Motivated by an interest in quantum sensing, we define carefully a degree of entanglement, starting with bipartite pure
states and building up to a definition applicable to any mixed state on any tensor product of finite-dimensional vector
spaces. For mixed states the degree of entanglement is defined in terms of a minimum over all possible decompositions of
the mixed state into pure states. Using a variational analysis we show a property of minimizing decompositions. Combined
with data about the given mixed state, this property determines the degrees of entanglement of a given mixed state. For
pure or mixed states symmetric under permutation of particles, we show that no partial trace can increase the degree of
entanglement. For selected less-than-maximally-entangled pure states, we quantify the degree of entanglement surviving
a partial trace.
Much of the theory of quantum computing assumes the capacity to apply a chosen sequence of unitary transformations
to the state of a quantum register (sometimes called a memory). It is widely recognized that this
"application of a unitary transformation" requires an external influence. Here we relate the physics of external
influences to the well established framework of quantum-mechanical scattering problems, in order to show how
scattering is conceptually necessary to quantum computers, even in the idealization of zero temperature and no
imperfections.
For a single-qubit quantum register, infinitely slow limiting cases are shown in which scattering indeed results
in a unitary transformation of the register. Implications for "transformation-induced decoherence" are developed
and related to questions of errors and error correction.
In the simplest problems of quantum communication, Alice transmits one of two quantum states, with equal probabilities, to Bob's receiver, modeled by a positive-operator-valued measure (POVM); one seeks the POVM that is optimal according to one or another criterion. We discuss four such criteria, the first three of which lead to distinctive types of POVM. By introducing a reciprocal basis for the state vectors, we shorten the derivations of known results for the two most popular criteria. A new optimization problem defined by a third criterion, intermediate between the first two, is formulated and solved. Then we turn to a fourth criterion, that of minimizing Bob's Renyi entropy for an arbitrary order α. Depending on the value of α and the separation of Alice's states, the POVM that minimizes Bob's entropy can be any of the preceding three types.
To a large extent, the power of a quantum computer comes from the possibility of operating on a number of quantum memories simultaneously. For example, if there are n linearly independent quantum states in each of N quantum memories, then the number of linearly independent quantum states is n^N when they are taken together. This exponential increase is the basis of the present interest in quantum computing. In the context of quantum mechanics as described by the Schrodinger equation (which is a partial differential equation), the most explicit model of the quantum memory is one with two linearly independent quantum states described in terms of the Fermi pseudo-potential at one point. In this model, operations on the quantum memory are accomplished through scattering with both symmetrical and anti-symmetrical incident waves. As a first step toward operating on more than one quantum memory, this model is generalized in two directions. (1) With the Fermi pseudo-potential at one point retained, three linearly independent quantum states are used for the quantum memory. This model of the quantum memory requires three external connections. (2) With a more general potential, operations on the quantum memory with two states are accomplished through scattering with either symmetrical or anti-symmetrical incident waves. This second generalization is important because it is not practical to keep the number of external connections equal to the number of linearly independent quantum states.
In quantum key distribution (QKD), Alice selects a sequence of characters from a finite alphabet and transmits the corresponding signals, each described by a quantum state, to Bob. Between Alice and Bob, Eve---with her own receiver and transmitter---can eavesdrop. Eve is assumed to know beforehand all the possible states from among which Alice chooses. The central question is: Can Eve gain significant information about the key without influencing what Bob receives in ways that Alice and Bob can
detect? This formulation implies that many options are available to Eve. It is the purpose here to discuss one of these options, where extensive use is made of the Schrodinger equation with spatial variables. This procedure of Eve---which consists of letting the quantum state of Alice scatter from a quantum memory and then, using the information thus obtained, sending a suitably chosen quantum state to Bob---is discussed in detail. Furthermore, there are ways for Alice to defeat this procedure by an unusual choice of her quantum states. This counter-measure is also presented and discussed.
