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Anthony Leverrier - One of the best experts on this subject based on the ideXlab platform.
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su p q coherent states and a gaussian De Finetti theorem
Journal of Mathematical Physics, 2018Co-Authors: Anthony LeverrierAbstract:We prove a generalization of the quantum De Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of consiDering the action of the permutation group on n copies of that space, we consiDer the action of the unitary group U(n) on the creation operators of the n moDes and Define a natural generalization of the symmetric subspace as the space of states invariant unDer unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-moDe Fock space. The (Gaussian) SU(p, q) coherent states resolve the iDentity on the symmetric subspace, which implies a Gaussian De Finetti theorem stating that tracing over a few moDes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this De Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of inDepenDent normal variables if n3 = O(m).We prove a generalization of the quantum De Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of consiDering the action of the permutation group on n copies of that space, we consiDer the action of the unitary group U(n) on the creation operators of the n moDes and Define a natural generalization of the symmetric subspace as the space of states invariant unDer unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-moDe Fock space. The (Gaussian) SU(p, q) coherent states resolve the iDentity on the symmetric subspace, which implies a Gaussian De Finetti theorem stating that tr...
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security of continuous variable quantum key distribution via a gaussian De Finetti reduction
Physical Review Letters, 2017Co-Authors: Anthony LeverrierAbstract:: Establishing the security of continuous-variable quantum key distribution against general attacks in a realistic finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. InDeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on De Finetti reductions only proviDe security for unrealistically large block lengths. We address this problem here by consiDering a new type of Gaussian De Finetti reduction, that exploits the invariance of some continuous-variable protocols unDer the action of the unitary group U(n) (instead of the symmetric group S_{n} as in usual De Finetti theorems), and by introducing generalized SU(2,2) coherent states. Crucially, combined with an energy test, this allows us to truncate the Hilbert space globally instead as at the single-moDe level as in previous approaches that failed to proviDe security in realistic conditions. Our reduction shows that it is sufficient to prove the security of these protocols against Gaussian collective attacks in orDer to obtain security against general attacks, thereby confirming rigorously the wiDely held belief that Gaussian attacks are inDeed optimal against such protocols.
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SU(p,q) coherent states and Gaussian De Finetti theorems
2017Co-Authors: Anthony LeverrierAbstract:We prove a generalization of the quantum De Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of consiDering the action of the permutation group on n copies of that space, we consiDer the action of the unitary group U(n) on the creation operators of the n moDes and Define a natural generalization of the symmetric subspace as the space of states invariant unDer unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p,q) of signature (p,q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p,q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p,q) on an n(p+q)-moDe Fock space. The (Gaussian) SU(p,q) coherent states resolve the iDentity on the symmetric subspace, which implies a Gaussian De Finetti theorem stating that tracing over a few moDes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this De Finetti theorem, we show that the n×n upper-left submatrix of an n×n Haar-invariant unitary matrix is close in total variation distance to a matrix of inDepenDent normal variables if n3=O(m).
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su p q coherent states and a gaussian De Finetti theorem
QIP 2017 - 20th Annual Conference on Quantum Information Processing, 2017Co-Authors: Anthony LeverrierAbstract:We prove a generalization of the quantum De Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of consiDering the action of the permutation group on $n$ copies of that space, we consiDer the action of the unitary group $U(n)$ on the creation operators of the $n$ moDes and Define a natural generalization of the symmetric subspace as the space of states invariant unDer unitaries in $U(n)$. Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group $SU(p,q)$ of signature $(p,q)$. More precisely, this construction yields a unitary representation of the noncompact simple real Lie group $SU(p,q)$. We therefore find a dual unitary representation of the pair of groups $U(n)$ and $SU(p,q)$ on an $n(p+q)$-moDe Fock space. The (Gaussian) $SU(p,q)$ coherent states resolve the iDentity on the symmetric subspace, which implies a Gaussian De Finetti theorem stating that tracing over a few moDes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this De Finetti theorem, we show that the $n\times n$ upper-left submatrix of an $n\times n$ Haar-invariant unitary matrix is close in total variation distance to a matrix of inDepenDent normal variables if $n^3 =O(m)$.
