Intrinsic Sensitivity Limits for Multiparameter Quantum Metrology

Aaron Z. Goldberg, Luis Sanchez-Soto, Hugo Ferretti

The quantum Cramér-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. In the multiparameter scenario, this bound becomes a matrix inequality, which can be cast to a scalar form with a properly chosen weight matrix. Multiparameter estimation thus elicits tradeoffs in the precision with which each parameter<br>can be estimated. We show that, if the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor<br>linked to the geometry of the underlying algebra su(n). This ensures an intrinsic bound that is independent of the choice of parametrization.<br>

Comment on “An encryption protocol for NEQR images based on one-particle quantum walks on a circle”

Markus Grassl

Rotation sensing at the ultimate limit

Aaron Z. Goldberg, Andrei B. Klimov, Gerd Leuchs, Luis Sanchez-Soto

Conventional classical sensors are approaching their maximum sensitivity<br>levels in many areas. Yet these levels still are far from the ultimate limits<br>dictated by quantum mechanics. Quantum sensors promise a substantial step ahead<br>by taking advantage of the salient sensitivity of quantum states to the<br>environment. Here, we focus on sensing rotations, a topic of broad application.<br>By resorting to the basic tools of estimation theory, we derive states that<br>achieve the ultimate sensitivities in estimating both the orientation of an<br>unknown rotation axis and the angle rotated about it. The critical enhancement<br>obtained with these optimal states should make of them an indispensable<br>ingredient in the next generation of rotation sensors that is now blossoming.<br>

Axial superlocalization with vortex beams

D. Koutny, Z. Hradil, J. Rehacek, Luis Sanchez-Soto

Improving axial resolution is of paramount importance for three-dimensional optical imaging systems. Here, we investigate the ultimate precision in axial<br>localization using vortex beams. For Laguerre-Gauss beams, this limit can be achieved with just an intensity scan. The same is not true for superpositions<br>of Laguerre-Gauss beams, in particular for those with intensity profiles that rotate on defocusing. Microscopy methods based on rotating vortex beams may thus benefit from replacing traditional intensity sensors with advanced mode-sorting techniques.

Effects of coherence on temporal resolution

Syamsundar De, Jano Gil-Lopez, Benjamin Brecht, Christine Silberhorn, Luis Sanchez-Soto, Z. Hradil, J. Rehacek

Measuring small separations between two optical sources, either in space or in time, constitute an important metrological challenge as standard<br>intensity-only measurements fail for vanishing separations. Contrarily, it has been established that appropriate coherent mode projections can appraise<br>arbitrarily small separations with quantum-limited precision. However, the question of whether the optical coherence brings any metrological advantage to<br>mode projections is still a point of debate. Here, we elucidate this problem by experimentally investigating the effect of varying coherence on estimating the<br>temporal separation between two single-photon pulses. We show that, for an accurate interpretation, special attention must be paid to properly normalize<br>the quantum Fisher information to account for the strength of the signal. Our experiment demonstrates that coherent mode projections are optimal for any<br>degree of coherence.<br>

Benchmarking quantum tomography completeness and fidelity with machine learning

Yong Siah Teo, Seongwook Shin, Hyunseok Jeong, Yosep Kim, Yoon-Ho Kim, Gleb I. Struchalin, Egor V. Kovlakov, Stanislav S. Straupe, Sergei P. Kulik, et al.

We train convolutional neural networks to predict whether or not a set of measurements is informationally complete to uniquely reconstruct any given quantum state with no prior information. In addition, we perform fidelity benchmarking based on this measurement set without explicitly carrying out state tomography. The networks are trained to recognize the fidelity and a<br>reliable measure for informational completeness through collective encoding of quantum measurements, data and target states into grayscale images. By<br>gradually accumulating measurements and data, these convolutional networks can efficiently certify a low-measurement-cost quantum-state characterization<br>scheme. We confirm the potential of this machine-learning approach by presenting experimental results for both spatial-mode and multiphoton systems<br>of large dimensions. These predictions are further shown to improve with noise recognition when the networks are trained with additional bootstrapped training sets from real experimental data.<br>

SU(1, 1) covariant s-parametrized maps

Andrei B. Klimov, Ulrich Seyfarth, Hubert de Guise, Luis Sanchez-Soto

We propose a practical recipe to compute the s-parametrized maps for systems with SU(1, 1) symmetry using a connection between the Q- and P-symbols through the action of an operator invariant under the group. This establishes equivalence relations between s-parametrized SU(1, 1)-covariant maps. The particular case of the self-dual (Wigner) phase-space functions, defined on the upper sheet of the two-sheet hyperboloid (or, equivalently, inside the Poincaré disc) are analysed.