The security of quantum key distribution against undetected eavesdropping depends on the key-sharing parties (Alice and Bob) making a probabilistic estimate of the ignorance of a maximally adept eavesdropper (Eve) concerning sifted, error-free bits from which Alice and Bob distill a key. For individual attacks on the BB84 protocol, we show how to generalize the defense function and the defense frontier of Slutsky et al. to take advantage of Cachin’s analysis of Renyi entropy of arbitrary order R, here called R-entropy. For a special case of an attack uniform over all bits, an optimum defense frontier is displayed. Evidence is discussed for the conjecture that this defense frontier in terms of R-entropy holds good not just for uniform attacks but for all individual attacks on BB84.
We also show how the entropy estimate fits in to the full suite of key-distillation protocols in a QKD system, in particular how it relates to privacy amplification. After privacy amplification, Eve will have, with high probability, no information about the remaining bits. By choosing the optimal Rényi order R, we can distill secure bits in the presence of a significantly higher error rate.
Users of a quantum cryptographic system face a problem of deciding on the ignorance of a maximally adroit eavesdropper concerning their key material. It is known that there can be no sure, positive, lower bound on any plausible measure of ignorance, and for this reason we characterize the problem as the making of an informed guess, meaning a guess that employs a rule that can be shown to work except in unlikely cases. As the measure of an eavesdropper's ignorance concerning n bits of sifted key material less some number k of bits found in error and discarded, we analyze Renyi entropy of arbitrary order R, for 1 ≤ R ≤ 2. We offer a rule for deciding on Renyi entropy based on a tighter bound on the relevant probability distributions than has been available. To this end, we employ a recently derived approximation to the cumulative binomial distribution which is uniformly accurate over a larger domain than previously available approximations. This results in a longer distilled key than that obtained from looser bounds, as well as generalizing the order R. Some numerical examples are presented.
An essential component of any quantum computer is the quantum memory, the content of which is a pure quantum state. A program to study the quantum memory is initiated here, where the spatial variables are of central importance. The presence of the spatial variables makes it possible to apply the powerful and well-developed theory of scattering: The fundamental operations of writing on, reading and resetting the quantum memory are all performed through scattering from the memory. The requirement that the quantum memory must remain in a pure state after scattering implies that the scattering is of a special type, and only certain incident waves are admissible. Models based on the coupled-channel Schrodinger equation for potential scattering are formulated, where there is indeed the required large collection of admissible incident waves. On the basis of these models, certain types of decoherence are unavoidable. Such decoherence and the necessity of using the relativistic Schrodinger equation are discussed. One of the implications of quantum memory is the possible lack of security for the quantum key distribution in quantum cryptography.
KEYWORDS: Waveguides, Microwave radiation, Polonium, Signal attenuation, Antennas, Polarization, Microsoft Foundation Class Library, Reflection, Electromagnetism, Metals
In order to transfer electromagnetic (EM) energy efficiently from a relativistic klystron amplifier (RKA) to a steerable super-array, the issue of mode conversion must be considered, i.e., the transfer of electromagnetic fields from a coaxial-line waveguide to a number of rectangular waveguides. A special converter has been designed to solve this matching problem. The converter consists of several sections including a multi-fin converter, a fan- shaped converter, and a twisted-rectangular converter. The mode conversion in each of these sections has been studied analytically and experimentally. The optimal parameters for minimum energy loss have been found for the different converter sections and the power breakdown in some of the sections has been studied. Formulas for engineering design are presented.
KEYWORDS: Waveguides, Microwave radiation, Antennas, Transformers, Transmission electron microscopy, Polarization, Maxwell's equations, Microsoft Foundation Class Library, Amplifiers, Wave propagation
In this paper, the design and properties of a multi-fin coaxial waveguide converter are examined in detail. The converter has been designed as a means of transferring power from a high-power Relativistic Klystron Amplifier (RKA) to an antenna system. The output from an RKA is a coaxial waveguide, but the radially polarized TEM mode propagating in a coaxial line is not suitable for radiation. In order to transmit high electromagnetic (EM) power to the elements of a transmitting antenna, the polarization of the electric fields in the feeding waveguide should be parallel. A rectangular waveguide excited in the H1,0-mode has such a linear polarization. Therefore, it is necessary to provide a suitable matching transformer between the coaxial and rectangular waveguides. The purpose of the transformer is to convert the TEM mode in the coaxial line into the H1,0-mode in a number of rectangular waveguides. These are then connected to the antennas.
