arXiv subjects

Amir Kalev

Publications and source records attributed to Amir Kalev.

51 records · Page 3Linked to original sources

arXiv2014

Quantum process tomography of unitary and near-unitary maps

We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a $d$-level system is completely characterized by a minimal set of $d^2{+}d$ elements associated with a collection of POVMs, in contrast to the $d^4{-}d^2$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with a particular set of $d$ pure states. We further compare the performance of different compressed sensing algorithms used to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Compressed sensing techniques can therefore be used both as indicators of error models and to validate the use of the prior assumptions.

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arXiv2014

Mutually unbiased measurements in finite dimensions

We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of MUB. We derive their general form, and show that in a finite, $d$-dimensional Hilbert space, one can construct a complete set of $d+1$ mutually unbiased measurements. Beside of their intrinsic link to MUB, we show, that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy non-trivial entropic uncertainty relation similar to $d+1$ MUB.

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arXiv2013

Quantum mechanical retrodiction through an extended mean King problem

The mean King problem is a conditional retrodiction problem. In this problem Alice prepares a two prime-dimensional particles state and avails one of the particles to the King who measures its state in one of mutually unbiased bases of his choice. The King tells Alice his choice of basis after she completes a control measurement on his particle. Conditioned on this knowledge, she now infers the state observed by the King by utilizing the outcome of her control measurement. In the extended mean King problem, studied in this paper, the King does not tell Alice his measurement basis, but instead both the King and Alice repeat their measurements. Proper ordering of these allows Alice to deduce both the basis used by the King and the outcome of his first measurement, with the King reticent throughout, i.e., this protocol effects a complete (almost) retrodiction of the King's first measurement.

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arXiv2013

Optimal approximate transpose map via quantum designs and its applications to entanglement detection

We show that quantum designs characterize the general structure of the optimal approximation of the transpose map on quantum states. Based on this characterization, we propose an implementation of the approximate transpose map by a measurement-and-preparation scheme. The results show that state-manipulation in quantum two-designs suffices for transpose-based quantu3 information applications. In particular, we present how these results can be applied to the framework of detecting multipartite entangled states, for instance, when local measurements or interferometry-based experimental approaches are applied.

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arXiv2013

A geometrical relation between symmetric operators and mutually unbiased operators

In this work we study the relation between the set of symmetric operators and the set of mutually unbiased operators from finite plane geometry point of view. Here symmetric operators are generalization of symmetric informationally complete probability-operator measurements (SIC POMs), while mutually unbiased operators are the operator generalization of mutually unbiased bases (MUB). We also discuss the implication of this relation to the particular cases of rank-1 SIC POMs and MUB.

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arXiv2012

Micromotion in trapped atom-ion systems

We examine the validity of the harmonic approximation, where the radio-frequency ion trap is treated as a harmonic trap, in the problem regarding the controlled collision of a trapped atom and a single trapped ion. This is equivalent to studying the effect of the micromotion since this motion must be neglected for the trapped ion to be considered as a harmonic oscillator. By applying the transformation of Cook and Shankland we find that the micromotion can be represented by two periodically oscillating operators. In order to investigate the effect of the micromotion on the dynamics of a trapped atom-ion system, we calculate (i) the coupling strengths of the micromotion operators by numerical integration and (ii) the quasienergies of the system by applying the Floquet formalism, a useful framework for studying periodic systems. It turns out that the micromotion is not negligible when the distance between the atom and the ion traps is shorter than a characteristic distance. Within this range the energy diagram of the system changes remarkably when the micromotion is taken into account, which leads to undesirable consequences for applications that are based on an adiabatic process of the trapped atom-ion system. We suggest a simple scheme for bypassing the micromotion effect in order to successfully implement a quantum controlled phase gate proposed previously and create an atom-ion macromolecule. The methods presented here are not restricted to trapped atom-ion systems and can be readily applied to studying the micromotion effect in any system involving a single trapped ion.

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arXiv2012

Quantum State Tomography Using Successive Measurements

We describe a quantum state tomography scheme which is applicable to a system described in a Hilbert space of arbitrary finite dimensionality and is constructed from sequences of two measurements. The scheme consists of measuring the various pairs of projectors onto two bases --which have no mutually orthogonal vectors--, the two members of each pair being measured in succession. We show that this scheme implies measuring the joint quasi-probability of any pair of non-degenerate observables having the two bases as their respective eigenbases. The model Hamiltonian underlying the scheme makes use of two meters initially prepared in an arbitrary given quantum state, following the ideas that were introduced by von Neumann in his theory of measurement.