Achieving the ultimate quantum timing resolution

Vahid Ansari, Benjamin Brecht, Jano Gil-López, John M. Donohue, Jaroslav Řeháček, Zdeněk Hradil, Luis Sanchez-Soto, Christine Silberhorn

Accurate time-delay measurement is at the core of many modern technologies.<br>Here, we present a temporal-mode demultiplexing scheme that achieves the<br>ultimate quantum precision for the simultaneous estimation of the temporal<br>centroid, the time offset, and the relative intensities of an incoherent<br>mixture of ultrashort pulses at the single-photon level. We experimentally<br>resolve temporal separations ten times smaller than the pulse duration, as well<br>as imbalanced intensities differing by a factor of 10^2. This represents an<br>improvement of more than an order of magnitude over the best standard methods<br>based on intensity detection.<br>

Quantum Fisher Information with Coherence

Zdeněk Hradil, Jaroslav Řeháček, Luis Sanchez-Soto, Berthold-Georg Englert

In recent proposals for achieving optical super-resolution, variants of the<br>Quantum Fisher Information (QFI) quantify the attainable precision. We find<br>that claims about a strong enhancement of the resolution resulting from<br>coherence effects are questionable because they refer to very small subsets of<br>the data without proper normalization. When the QFI is normalized, accounting<br>for the strength of the signal, there is no advantage of coherent sources over<br>incoherent ones. Our findings have a bearing on further studies of the<br>achievable precision of optical instruments.<br>

Compressively certifying quantum measurements

I. Gianani, Y. S. Teo, V. Cimini, H. Jeong, Gerd Leuchs, M. Barbieri, Luis Sanchez-Soto

We introduce a reliable compressive procedure to uniquely characterize any<br>given low-rank quantum measurement using a minimal set of probe states that is<br>based solely on data collected from the unknown measurement itself. The<br>procedure is most compressive when the measurement constitutes pure detection<br>outcomes, requiring only an informationally complete number of probe states<br>that scales linearly with the system dimension. We argue and provide numerical<br>evidence showing that the minimal number of probe states needed is even<br>generally below the numbers known in the closely-related classical<br>phase-retrieval problem because of the quantum constraint. We also present<br>affirmative results with polarization experiments that illustrate significant<br>compressive behaviors for both two- and four-qubit detectors just by using<br>random product probe states.<br>

Wigner function for SU(1,1)

Ulrich Seyfarth, Andrei B. Klimov, Hubert de Guise, Gerd Leuchs, Luis Sanchez-Soto

In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples.

Fundamental quantum limits in ellipsometry

L. Rudnicki, Luis Sanchez-Soto, Gerd Leuchs, R. W. Boyd

We establish the ultimate limits that quantum theory imposes on the accuracy attainable in optical ellipsometry. We show that the standard quantum limit, as usual reached when the incident light is in a coherent state, can be surpassed with the use of appropriate squeezed states and, for tailored beams, even pushed to the ultimate Heisenberg limit.

Universal compressive characterization of quantum dynamics

Yosep Kim, Yong Siah Teo, Daekun Ahn, Dong-Gil Im, Young-Wook Cho, Gerd Leuchs, Luis Sanchez-Soto, Hyunseok Jeong, Yoon-Ho Kim

Recent quantum technologies utilize complex multidimensional processes that<br>govern the dynamics of quantum systems. We develop an adaptive<br>diagonal-element-probing compression technique that feasibly characterizes any<br>unknown quantum processes using much fewer measurements compared to<br>conventional methods. This technique utilizes compressive projective<br>measurements that are generalizable to arbitrary number of subsystems. Both<br>numerical analysis and experimental results with unitary gates demonstrate low<br>measurement costs, of order O(d^2) for d-dimensional systems, and<br>robustness against statistical noise. Our work potentially paves the way for a<br>reliable and highly compressive characterization of general quantum devices.<br>

Quantum concepts in optical polarization

Aaron Z. Goldberg, Pablo de la Hoz, Gunnar Bjork, Andrei B. Klimov, Markus Grassl, Gerd Leuchs, Luis Sanchez-Soto