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Security of Continuous-Variable Quantum Key Distribution via a Gaussian De Finetti Reduction
Physical Review Letters, 2017Co-Authors: Anthony LeverrierAbstract:Establishing the security of continuous-variable quantum key distribution against general attacks in a realistic finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. InDeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on De Finetti reductions only proviDe security for unrealistically large block lengths. We address this problem here by consiDering a new type of Gaussian De Finetti reduction, that exploits the invariance of some continuous-variable protocols unDer the action of the unitary group $U(n)$ (instead of the symmetric group $S_n$ as in usual De Finetti theorems), and by introducing generalized $SU(2,2)$ coherent states. Our reduction shows that it is sufficient to prove the security of these protocols against Gaussian collective attacks in orDer to obtain security against general attacks, thereby confirming rigorously the wiDely held belief that Gaussian attacks are inDeed optimal against such protocols.
Nicolas Cerf - One of the best experts on this subject based on the ideXlab platform.
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security of continuous variable quantum key distribution towards a De Finetti theorem for rotation symmetry in phase space
New Journal of Physics, 2009Co-Authors: Anthony Leverrier, Evgueni Karpov, Philippe Grangier, Nicolas CerfAbstract:Proving the unconditional security of quantum key distribution (QKD) is a highly challenging task as one needs to Determine the most efficient attack compatible with experimental data. This task is even more Demanding for continuous-variable QKD as the Hilbert space where the protocol is Described is infinite dimensional. A possible strategy to address this problem is to make an extensive use of the symmetries of the protocol. In this paper, we investigate a rotation symmetry in phase space that is particularly relevant to continuous-variable QKD, and explore the way towards a new quantum De Finetti theorem that would exploit this symmetry and proviDe a powerful tool to assess the security of continuous-variable protocols. As a first step, a single-party asymptotic version of this quantum De Finetti theorem in phase space is Derived.
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Quantum De Finetti theorem in phase-space representation
Physical Review A, 2009Co-Authors: Anthony Leverrier, Nicolas CerfAbstract:The quantum versions of De Finetti's theorem Derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant unDer permutations of their n parties, toward probabilistic mixtures of inDepenDent and iDentically distributed (IID) states of the form {sigma}{sup xn}. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by consiDering invariance unDer orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum De Finetti theorem particularly relevant to continuous-variable systems. Specifically, an n-moDe bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, n iDentical thermal states)
Renato Renner - One of the best experts on this subject based on the ideXlab platform.
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De Finetti reductions for correlations
Journal of Mathematical Physics, 2015Co-Authors: Rotem Arnon-friedman, Renato RennerAbstract:When analysing quantum information processing protocols one has to Deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to iDentify some additional structure. De Finetti theorems proviDe such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant unDer permutations of subsystems to states in which the subsystems are inDepenDent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known De Finetti theorems usually refer to the internal quantum state of a system and Depend on its dimension. Here we prove a different De Finetti theorem where systems are moDelled in terms of their statistics unDer measurements. This is necessary for a large class of applications wiDely consiDered today, such as Device inDepenDent protocols, where the unDerlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.
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De Finetti reductions beyond quantum theory
arXiv: Quantum Physics, 2013Co-Authors: Rotem Arnon-friedman, Renato RennerAbstract:De Finetti-type theorems enable a substantially simplified analysis of information-processing tasks in various areas, such as quantum cryptography and quantum tomography. The iDea is that instead of carrying out the analysis for any possible state it is sufficient to consiDer one particular De Finetti state, that is, a convex combination of i.i.d. states. It is thus interesting to see whether such De Finetti-type theorems are unique for quantum theory or whether they hold for more general theories. Here we prove a De Finetti-type theorem in the framework of conditional probability distributions. In this framework, a system is Described by a conditional probability distribution P_A|X where X Denotes the measurement and A the outcome. We show that any permutation invariant system P_A|X can be reduced to a De Finetti system. We then Demonstrate how the theorem can be applied to security proofs of Device inDepenDent cryptographic protocols.
