The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
Matthias Christandl
备注：
PhD thesis, February 2006, University of Cambridge. Part I contains results from quant-ph/0409016 and quant-ph/0511029, and analyses Horn's problem in this context. Part II reviews entanglement measures, presents results from quant-ph/0308088 and quant-ph/0501090, and provides new material on entanglement measures, information-gain versus disturbance tradeoffs and cheat sensitive quantum string commitment
This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.
Recently the explicit applicability of bound entanglement in quantum cryptography has been shown. In this paper some of recent results respecting this topic are reviewed. In particular relevant notions and definitions are reminded. The new construction of bound entangled states containing secure correlations is presented. It provides low dimensional 6\otimes6 bound entangled states with nonzero distillable key.
Quantum discord is the minimal bipartite resource which is needed for a secure quantum key distribution, being a cryptographic primitive equivalent to non-orthogonality. Its role becomes crucial in device-dependent quantum cryptography, where the presence of preparation and detection noise (inaccessible to all parties) may be so strong to prevent the distribution and distillation of entanglement. The necessity of entanglement is re-affirmed in the stronger scenario of device-independent quantum cryptography, where all sources of noise are ascribed to the eavesdropper.
Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the non-local properties of the state and can be "seen" by looking at one party's density matrix only. Pure states of three and more parties however lack such a simple form. They have more invariants under local unitary transformations than any one party can "see" on their sub-system. These "hidden non-localities" will allow us to exhibit a class of multipar…
There has been spectacular progress in the field of quantum information in recent decades. The development of this field highlights the importance of the role of entanglement in quantum computing, quantum teleportation and quantum cryptography. These notes serve to provide a gentle introduction to the entanglement of bipartite states. In these notes, we introduce Bell's theorem in the form derived by Clauser, Horne, Shimony and Holt. We discuss the Schmidt decomposition and the Peres-Horodecki criterion in the entanglement of pure and mixed bipartite states. Finally, we describe a teleportatio…
Granting information privacy is of crucial importance in our society, notably in fiber communication networks. Quantum cryptography provides a unique means to establish, at remote locations, identical strings of genuine random bits, with a level of secrecy unattainable using classical resources. However, several constraints, such as non-optimized photon number statistics and resources, detectors' noise, and optical losses, currently limit the performances in terms of both achievable secret key rates and distances. Here, these issues are addressed using an approach that combines both fundamenta…
We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be impleme…
In order to study multipartite quantum cryptography, we introduce quantities which vanish on product probability distributions, and which can only decrease if the parties carry out local operations or carry out public classical communication. These ``secrecy monotones'' therefore measure how much secret correlations are shared by the parties. In the bipartite case we show that the mutual information is a secrecy monotone. In the multipartite case we describe two different generalisations of the mutual information, both of which are secrecy monotones. The existence of two distinct secrecy monot…
We propose some protocols to implement various classes of bipartite unitary operations on two remote parties with the help of repeater nodes in-between. We also present a protocol to implement a single-qubit unitary with parameters determined by a remote party with the help of up to three repeater nodes. It is assumed that the neighboring nodes are connected by noisy photonic channels, and the local gates can be performed quite accurately, while the decoherence of memories is significant. A unitary is often a part of a larger computation or communication task in a quantum network, and to reduc…
Semi-quantum communication, a model introduced in 2007 by M. Boyer, D. Kenigsberg, and T. Mor (PRL 99 140501), involves the use of fully-quantum users and semi-quantum, or "classical" users. These restricted users are only allowed to interact with the quantum channel in a limited manner. Originally introduced to study the key-distribution problem, semi-quantum research has since expanded, and continues to grow, with new protocols, security proof methods, experimental implementations, and new cryptographic applications beyond key distribution. Research in the field of semi-quantum cryptography …
Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several…
We report investigation on quantum discord in classical second-order interference. In particular, we theoretically show that a bipartite state with D = 0.311 of discord can be generated via classical second-order interference. We also experimentally verify the theory by obtaining D = 0.197 +- 0.060 of non-zero discord state. Together with the fact that non-classicalities originated from physical constraints and information theoretic perspectives are not equivalent, this result provides an insight to understand the nature of quantum discord.