High-power microwave pulses can destroy electronics of targets at altitudes of 100 km or higher, and preliminary designs of microwave antennas driven by Relativistic Klystron Amplifiers have been sketched. This paper discusses: (1) the susceptibility of the atmosphere to microwave breakdown, and (2) the constraint on the design of a microwave weapon imposed by the need to avoid breakdown.
Antenna arrays are potentially useful for both the transmission and the reception of trains of electromagnetic pulses. In this paper, we formulate an aiming problem pertaining to an RKA- driven launcher of microwave pulses, and draw on a study of quantum measurements to develop an approach to aiming. Appendix A reviews the behavior of microwave radiation focused at a target in the Fresnel zone. Appendix B derives the antenna current for optimal angular sensitivity of a monopulse radar for a target in the Fresnel zone, very close to the axis.
35The resonant properties of large circular arrays of parallel, nonstaggered dipoles have been analyzed specifically when only one element is driven. The approach involves using tubular cylindrical dipoles and the integral equations of antenna theory. The parameters that are involved in the adjustment to any one of many possible resonant distributions of current around the array include the length and radius of the (Nu) identical tubular dipoles and the distance between them as fractions of the operating wavelength. Extremely sharp resonances have been obtained at properly selected values of the parameters. The very large resonant amplitudes of the currents in the elements can have any one of a variety of different symmetrical distributions of phase around the array including 180 degree(s) shifts from element to adjacent element. Typical field patterns consist of many extremely sharp peaks. The study of the circular array has been carried out using the method of symmetrical components. In order to study closed loops of dipoles other than the circle, e.g., ellipses, egg-shaped curves, etc., the analysis is being generalized.
An electromagnetic (EM) missile launched from a hyperboloidal lens has been investigated experimentally. The system includes a hyperboloidal lens made of wax and a V-conical antenna (with a 30 degree(s) apex angle) located at the focal point of the lens. As expected, the lens system performs like an aperture field similar to that from a parabolic reflector. Both the measurement and simulation show that the EM missile launched from the hyperboloidal lens has a slow rate of energy decay and a waveform consisting of a pair of opposite pulses--properties similar to those of a parabolic reflector.
Conventional radar pulses spread so that the energy reaching a reflecting target at range r decreases as r exp -2. The energy echoed from the target also decreases as r exp -2, so that the energy backscattered to the transmitter is proportional to the product, i.e., to r exp -4. Pulses of finite total energy that satisfy Maxwell''s equations need not decrease as r exp -2, but can instead decrease much more slowly, for instance as r exp -epsilon where epsilon is a positive number that can be chosen as small as desired. Such pulses are referred to as electromagnetic missiles. Here, the echo backscattered when an electromagnetic missile encounters a plate target is studied. By analogy with conventional radar, it might be expected that the energy of the echo would be proportional to r exp -2(epsilon). Instead, the reflected energy decreases even slower, proportional to r exp -epsilon, so that no squaring occurs, resulting in a larger than expected radar return.
Electromagnetic pulses of finite total radiated energy can deliver energy to a distant receiver that decreases with distance much more slowly than the usual r exp -2. Such electromagnetic missiles can be generated by any of a class of current distributions over one or another set of points contained in a bounded region of a two-dimensional or three-dimensional space. For a given currrent-bearing set of points as a transmitter, one can explore various time and space dependencies of the transmitting current. Over some classes of dependencies, the energy reaching the receiver decreases as r exp -epsilon, where epsilon is a positive number that can vary from one current to another. Current-bearing sets so far explored have dimension 0, 1 or 2. For these, the requirement of finite total radiated energy establishes a greatest lower bound on epsilon, dependent on the dimensions of both the set and the space in which it resides. This infimum defines what can be called the pulse index of a set relative to the space containing it. One might conjecture that the pulse index can be defined for sets of fractional Hausdorff dimension. It will be shown that this is so, and that for certain sets of fractional dimension, the index is related to the dimension just as it is for integral dimensions.
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