Symmetry of large physical systems implies independence of subsystems

By Renato Renner. 2007

259 citations recorded by the source corpus.

Composite systems consisting of a large number of similar subsystems play an important role in many areas of physics as well as in information theory. Their analysis, however, often relies on the assumption that the subsystems are mutually independent (or only weakly correlated). Here, we show that this assumption is generally justified for quantum systems that are symmetric, that is, invariant under permutations of the subsystems. Because symmetry is often implied by natural properties, for example, the indistinguishability of identical particles, the result has a wide range of consequences. In particular, it implies that global properties of a large composite system can be estimated by measurements applied to a limited number of (randomly chosen) sample subsystems, a fact that is important for the interpretation of experimental data. Moreover, it generalizes statements in quantum information theory and cryptography, which previously have only been known to hold under certain independence assumptions.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/6cc94916d89bb1d5d558dba5257a733983d6bb3e

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Citations in the corpus

Secure quantum key distribution with realistic devices (2020, background): “…can be extended to the case of coherent attack by further analysis (Coles et al., 2017; Tomamichel et al., 2012), such as the de Finetti theorem (Renner, 2007), the post-selection technique (Christandl et al., 2009) and uncertainty relation for smooth entropies (Tomamichel and Renner, 2011).”
The security of practical quantum key distribution (2008, methodology): “2006; Gottesman, Lo, Lütkenhaus and Preskill, 2004; Kraus, Branciard and Renner, 2007).”
Faithful Squashed Entanglement (2010, result): “In Corollary II.2 of [27], a similar result was shown using the exponential de Finetti theorem [ 83 ].”
Free randomness can be amplified (2011, unknown): “Following arguments by de Finetti [2, 3], this symmetry implies that we can treat these particles as independent particles of the same type.”
Entanglement theory and the second law of thermodynamics (2008, background): “4 ), which we generalize to tackle also non-identical-and-independently-distributed instances, together with ideas from convex optimization and duality, applied to the cone of separable states, recent results on the characterization of symmetric state”
High-rate measurement-device-independent quantum cryptography (2013, background): “However, this can always be reduced to the previous attack, coherent within the single use, by assuming that Alice and Bob perform random permutations on their data [34, 35].”
A Generalization of Quantum Stein’s Lemma (2009, methodology): “The idea is to combine Theorem I with the exponential de Finetti theorem [20, 21 ].”
Advances in quantum cryptography (2019, unknown): “In the asymptotic scenario, it has been proved [52] that this attack can be reduced to a collective one by running a random symmetrization routine which exploits the quantum de Finetti theorem [52–54].”
Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations (2017, background): “y this is remarkable, recall that permutation symmetry-based de Finetti representations achieve exponential convergence if they relax the form of allowed states to include \almost product states"[Ren07] or\high weight vectors"[KM09]. Such a relaxation is necessary { in fact, already for cl”
Tight finite-key analysis for quantum cryptography (2011, methodology): “It therefore avoids various estimates, including the de Finetti Theorem [24] and the Post-Selection technique [25], that have previously led to too pes-simistic bounds.”
The Church of the Symmetric Subspace (2013, background): “See [32] for more discussion of this point, [31] for an application to quantum key distribution or [16] for an application to mean-ﬁeld Hamiltonians.”
Less is Different: Emergence and Reduction Reconciled (2011, background): “…on N systems, by tracing out (partial tracing over) N −M systems, can be approximated by a mixture of product states on the M systems (i.e. states ω⊗M), with an error that goes to zero with the ratio M/N , e.g. an error like O(M/N); (Koenig and Renner 2005, Renner 2007, Koenig and Mitchison 2007).”
Quantum Conditional Mutual Information and Approximate Markov Chains (2014, methodology): “The second is an extension of the method of de Finetti reductions [14,15,45,46] (Sect.”