quantum bits [10,11]. Both examples are embeddable in standard complex quantum theory. An interesting approach is presented in [12], where a hierarchy of theories exhibiting invariant reversible dynamics is introduced. However, apart from quantum theory, it is not known whether there are other examples that can be promoted to a full, mathematically consistent theory.

4. G-bit theories. Particularly interesting results can be obtained in the case that the local state spaces are d-dimensional Bloch balls, i.e.\ \Omega_A=\{(1,\vec r)\,\,|\,\,|\vec r|\leq 1,\vec r \in\mathbb{R}^d\}. In this case, it has been shown in [7] that the only composite state space that satisfies the assumptions of the Open Problem above is the quantum state space of two qubits for d=3. No other pairs of Bloch balls with $d\neq 3$ allow a composite with any non-product reversible transformation (assuming local tomography and continuity of the group of transformations). This has recently been generalized to three and more Bloch balls [8].

5. “Boxworld”. In the case of the so-called “boxworld” (e.g.\ state spaces that contain all non-signalling correlations) [1], it has been shown that all reversible transformations for any number of parties, measurements, and outcomes are trivial, i.e. the simple product transformations [3]. Sabri Al-Safi and co-authors have generalized this result in interesting ways [4,5,6].


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[4] S. W. Al-Safi and A. J. Short, Reversible Dynamics in Strongly Non Local Boxworld Systems, J. Phys. A: Math. Theor. 47, 325303 (2014).
[5] S. W. Al-Safi and J. Richens, Reversibility and the structure of the local state space, New J. Phys. 17, 123001 (2015).
[6] J. G. Richens, J. H. Selby, and S. W. Al-Safi, Entanglement is an inevitable feature of any non-classical theory, arXiv:1610.00682 (2016).
[7] Ll. Masanes, M. P. Müller, D. Pérez-García, and R. Augusiak, Entanglement and the three-dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014).
[8] M. Krumm and M. P. Müller, in preparation (2017).
[9] L. Hardy, and W. K. Wootters, Limited Holism and Real-Vector-Space Quantum Theory, Found. Phys. 42 (3), 454-473 (2012).
[10] H. Barnum, M. A. Graydon, and A. Wilce, Some Nearly Quantum Theories, in Proceedings of QPL 2015, EPTCS 195, 59-70 (2015).
[11] H. Barnum, M. A. Graydon, and A. Wilce, Composites and Categories of Euclidean Jordan Algebras, arXiv:1606.09331 (2016).
[12] B. Dakić, and Č. Brukner, The Classical Limit of a Physical Theory and the Dimensionality of Space, In “Quantum Theory: Informational Foundations and Foils”, Eds. G. Chiribella, and R. Spekkens, Springer (2016).
[13] R. Kueng, H. Zhu, and D. Gross, Low rank matrix recovery from Clifford orbits, arXiv:1610.08070 (2016).