We comprehensively review the quantum theory of the polarization properties<br>of light. In classical optics, these traits are characterized by the Stokes<br>parameters, which can be geometrically interpreted using the Poincaré sphere.<br>Remarkably, these Stokes parameters can also be applied to the quantum world,<br>but then important differences emerge: now, because fluctuations in the number<br>of photons are unavoidable, one is forced to work in the three-dimensional<br>Poincaré space that can be regarded as a set of nested spheres. Additionally,<br>higher-order moments of the Stokes variables might play a substantial role for<br>quantum states, which is not the case for most classical Gaussian states. This<br>brings about important differences between these two worlds that we review in<br>detail. In particular, the classical degree of polarization produces<br>unsatisfactory results in the quantum domain. We compare alternative quantum<br>degrees and put forth that they order various states differently. Finally,<br>intrinsically nonclassical states are explored and their potential applications<br>in quantum technologies are discussed.

Objective Compressive Quantum Process Tomography

Y. S. Teo, G. I. Struchalin, E. V. Kovlakov, D. Ahn, H. Jeong, S. S. Straupe, S. P. Kulik, Gerd Leuchs, Luis Sanchez-Soto

We present a compressive quantum process tomography scheme that fully<br>characterizes any rank-deficient completely-positive process with no a priori<br>information about the process apart from the dimension of the system on which<br>the process acts. It uses randomly-chosen input states and adaptive output von<br>Neumann measurements. Both entangled and tensor-product configurations are<br>flexibly employable in our scheme, the latter which naturally makes it<br>especially compatible with many-body quantum computing. Two main features of<br>this scheme are the certification protocol that verifies whether the<br>accumulated data uniquely characterize the quantum process, and a compressive<br>reconstruction method for the output states. We emulate multipartite scenarios<br>with high-order electromagnetic transverse modes and optical fibers to<br>positively demonstrate that, in terms of measurement resources, our<br>assumption-free compressive strategy can reconstruct quantum processes almost<br>equally efficiently using all types of input states and basis measurement<br>operations, operations, independent of whether or not they are factorizable<br>into tensor-product states.<br>

Efficient generation of temporally shaped photons using nonlocal spectral filtering

V. Averchenko, D. Sych, C. Marquardt, G. Leuchs

We study the generation of single-photon pulses with the tailored temporal shape via nonlocal spectral filtering. A shaped photon is heralded from a time-energy entangled photon pair upon spectral filtering and time-resolved detection of its entangled counterpart. We show that the temporal shape of the heralded photon is defined by the time-inverted impulse response of the spectral filter and does not depend on the heralding instant. Thus one can avoid postselection of particular heralding instants and achieve a substantially higher heralding rate of shaped photons as compared to the generation of photons via nonlocal temporal modulation. Furthermore, the method can be used to generate shaped photons with a coherence time in the ns-μs range and is particularly suitable to produce photons with the exponentially rising temporal shape required for efficient interfacing to a single quantum emitter in free space.

Quasiprobability currents on the sphere

I. Valtierra, A. B. Klimov, Gerd Leuchs, Luis Sanchez-Soto

We present analytic expressions for the s-parametrized currents on the sphere for both unitary and dissipative evolutions. We examine the spatial distribution of the flow generated by these currents for quadratic Hamiltonians. The results are applied for the study of the quantum dissipative dynamics of the time-honored Kerr and Lipkin models, exploring the appearance of the semiclassical limit in stable, unstable and tunnelling regimes.

R&D advances for quantum communication systems

Gerd Leuchs, Christoph Marquardt, Luis Sanchez-Soto, Dmitry V. Strekalov, Alan E. Willner

Understanding the nature of light leads to the question of how the principles of quantum physics can be harnessed in practical optical communication. A deeper understanding of fundamental physics has always advanced technology. However, the quantum principles certainly have a distinctly limiting character when looked upon from the engineering point of view. A particle cannot have well-defined momentum and position at the same time. An informative measurement will unpredictably alter the state of a quantum object. One cannot reliably clone an arbitrary quantum state. These and a number of other similar principles give rise to what is commonly known as the quantum “no-go theorems”—a disconcerting term when it comes to building something practical. And yet a search for novel principles of communication enabled by quantum physics began already in its early days and has only intensified since. On this path physicists are faced with a remarkable challenge: to turn a series of negative statements into new technological recipes.