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arXiv2012

Experimental proposal for symmetric minimal two-qubit state tomography

We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The basis of the first measurement and the four bases of the second measurements are pairwise unbiased --- a hint at a possibly profound link between SIC POMs and mutually unbiased bases.

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arXiv2012

Symmetric minimal quantum tomography by successive measurements

We consider the implementation of a symmetric informationally complete probability-operator measurement (SIC POM) in the Hilbert space of a d-level system by a two-step measurement process: a diagonal-operator measurement with high-rank outcomes, followed by a rank-1 measurement in a basis chosen in accordance with the result of the first measurement. We find that any Heisenberg-Weyl group-covariant SIC POM can be realized by such a sequence where the second measurement is simply a measurement in the Fourier basis, independent of the result of the first measurement. Furthermore, at least for the particular cases studied, of dimension 2, 3, 4, and 8, this scheme reveals an unexpected operational relation between mutually unbiased bases and SIC POMs; the former are used to construct the latter. As a laboratory application of the two-step measurement process, we propose feasible optical experiments that would realize SIC POMs in various dimensions.

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arXiv2010

Entanglement discontinuity

We identify a class of two-mode squeezed states which are parametrized by an angular variable ${0\leθ<2π}$ and a squeezing parameter $r$. We show that, for a large squeezing value, these states are either (almost) maximally entangled or product states depending on the value of $θ$. This peculiar behavior of entanglement is unique for infinite dimensional Hilbert space and has consequences for the entangling power of unitary operators in such systems. Finally, we show that, at the limit ${r\to\infty}$ these states demonstrate a discontinuity attribute of entanglement.

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arXiv2010

Encoding many qubits in a rotor

We propose a scheme for encoding many qubits in a single rotor, that is, a continuous and periodic degree of freedom. A key feature of this scheme is its ability to manipulate and entangle the encoded qubits with a single operation on the system. We also show, using quantum error-correcting codes, how to protect the qubits against small errors in angular position and momentum which may affect the rotor. We then discuss the feasibility of this scheme and suggest several candidates for its implementation. The proposed scheme is immediately generalizable to qudits of any finite dimension.

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arXiv2008

Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions

We study the possibility of giving a classical interpretation to quantum projective measurements for a particle described by a pure Gaussian state whose Wigner function is non-negative. We analyze the case of a projective measurement which gives rise to a proper Wigner function, i.e., taking on, as its values, the eigenvalues of the projector. We find that, despite having this property, this kind of projector produces a state whose Wigner function ceases to be non-negative and hence precludes its interpretation as a classical probability density. We also study the general case in which the projected state has a non-negative Wigner function; but then we find that the Wigner function of the projector is not a proper one. Thus, we conclude that a non-negative Wigner function is inadequate to serve as a hidden variable model for quantum processes in which projective measurements take place.

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arXiv2008

Equation of Motion for the Quantum Characteristic Functions

In this paper, we derive equations of motion for the normal-order, the symmetric-order and the antinormal-order quantum characteristic functions, applicable for general Hamiltonian systems. We do this by utilizing the `characteristic form' of both quantum states and Hamiltonians. The equations of motion we derive here are rather simple in form and in essence, and as such have a number of attractive features. As we shall see, our approach enables the descriptions of quantum and classical time evolutions in one unified language. It allows for a direct comparison between quantum and classical dynamics, providing insight into the relations between quantum and classical behavior, while also revealing a smooth transition between quantum and classical time evolutions. In particular, the $\hbar\to 0$ limit of the quantum equations of motion instantly recovers their classical counterpart. We also argue that the derived equations may prove to be very useful in numerical simulations.

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arXiv2008

Quantum Teleportation and Hidden Variables

In this paper we address the question as to what extent the quantum-mechanical nature of the process is relevant for teleportation of A spin-1/2 state. For this purpose we analyze the possibility of underpinning teleportation with a local-hidden-variable model. The nature of the models, which we consider as legitimate candidates, guarantees the classical character of all the probabilities which can be deduced from them. When we try to describe the teleportation process following two different mathematical routes, we find two different hidden-variable densities, which thus end up having a doubtful physical significance within the "reality" that a hidden-variable model tries to restore. This result we consider as a "no-go theorem" for the hidden-variable description of the teleportation process. We also show that this kind of conflict arises when considering successive measurements (one of which is selective projective) for one spin-1/2 particle.

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arXiv2004

Measurement of an integral of a classical field with a single quantum particle

A method for measuring an integral of a classical field via local interaction of a single quantum particle in a superposition of 2^N states is presented. The method is as efficient as a quantum method with N qubits passing through the field one at a time and it is exponentially better than any known classical method that uses N bits passing through the field one at a time. A related method for searching a string with a quantum particle is proposed.

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