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De Finetti representation theorem for infinite dimensional quantum systems and applications to quantum cryptography
Physical Review Letters, 2009Co-Authors: Renato Renner, J I CiracAbstract:We show that the quantum De Finetti theorem holds for states on infinite-dimensional systems, proviDed they satisfy certain experimentally verifiable conditions. This result can be applied to prove the security of quantum key distribution based on weak coherent states or other continuous variable states against general attacks.
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One-and-a-half quantum De Finetti theorems
Communications in Mathematical Physics, 2007Co-Authors: Matthias Christandl, Robert Koenig, Graeme Mitchison, Renato RennerAbstract:We prove a new kind of quantum De Finetti theorem for representations of the unitary group U(d). ConsiDer a pure state that lies in the irreducible representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the symmetric representation, this yields the conventional quantum De Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second De Finetti-style theorem (our 'half' theorem); the De Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also proviDes some useful examples, and gives some insight into the structure of the set of convex combinations of product states.
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one and a half quantum De Finetti theorems
Communications in Mathematical Physics, 2007Co-Authors: Matthias Christandl, Graeme Mitchison, Robert König, Renato RennerAbstract:When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum De Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by \({2\frac{kd^2}{n}}\) , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. ConsiDer a pure state that lies in the irreducible representation \({U_{\mu +\nu} \subset U_\mu \otimes U_\nu}\) of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out Uν. Then ξμ is close to a convex combination of the coherent states \({U_\mu(g)|{v_\mu\rangle}}\) , where \({g\in U(d)}\) and \({|v_\mu\rangle}\) is the highest weight vector in Uμ.
Ruediger Schack - One of the best experts on this subject based on the ideXlab platform.
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finite De Finetti theorem for infinite dimensional systems
Physical Review Letters, 2007Co-Authors: Christian Dcruz, Tobias J Osborne, Ruediger SchackAbstract:We formulate and prove a De Finetti representation theorem for finitely exchangeable states of a quantum system consisting of $k$ infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state $|\ensuremath{\Psi}⟩⟨\ensuremath{\Psi}|$ chosen from a family of subsets ${{\mathcal{C}}_{n}}$ of the full symmetric subspace for $n$ subsystems. We show that such states become arbitrarily close to mixtures of pure power states as $n$ increases. We give a second equivalent characterization of the family ${{\mathcal{C}}_{n}}$.
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Unknown Quantum States: The Quantum De Finetti Representation
Journal of Mathematical Physics, 2002Co-Authors: Carlton M. Caves, Christopher A. Fuchs, Ruediger SchackAbstract:We present an elementary proof of the quantum De Finetti representation theorem, a quantum analog of De Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical De Finetti theorem proviDes an operational Definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be Degrees of belief instead of objective states of nature. The quantum De Finetti theorem, in a closely analogous fashion, Deals with exchangeable Density-operator assignments and proviDes an operational Definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than...
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unknown quantum states the quantum De Finetti representation
arXiv: Quantum Physics, 2001Co-Authors: Carlton M. Caves, Christopher A. Fuchs, Ruediger SchackAbstract:We present an elementary proof of the quantum De Finetti representation theorem, a quantum analogue of De Finetti's classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical De Finetti theorem proviDes an operational Definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be Degrees of belief instead of objective states of nature. The quantum De Finetti theorem, in a closely analogous fashion, Deals with exchangeable Density-operator assignments and proviDes an operational Definition of the concept of an ``unknown quantum state'' in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further Demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.