Reading out Fisher information from the zeros of the point spread function

M. Paúr, B. Stoklasa, D. Koutný, J. Řeháček, Z. Hradil, J. Grover, A. Krzic, Luis Sanchez-Soto

We show that, for optical systems whose point spread functions exhibit isolated zeros, the information one can gain about the separation between two incoherent point light sources does not scale quadratically with the separation (which is the distinctive dependence causing Rayleigh’s curse) but only linearly. Moreover, the dominant contribution to the separation information comes from regions in the vicinity of these zeros. We experimentally confirm this idea, demonstrating significant superresolution using natural or artificially created spectral doublets.

Compressed sensing of twisted photons

F. Bouchard, D. Koutny, F. Hufnagel, Z. Hradil, J. Rehacek, Y. S. Teo, D. Ahn, H. Jeong, Luis Sanchez-Soto, et al.

The ability to completely characterize the state of a quantum system is an essential element for the emerging quantum technologies. Here, we present a<br>compressed-sensing inspired method to ascertain any rank-deficient qudit state, which we experimentally encode in photonic orbital angular momentum. We efficiently reconstruct these qudit states from a few scans with an intensified CCD camera. Since it requires only a few intensity measurements, our technique<br>would provide an easy and accurate way to identify quantum sources, channels, and systems.

Adaptive Compressive Tomography with No a priori Information

Daekun Ahn, Yong Siah Teo, Hyunseok Jeong, Frédéric Bouchard, Felix Hufnagel, Ebrahim Karimi, D Koutny, Jarda Rehacek, Zdenek Hradil, et al.

Quantum state tomography is both a crucial component in the field of quantum information and computation and a formidable task that requires an incogitable number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc.) about the state, apart from its dimension.

Tomography from collective measurements

A. Muñoz, A. B. Klimov, Markus Grassl, Luis Sanchez-Soto

We discuss the tomography of N-qubit states using collective measurements.The method is exact for symmetric states, whereas for not completely symmetric states the information accessible can be arranged as a mixture of irreducible SU(2) blocks. For the fully symmetric sector, the reconstruction protocol can be reduced to projections onto a canonically chosen set of pure states.<br>

Quantum-limited time-frequency estimation through mode-selective photon measurement

John M. Donohue, Vahid Ansari, Jaroslav Řeháček, Zdeněk Hradil, Bohumil Stoklasa, Martin Paúr, Luis Sanchez-Soto, Christine Silberhorn

By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small separations between objects with quantum-limited precision,<br>free of uncertainty arising from overlapping intensity profiles. Here we extend these techniques to the time-frequency domain using mode-selective sum-frequency generation with shaped ultrafast pulses. We experimentally resolve temporal and spectral separations between incoherent mixtures of<br>single-photon level signals ten times smaller than their optical bandwidths with a ten-fold improvement in precision over the intensity-only Cramér-Rao<br>bound.

Testing for entanglement with periodic coarse-graining

Daniel S. Tasca, Łukasz Rudnicki, Reuben S. Aspden, Miles J. Padgett, Paulo H. Souto Ribeiro, Stephen P. Walborn

Continuous variables systems find valuable applications in quantum<br>information processing. To deal with an infinite-dimensional Hilbert space, one<br>in general has to handle large numbers of discretized measurements in tasks<br>such as entanglement detection. Here we employ the continuous transverse<br>spatial variables of photon pairs to experimentally demonstrate novel<br>entanglement criteria based on a periodic structure of coarse-grained<br>measurements. The periodization of the measurements allows for an efficient<br>evaluation of entanglement using spatial masks acting as mode analyzers over<br>the entire transverse field distribution of the photons and without the need to<br>reconstruct the probability densities of the conjugate continuous variables.<br>Our experimental results demonstrate the utility of the derived criteria with a<br>success rate in entanglement detection of $\sim60\%$ relative to $7344$ studied<br>cases.<br>

Gauge invariant information concerning quantum channels

Łukasz Rudnicki, Zbigniew Puchała, Karol Zyczkowski

Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

Partially coherent axiconic surface plasmon polariton fields

Yahong Chen, Andreas Norrman, Sergey A. Ponomarenko, Ari T. Friberg

We introduce a class of structured polychromatic surface electromagnetic fields, reminiscent of conventional optical axicon fields, through a judicious superposition of partially correlated surface plasmon polaritons. We show that such partially coherent axiconic surface plasmon polariton fields are structurally stable and statistically highly versatile with regard to spectral density, polarization state, energy flow, and degree of coherence. These fields can be created by plasmon coherence engineering and may prove instrumental broadly in surface physics and in various nanophotonics applications.

