Abstract
Quantum mechanics dramatically differs from classical physics. An interesting consequence of this fact is that quantum resources offer an advantage over classical resources in many information-theoretic tasks. In quantum information, the goal of which is to understand information processing from a quantum perspective, it is thus natural to focus on tasks where quantum resources provide an advantage over classical ones, and to overlook tasks where quantum mechanics provides no advantage. But are the latter tasks really useless from a more general perspective? In this review we focus on a simple information-theoretic game called ‘guess your neighbour’s input’, for which classical and quantum players perform equally well. Interestingly, this seemingly innocuous game turns out to be useful in various contexts. From a fundamental point of view, the game provides a sharp separation between quantum mechanics and other more general physical theories, hence bringing a deeper understanding of the foundations of quantum mechanics. The game also finds unexpected applications in quantum foundations and quantum information theory, related to Gleason’s theorem, and to bound entanglement and unextendible product bases.
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Notes
- 1.
By simplicity, we consider \(\lambda \) to be discrete, but all the formulation can be extended to the continuous case.
- 2.
Note that for 2 players, no-signaling correlation provide no advantage.
- 3.
To see this explicitly let us first notice that for a normalized witness W it holds that \(W=(I\otimes \varLambda )(\varPhi )\) with trace-preserving \(\varLambda \) iff \(W_A=\text{ tr }_B W=\mathbbm {1}/d\). Then, if \(W_A\ne \mathbbm {1}/d\) but it is of full rank, one introduces another witness \(\widetilde{W}=(1/d)(W_A^{-1/2}\otimes {\mathbbm {1}})W(W_A^{-1/2}\otimes \mathbbm {1})\). Clearly, \(\widetilde{W}_A=\mathbbm {1}/d\) and thus \(\widetilde{W}\) is isomorphic to a trace-preserving positive map \(\widetilde{\varLambda }\). Consequently,
where \(\varPsi \) denotes a projector onto some normalized pure state \(| \varPsi \rangle =\sqrt{d}(\sqrt{W_A}\otimes \mathbbm {1})| \varPhi \rangle \) of full Schmidt rank. Finally, if \(W_A\) is rank-deficient, one constructs yet another witness \(W'=W+\mathscr {P}_A^{\perp }\otimes \mathbbm {1}\), where \(\mathscr {P}_A^{\perp }=\mathbbm {1}-\mathscr {P}_A\) with \(\mathscr {P}_A\) denoting a projector onto the support of \(W_A\). Then, \(W'_A\) is of full-rank and therefore \(W'\) admits the form (29). To complete the proof, it suffices to notice that \(W=(\mathscr {P}_A\otimes \mathbbm {1})W'(\mathscr {P}_A\otimes \mathbbm {1})\), and hence W also assumes the form (29) with a normalized pure state \(| \varPsi \rangle =\sqrt{d}[\mathscr {P}_A(W_A')^{1/2}\otimes \mathbbm {1}]| \varPhi \rangle =\sqrt{d}(W_A^{1/2}\otimes \mathbbm {1})| \varPhi \rangle \) which is now not of full Schmidt rank.
- 4.
The dual map \(\varLambda ^*\) of \(\varLambda \) is the map such that \(\text{ tr }(A\varLambda (B))=\text{ tr }[\varLambda ^*(A)B]\).
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Acknowledgments
Discussions with T. Fritz are acknowledged. This work was supported by the ERC starting grant PERCENT, the EU AQUTE and QCS projects, the Spanish CHIST-ERA DIQIP, FIS2008-00784 and FIS2010-14830 projects, and the UK EPSRC. R. A. is supported by the Spanish MINCIN through the Juan de la Cierva program.
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Acín, A., Almeida, M.L., Augusiak, R., Brunner, N. (2016). Guess Your Neighbour’s Input: No Quantum Advantage but an Advantage for Quantum Theory. In: Chiribella, G., Spekkens, R. (eds) Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7303-4_14
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