A single photon source (SPS) is very important for quantum computation. In particular, it is essential for secured quantum cryptography. But there is no perfect SPS in reality. Therefore, probabilistic SPS where probability of simultaneous emission of two, three, four and more photon is less than the emission of a single photon are used. Since classical photon always comes in bunch, the required single photon source must be nonclassical. In the well-known antibunched state the rate of simultaneous emission of two photon is less than that of single photon. But the requirement of quantum cryptography is a multiphoton version of the antibunched state or the higher order antibunched state. Recently we have reported a mathematical criterion for higher order antibunching. Here we have shown that any proposal for SPS to be used in quantum cryptography should satisfy this criterion. We have studied four wave mixing as a possible candidate of single photon source.
We give a mathematical proof for an identification criterion by a probability measure for the ground state among an infinite number of available states, or a finitely truncated number with appropriate boundary conditions, in a quantum adiabatic algorithm for Hilbert's tenth problem.
Single photon sources to be used in quantum cryptography must show higher order antibunching (HOA). HOA is reported by us in several many wave mixing processes. In the present work we have investigated the possibility of observing HOA in multiwave mixing processes in general. The generalized Hamiltonian is solved for several particular cases in Heisenberg picture and possibility of observing HOA is investigated with the help of criterion of Pathak and Garcia. Several particular cases of the generalized Hamiltonian are solved with the help of short time approximation technique and HOA is report…
We analyze tolerable quantum bit error rates in secret sharing protocols, and show that using entangled encoding states is advantageous in the case when the eavesdropping attacks are local. We also provide a criterion for security in secret sharing -- a parallel of the Csiszar-Korner criterion in single-receiver cryptography.
Quantum cryptography uses techniques and ideas from physics and computer science. The combination of these ideas makes the security proofs of quantum cryptography a complicated task. To prove that a quantum-cryptography protocol is secure, assumptions are made about the protocol and its devices. If these assumptions are not justified in an implementation then an eavesdropper may break the security of the protocol. Therefore, security is crucially dependent on which assumptions are made and how justified the assumptions are in an implementation of the protocol. This thesis is primarily a review…
With the rapid development of quantum computers the currently secure cryptographic protocols may not stay that way. Quantum mechanics provides means to create an inherently secure communication channel that is protected by the laws of physics and not by the computational hardness of certain mathematical problems. This paper is a non-technical overview of quantum key distribution, one of the most well-known application of quantum cryptography, a type of cryptography poised to exploit the laws of quantum mechanics directly.
Although practised as an art and science for ages, cryptography had to wait until the mid-twentieth century before Claude Shannon gave it a strong mathematical foundation. However, Shannon's approach was rooted is his own information theory, itself inspired by the classical physics of Newton and Einstein. But our world is ruled by the laws of quantum mechanics. When quantum-mechanical phenomena are taken into account, new vistas open up both for codemakers and codebreakers. Is quantum mechanics a blessing or a curse for the protection of privacy? As we shall see, the jury is still out!
Quantum key distribution, first proposed by Bennett and Brassard, provides a possible key distribution scheme whose security depends only on the quantum laws of physics. So far the protocol has been proved secure even under channel noise and detector faults of the receiver, but is vulnerable if the photon source used is imperfect. In this paper we propose and give a concrete design for a new concept, {\it self-checking source}, which requires the manufacturer of the photon source to provide certain tests; these tests are designed such that, if passed, the source is guaranteed to be adequate fo…
We describe a new error reconciliation protocol {\it Winnow} based on the exchange of parity and Hamming's ``syndrome'' for $N-$bit subunits of a large data set. {\it Winnow} was developed in the context of quantum key distribution and offers significant advantages and net higher efficiency compared to other widely used protocols within the quantum cryptography community. A detailed mathematical analysis of Winnow is presented in the context of practical implementations of quantum key distribution; in particular, the information overhead required for secure implementation is one of the most im…
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…
An arbitrary polarization state of a single-mode biphoton is considered. The operationalistic criterion is formulated for the orthogonality og these states. It can be used to separate a biphoton with an arbitrary degree of polarization from a set of biphotons orthogonal to it. This is necessary fro the implementation of quantum cryptography protocol based on three-level systems. The experimental test of this criterion amounts to the observation of the anticorrelation effect for a biphoton with an arbitraty polarization state.