Nonregularity of three-dimensional polarization states

José J. Gil, Andreas Norrman, Ari T. Friberg, Tero Setälä

Regular states of polarization are defined as those that can be decomposed into a pure state (fully polarized), a two-dimensional (2D) unpolarized state (a state whose polarization ellipse evolves fully randomly in a fixed plane), and a three-dimensional (3D) unpolarized state (a state whose polarization ellipse evolves fully randomly in the 3D space) \[Phys. Rev. A95, 053856 (2017)PLRAAN1050-294710.1103/PhysRevA.95.053856\]. For nonregular states, the middle component can be considered as an equiprobable mixture of two pure states, whose polarization ellipses lie in different planes. In this work, we identify a perfect nonregular state and introduce the degree of nonregularity as a measure of the proximity of a nonregular state to regularity. We also analyze and interpret the notion of polarization-state regularity in terms of polarimetric parameters. Our results bring new insights into the polarimetric structure of 3D light fields.

Off-resonant emission of photon pairs in nonlinear optical cavities

Valentin Averchenko, Gerhard Schunk, Michael Förtsch, Martin Fischer, Dmitry Strekalov, Gerd Leuchs, Christoph Marquardt

Cavity-assisted spontaneous parametric down-conversion (SPDC) and spontaneous four-wave mixing (SFWM) in nonlinear optical materials are practical and versatile methods to generate narrowband time-energy entangled photon pairs. Time- energy entangled photons with tailored spectro-temporal properties are particularly useful for efficient quantum optical interfaces. In this work we study the generation of photon pairs in cavity-assisted SPDC and SFWM for the general case of off-resonant conversion, namely, when the frequencies of the generated photons do not match the cavity resonances. Such a frequency mismatch in particular depends on temperature and requires an additional control in the experiment. First, we propose a generic model, for description of cavity-assisted SPDC and SFWM. We show that in both processes the mismatch reduces the generation rate of photons, distorts the spectrum and the auto-correlation function of the generated fields, as well as affects the photon generation dynamics. Second, we verify the results experimentally using parametric generation of photon pairs in a nonlinear whispering gallery mode resonator (WGMR) as an experimental platform with controlled frequency mismatch. Our work reveals the role of the frequency mismatch in the photon generation process and shows a way to control it. Obtained results constitute one more step in the direction of full control over the spectro-temporal properties of entangled photon pairs and the heralded generation of single-photon pulses with a tailored temporal mode.

The Geometrical Basis of PT Symmetry

Luis Sanchez-Soto, Juan J. Monzon

We reelaborate on the basic properties of PT symmetry from a geometrical perspective. The transfer matrix associated with these systems induces a Mobius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

Mutual Unbiasedness in Coarse-Grained Continuous Variables

Daniel S. Tasca, Piero Sánchez, Stephen P. Walborn, Łukasz Rudnicki

The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of “standard” coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.

Uncertainty Relations for Coarse-Grained Measurements: An Overview

Fabricio Toscano, Daniel S. Tasca, Łukasz Rudnicki, Stephen P. Walborn

Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.

Quantum metrology at the limit with extremal Majorana constellations

F. Bouchard, P. de la Hoz, G. Bjork, R. W. Boyd, Markus Grassl, Z. Hradil, E. Karimi, A. B. Klimov, Gerd Leuchs, et al.

Quantum metrology allows for a tremendous boost in the accuracy of measurement of diverse physical parameters. The estimation of a rotation<br>constitutes a remarkable example of this quantum-enhanced precision. The recently introduced Kings of Quantumness are especially germane for this task<br>when the rotation axis is unknown, as they have a sensitivity independent of that axis and they achieve a Heisenberg-limit scaling. Here, we report the<br>experimental realization of these states by generating up to 21-dimensional orbital angular momentum states of single photons, and confirm their high metrological abilities.<br>

Fibonacci-Lucas SIC-POVMs

Markus Grassl, Andrew J. Scott

We present a conjectured family of SIC-POVMs which have an additional symmetry group whose size is growing with the dimension. The symmetry group is related to Fibonacci numbers, while the dimension is related to Lucas numbers. The conjecture is supported by exact solutions for dimensions d=4,8,19,48,124,323, as well as a numerical solution for dimension d=844.

