According to the quantum de Finetti theorem, if the state of an N-partite system is invariant under permutations of the subsystems then it can be approximated by a state where almost all subsystems are identical copies of each other, provided N is sufficiently large compared to the dimension of the subsystems. The de Finetti theorem has various applications in physics and information theory, where it is for instance used to prove the security of quantum cryptographic schemes. Here, we extend de Finetti's theorem, showing that the approximation also holds for infinite dimensional systems, as long as the state satisfies certain experimentally verifiable conditions. This is relevant for applications such as quantum key distribution (QKD), where it is often hard - or even impossible - to bound the dimension of the information carriers (which may be corrupted by an adversary). In particular, our result can be applied to prove the security of QKD based on weak coherent states or Gaussian states against general attacks.
We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for any arbitrary input, it is sufficient to consider the special case where the input is a particular de Finetti-type state, i.e., a state which consists of n identical and independent copies of an (unknown) state on a single subsystem. A similar statement holds for more general channels which are covariant with respect to the action of an arbitrary finite …
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 chosen from a family of subsets 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 C_n.
According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle nu…
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, t…
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti …
In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner's "exponential" approximation by "almost-product" states, a theorem which deals with certain triples of representations of the unitary group, and D'Cruz et al.'s result for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance correspondi…
We derive a bound for the security of QKD with finite resources under one-way post-processing, based on a definition of security that is composable and has an operational meaning. While our proof relies on the assumption of collective attacks, unconditional security follows immediately for standard protocols like Bennett-Brassard 1984 and six-states. For single-qubit implementations of such protocols, we find that the secret key rate becomes positive when at least N\sim 10^5 signals are exchanged and processed. For any other discrete-variable protocol, unconditional security can be obtained us…
We point out that the quantum de Finetti representation, unique for infinitely extendable exchangeable systems, assigns a non-zero Quantum Discord to uncorrelated systems and thus cannot serve as an universal prior distribution in the Bayesian Quantum Tomography. This apparent paradox stems from linearity of the Born rule for the probability assignment in Quantum Mechanics, which results in mixing of one's knowledge about the quantum state and the representative of the state in one density matrix.
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, towards probabilistic mixtures of independent and identically distributed (i.i.d.) states. 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 …
A practical quantum key distribution (QKD) protocol necessarily runs in finite time and, hence, only a finite amount of communication is exchanged. This is in contrast to most of the standard results on the security of QKD, which only hold in the limit where the number of transmitted signals approaches infinity. Here, we analyze the security of QKD under the realistic assumption that the amount of communication is finite. At the level of the general formalism, we present new results that help simplifying the actual implementation of QKD protocols: in particular, we show that symmetrization ste…
In the realistic quantum key distribution (QKD), Alice and Bob respectively get a quantum state from an unknown channel, whose dimension may be unknown. However, while discussing the security, sometime we need to know exact dimension, since current exponential de Finetti theorem, crucial to the information-theoretical security proof, is deeply related with the dimension and can only be applied to finite dimensional case. Here we address this problem in detail. We show that if POVM elements corresponding to Alice and Bob's measured results can be well described in a finite dimensional subspace …
De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only a matrix inequality between the symmetric state and one of de Finetti form) is enough, and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini and Winter. In particular we show tha…
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…
Relativistic quantum field theory imposes additional fundamental restrictions on the distinguishability of quantum states. Because of the unavoidable delocalization of the quantum field states in the Minkowski space-time, the reliable (with unit probability) distinguishability of orthogonal states formally requires infinite time. For the cryptographic protocols which are finite in time the latter means that the effective ``noise'' is present even in the ideal communication channel because of the non-localizability of the quantum field states.
We present the first device-independent quantum cryptography protocol for continuous variables. Our scheme is based on the Gottesman-Kitaev-Preskill encoding scheme whereby a qubit is embedded in the infinite-dimensional space of a quantum harmonic oscillator. The novel application of discrete-variable device-independent quantum key distribution to this encoding enables a continuous-variable analogue. Since the security of this protocol is based on discrete-variables we inherit by default security against collective attacks and, under certain memoryless assumptions, coherent attacks. We find t…
In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in such infinite group platforms are not yet known.
From Worst-Case Hardness of $\mathsf{NP}$ to Quantum Cryptography via Quantum Indistinguishability Obfuscation
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；NTT Social Informatics Laboratories（NTT社会信息学实验室）；NTT Research Center for Theoretical Quantum Information（NTT理论量子信息中心）
Quantum steering means that in some bipartite quantum systems, the local measurements on one side can determine the state of the other side. Here we show that in high-dimensional systems, there exists a specific entangled state which can display a kind of chaos effect when being adopted for steering. That is, a subtle difference in the measurement results on one side can steer the other side into completely orthogonal states. Moreover, by expanding the result to infinite-dimensional systems, we find two sets of states for which, contrary to common belief, even though their density matrices app…
Unconditional Security of Discrete-Modulated CV-QKD from Infinite-Dimensional MEAT
University of Waterloo（滑铁卢大学）；National University of Singapore（新加坡国立大学）；Technische Universität Wien（维也纳工业大学）；AIT Austrian Institute of Technology（奥地利技术研究院）
We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate `complexity metric' must be used that takes into account the relative difficulty of performing `nonlocal' operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable sa…
Prepare-and-measure (P&M) quantum networks are the basic building blocks of quantum communication and cryptography. These networks crucially rely on non-orthogonal quantum encodings to distribute quantum correlations, thus enabling superior communication rates and information-theoretic security. Here, we present a computational toolbox that is able to efficiently characterise the set of input-output probability distributions for any discrete-variable P&M quantum network, assuming only the inner-product information of the quantum encodings. Our toolbox is thus highly versatile and can be used t…
Continuous-variable quantum information, encoded into infinite-dimensional quantum systems, is a promising platform for the realization of many quantum information protocols, including quantum computation, quantum metrology, quantum cryptography, and quantum communication. To successfully demonstrate these protocols, an essential step is the certification of multimode continuous-variable quantum states and quantum devices. This problem is well studied under the assumption that multiple uses of the same device result into identical and independently distributed (i.i.d.) operations. However, in …
The degree of freedom of orbital angular momentum (OAM) is an important resource in high-dimensional quantum information processing, as the quantum number of OAM can be infinite. The Dove prism (DP) is a most common tool to manipulate the OAM light, such as in interferometers. However, the Dove prism does not preserve the polarization of the photon states and decreases the sorting fidelity of the interferometer. In this work, we analyze the polarization-dependent effect of the DP on single-path Sagnac interferometers. The results are instructive to quantum information processing with OAM light…
This thesis deals with the study of quantum communication protocols with Continuous Variable (CV) systems. Continuous Variable systems are those described by canonical conjugated coordinates x and p endowed with infinite dimensional Hilbert spaces, thus involving a complex mathematical structure. A special class of CV states, are the so-called Gaussian states. With them, it has been possible to implement certain quantum tasks as quantum teleportation, quantum cryptography and quantum computation with fantastic experimental success. The importance of Gaussian states is two-fold; firstly, its st…
We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the poss…
Entanglement is widely considered the cornerstone of quantum information and an essential resource for relevant quantum effects, such as quantum teleportation, quantum cryptography, or the speed-up of quantum computing, as in Shor's algorithm. However, up to now, there is no general characterization of entanglement for many-body systems. In this sense, it is encouraging that quantum states connected by stochastic local operations assisted with classical communication (SLOCC), which perform probabilistically the same quantum tasks, can be collected into entanglement classes. Nevertheless, there…