Quantum Cryptography and Autoformalization in Lean
MIT Center for Theoretical Physics – a Leinweber Institute（麻省理工学院理论物理中心——莱因韦伯研究所）；Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院电子研究实验室）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程和计算机科学系）
In a previous paper certain measurable criteria have been derived, that are sufficient to demonstrate the existence of Einstein-Podolsky-Rosen (EPR) correlations for measurements with continuous variable outcomes. Here it is shown how such EPR criteria, which do not demand perfect EPR correlations, can be used to prove the extent of security for continuous variable quantum cryptographic schemes (in analogy to that proposed by Ekert) where Alice and Bob hope to construct a secure sequence of values from measurements performed on continuous-variable EPR-correlated fields sent from a distant sour…
In contemporary cryptographic systems, secret keys are usually exchanged by means of methods, which suffer from mathematical and technology inherent drawbacks. That could lead to unnoticed complete compromise of cryptographic systems, without a chance of control by its legitimate owners. Therefore a need for innovative solutions exists when truly and reliably secure transmission of secrets is required for dealing with critical data and applications. Quantum Cryptography (QC), in particular Quantum Key Distribution (QKD) can answer that need. The business white paper (BWP) summarizes how secret…
This research note suggests a new way to realize a high speed direct encryption based on quantum detection theory. The conventional cipher is designed by a mathematical algorithm and its security is evaluated by the complexity of the algorithm for cryptanalysis and ability of computers. This kind of cipher cannot exceed the Shannon limit of cryptography,and it can be decrypted with probability one in principle by trying all the possible keys against the data length equal to the secret key length. A cipher with quantum effect in physical layer may exceed the Shannon limit of cryptography. The q…
Proofs of the security of quantum key distribution are propositions about models written in the mathematical language of quantum mechanics, and the issue is the linking of such models to actual devices in an experiment on security. To explore this issue, we adapt Wittgenstein's method of language games to view quantum language in its application to experimental activity involving transmitting and receiving devices. We sketch concepts with which to think about models in relation to experiments, without assuming the experiments accord with any model included is a concept of one quantum-mechanica…
Based on the firm laws of physics rather than unproven foundations of mathematical complexity, quantum cryptography provides a radically different solution for encryption and promises unconditional security. Quantum cryptography systems are typically built between two nodes connected to each other through fiber optic. This chapter focuses on quantum cryptography systems operating over free-space optical channels as a cost-effective and license-free alternative to fiber optic counterparts. It provides an overview of the different parts of an experimental free-space quantum communication link de…
The quest for practical cryptographic primitives that are robust against quantum computers is of vital importance for the field of cryptography. Among the abundance of different cryptographic primitives one may consider, one-way functions stand out as fundamental building blocks of more complex cryptographic protocols, and they play a central role in modern asymmetric cryptography. We propose a mathematical one-way function, which relies on coarse-grained boson sampling. The evaluation and the inversion of the function are discussed in the context of classical and quantum computers. The presen…
We give a mathematical criterion for the concept of information flow within closed quantum systems described by quantum registers. We define the concepts of separations and entanglements over quantum registers and use them with the quantum zip properties of inner products over quantum registers to establish the concept of partition change, which is fundamental to our criterion of endophysical information exchange within such quantum systems.
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entan…
In this Thesis, several results in quantum information theory are collected, most of which use entropy as the main mathematical tool. *While a direct generalization of the Shannon entropy to density matrices, the von Neumann entropy behaves differently. A long-standing open question is, whether there are quantum analogues of unconstrained non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a step towards a conjectured unconstrained inequality by Linden and Winter. *IID quantum state merging can be optimally achieved using the decoupling technique. …
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…
Quantum computers are expected to be able to solve mathematical problems that cannot be solved using conventional computers. Many of these problems are of practical importance, especially in the areas of cryptography and secure communications. APL is developing an optical approach to quantum computing in which the bits, or "qubits", are represented by single photons. Our approach allows the use of ordinary (linear) optical elements that are available for the most part as off-the-shelf components. Recent experimental demonstrations of a variety of logic gates for single photons, a prototype mem…
All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schrödinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it …
Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, c…
One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answ…