Weihua Liu - One of the best experts on this subject based on the ideXlab platform.
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General De Finetti Type Theorems in Noncommutative Probability
Communications in Mathematical Physics, 2019Co-Authors: Weihua LiuAbstract:We prove general De Finetti type theorems for classical and free inDepenDence. The De Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We Determine maximal distributional symmetries which means the corresponding De Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we Define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general De Finetti type theorem for Boolean inDepenDence.
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Noncommutative Distributional Symmetries and Their Related De Finetti Type Theorems
2016Co-Authors: Weihua LiuAbstract:The main theme of this thesis is to Develop De Finetti type theorems in noncommutative probability. In noncommutative area, there are inDepenDence relations other than classi- cal inDepenDence, e.g. Voiculescu’s free inDepenDence, Boolean inDepenDence and Muraki’s monotone inDepenDence. Free analogues of De Finetti type theorems were discovered by Ko stler and Speicher and were Developed by Banica, Curran and Speicher. Here, we will Define noncommutative distributional symmetries for Boolean and monotone inDepenDence and we will prove De Finetti type theorems for them. These distributional symmetries are Defined via coactions of quantum structures including Woronowicz C∗-algebra and So ltan’s quantum families of maps. We show that the joint distribution of an infinite sequence of noncommutative random variables satisfies boolean exchangeability is equivalent to the fact that the sequence of the random variables is iDentically distributed and boolean inDepenDent with respect to the conditional expectation onto its tail algebra. Then, we Define noncom- mutative versions of spreadability and show Ryll-Nardzewski type theorems for monotone inDepenDence and boolean inDepenDence. We will show that, roughly speaking, an infinite bilateral sequence of random variables is monotonically(boolean) spreadable if and only if the variables are iDentically distributed and monotone(boolean) with respect to the conditional expectation onto its tail algebra. In the end of this thesis, we will prove general De Finetti theorems for classical, free and boolean inDepenDence. Our general De Finetti theorems work for non-easy quantum groups, which generalizes a recent work of Banica, Curran and Spe- icher. For infinite sequences, we Determine maximal distributional symmetries which means the corresponding De Finetti theorem fails if the sequence satisfies more symmetries other than the maximal one.
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On noncommutative distributional symmetries and De Finetti type theorems associated with them
arXiv: Operator Algebras, 2015Co-Authors: Weihua LiuAbstract:We prove general De Finetti theorems for classical, free and boolean inDepenDence. Our general De Finetti theorems work for non-easy quantum groups, which generalizes a recent work of Banica, Curran and Speicher. For infinite sequences, we will Determine maximal distributional symmetries which means the corresponding De Finetti theorem fails if the sequence satisfies more symmetries other than the maximal one. In addition, we Define boolean quantum semigroups in analogue of easy quantum groups by universal conditions on matrix coordinate generators and show some boolean analogue of De Finetti theorems.
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a noncommutative De Finetti theorem for boolean inDepenDence
Journal of Functional Analysis, 2015Co-Authors: Weihua LiuAbstract:Abstract We introduce a family of quantum semigroups and its natural coactions on noncommutative polynomials. We Define three invariance conditions for the joint distribution of sequences of selfadjoint noncommutative random variables associated with these coactions. For one of the invariance conditions, we show that the joint distribution of an infinite sequence of noncommutative random variables satisfy it is equivalent to the fact that the sequence of the random variables is iDentically distributed and boolean inDepenDent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of De Finetti theorem on exchangeable sequences. In the end of the paper, we also discuss the other two invariance conditions which lead to some trivial results.
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a noncommutative De Finetti theorem for boolean inDepenDence
arXiv: Operator Algebras, 2014Co-Authors: Weihua LiuAbstract:We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are iDentically distributed and boolean inDepenDent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of De Finetti theorem on exchangeable sequences. In the end of the paper, we will discuss the other two invariance conditions which lead to some trivial results.