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids…
The study of stronger-than-quantum effects is a fruitful line of research that provides valuable insight into quantum theory. Unfortunately, traditional bipartite steering scenarios can always be explained by quantum theory. Here we show that, by relaxing this traditional setup, bipartite steering incompatible with quantum theory is possible. The two scenarios we describe, which still feature Alice remotely steering Bob's system, are: (i) one where Bob also has an input and operates on his subsystem, and (ii) the `instrumental steering' scenario. We show that such bipartite post-quantum steeri…
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…
Can a classical system command a general adversarial quantum system to realize arbitrary quantum dynamics? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. Here we report a technique by which a classical system can certify the joint, entangled state of a bipartite quantum system, as well as command the application of specific operators on each subsyste…
The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated to a hyperbolic space with a Minkowski metric, which is divided in two regions - separablelike and entangledlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distances measures - one related to the purity and another related to amount of entanglement. The second …
Computational Cryptography from Pseudoentanglement
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；INESC-ID, Instituto Superior Técnico, Universidade de Lisboa（葡萄牙里斯本大学高等技术学院INESC-ID）
The commodity-based cryptography is an alternative approach to realize conventionally impossible cryptographic primitives such as unconditionally secure bit-commitment by consuming pre-established correlation between distrustful participants. A unit of such classical correlation is known as the one-time table (OTT). In this paper, we introduce a new example besides quantum key distribution in which quantum correlation is useful for cryptography. We propose a scheme for unconditionally secure qubit-commitment, a quantum cryptographic primitive forbidden by the recently proven no-masking theorem…
The field of quantum information is becoming more known to the general public. However, effectively demonstrating the concepts underneath quantum science and technology to the general public can be a challenging job. We investigate, extend, and much expand here "quantum candies" (invented by Jacobs), a pedagogical model for intuitively describing some basic concepts in quantum information, including quantum bits, complementarity, the no-cloning principle, and entanglement. Following Jacob's quantum candies description of the well known quantum key distribution protocol BB84, we explicitly demo…
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Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartit…
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In the counterfactual cryptography scheme proposed by Noh (2009), the sender Alice probabilistically transmits classical information to the receiver Bob without the physical travel of a particle. Here we generalize this idea to the distribution of quantum entanglement. The key insight is to replace their classical input choices with quantum superpositions. We further show that the scheme can be generalized to counterfactually distribute multi-partite cat states.
Unconventional Quantum Criticality in Long-Range Spin-1 Chains: Insights from Entanglement Entropy and Bipartite Fluctuations
The University of Hong Kong（香港大学）；HK Institute of Quantum Science & Technology, The University of Hong Kong（香港大学量子科学与技术研究院）；State Key Laboratory of Optical Quantum Materials, The University of Hong Kong（香港大学光学量子材料国家重点实验室）；The Chinese University of Hong Kong（香港中文大学）；Univ Toulouse, CNRS, LPT, Toulouse, France（法国图卢兹大学CNRS理论物理实验室）
In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call "squashed entanglement": it is a lower bound on entanglement of formation and an upper bound on distillable entanglement. Furthermore, it is convex, additive on tensor products, and superadditive in general. Continuity in the state is the only proper…
Phase-controlled bipartite and tripartite entanglement and Bell nonlocality in a closed-loop optomechanical system
Beijing Institute of Radio Metrology and Measurement（北京无线电计量测试研究所）；Science and Technology on Metrology and Calibration Laboratory, Beijing Institute of Radio Metrology and Measurement（北京无线电计量测试研究所计量与校准重点实验室）
Reproducing with elementary resources the correlations that arise when a quantum system is measured (quantum state simulation), allows one to get insight on the operational and computational power of quantum correlations. We propose a family of models that can simulate von Neumann measurements in the x-y plane of the Bloch sphere on n-partite GHZ states using only bipartite nonlocal boxes. For the tripartite and fourpartite states, the models use only bipartite nonlocal boxes; they can be translated into classical communication schemes with finite average communication cost.
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The research field of quantum entanglement theory is comparatively new. While a basic understanding of the most simple systems in question (i.e. bipartite systems) has been established over the past few decades, multipartite entanglement still holds many unsolved questions and intriguing riddles. In particular, it is completely unclear how several concepts from the bipartite case can be generalised in a meaningful way to multipartite scenarios. In this work, the main issues of multipartite entanglement detection, characterisation and classification are discussed. The differences and similariti…
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Einstein-Podolsky-Rosen (EPR) steering, a category of quantum nonlocal correlations describing the ability of one observer to influence another party's state via local measurements, is different from both entanglement and Bell nonlocality by possessing an asymmetric property. For multipartite EPR steering, the monogamous situation, where two observers cannot simultaneously steer the state of the third party, has been investigated rigorously both in theory and experiment. In contrast to the monogamous situation, the shareability of EPR steering in reduced subsystems allows the state of one part…
Anomalous Local Heat Capacity and Bipartite Entanglement
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien（维也纳量子科学与技术中心，原子研究所，维也纳工业大学）；Center for Macroscopic Quantum States (bigQ), Department of Physics, Technical University of Denmark（宏观量子态研究中心（bigQ），物理系，丹麦技术大学）
Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky and Rosen [18]. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication and quantum cryptography ([16],[17],[6]). In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states in bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are n…
We review a recently developed theoretical approach to the experimental detection and quantification of bipartite quantum correlations between a qubit and a d dimensional system. Specifically, introducing a properly designed measure Q, the presented scheme allows us to quantify general quantum correlations for arbitrary states of 2 x d systems without the need to fully reconstruct them by tomographic techniques. We take in exam the specifics of the required experimental architecture in nuclear magnetic resonance and optical settings. Finally we extend this approach to models of open system dyn…
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