Multiparameter Quantum Metrology of Incoherent Point Sources: Towards Realistic Superresolution

J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, Luis Sanchez-Soto

We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using alinear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the<br>conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can<br>gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.

Efficient tomography with unknown detectors

L. Motka, M. Paur, J. Rehacek, Z. Hradil, L. L. Sanchez-Soto

We compare the two main techniques used for estimating the state of a physical system from unknown measurements: standard detector tomography and data-pattern tomography. Adopting linear inversion as a fair benchmark, we show that the difference between these two protocols can be traced back to the nonexistence of the reverse-order law for pseudoinverses. We capitalize on this fact to identify regimes where the data-pattern approach outperforms the standard one and vice versa. We corroborate these conclusions with numerical simulations of relevant examples of quantum state tomography.

Coarse graining the phase space of N qubits

Olivia Di Matteo, Luis Sanchez-Soto, Gerd Leuchs, Markus Grassl

We develop a systematic coarse graining procedure for systems of N qubits. We exploit the underlying geometrical structures of the associated discrete<br>phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In<br>particular, our procedure naturally yields a subset of the original measurement operators, which can be used to construct a coarse discrete Wigner function. These operators also constitute a systematic choice of incomplete measurements for the tomographer wishing to probe an intractably large system.

Codes for Simultaneous Transmission of Quantum and Classical Information

M. Grassl, Sirui Lu, Bei Zeng

We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes $[\![n,k{: }m,d]\!]_q$ with length $n$ and distance $d$, that simultaneously transmit $k$ qudits and $m$ symbols from a classical alphabet of size $q$. Many good codes such as $[\![7,1{: }1,3]\!]_2$, $[\![9,2{: }2,3]\!]_2$, $[\![10,3{: }2,3]\!]_2$, $[\![11,4{: }2,3]\!]_2$, $[\![11,1{: }2,4]\!]_2$, $[\![13,1{: }4,4]\!]_2$, $[\![13,1{: }1,5]\!]_2$, $[\![14,1{: }2,5]\!]_2$, $[\![15,1{: }3,5]\!]_2$, $[\![19,9{: }1,4]\!]_2$, $[\![20,9{: }2,4]\!]_2$, $[\![21,9{: }3,4]\!]_2$, $[\![22,9{: }4,4]\!]_2$ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.

Temporal and spectral properties of quantum light

B. Stiller, U. Seyfarth, G. Leuchs, C. Fabre, V. Sandoghdar, N. Treps, L.F. Cugliandolo

Optimal measurements for resolution beyond the Rayleigh limit

J. Rehacek, M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto

We establish the conditions to attain the ultimate resolution predicted by quantum estimation theory for the case of two incoherent point sources using a linear imaging system. The solution is closely related to the spatial symmetries of the detection scheme. In particular, for real symmetric point spread functions, any complete set of projections with definite parity achieves the goal. (C) 2017 Optical Society of America

Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes

Nuh Aydin, Nicholas Connolly, Markus Grassl

One of the most important and challenging problems in coding theory is to construct codes with good parameters. There are various methods to construct codes with the best possible parameters. A promising and fruitful approach has been to focus on the class of quasi-twisted (QT) codes which includes constacyclic codes as a special case. This class of codes is known to contain many codes with good parameters. Although constacyclic codes and QT codes have been the subject of numerous studies and computer searches over the last few decades, we have been able to discover a new fundamental result about the structure of constacyclic codes which was instrumental in our comprehensive search method for new QT codes over GF(7). We have been able to find 41 QT codes with better parameters than the previously best known linear codes. Furthermore, we derived a number of additional new codes via Construction X as well as standard constructions, such as shortening and puncturing.

New self-dual additive F_4-codes constructed from circulant graphs

Markus Grassl, Masaaki Harada

In order to construct quantum [[n, 0, d]] codes for (n, d) = (56, 15), (57, 15), (58, 16), (63, 16), (67, 17), (70, 18), (71, 18), (79, 19), (83, 20), (87, 20), (89, 21), (95, 20), we construct self-dual additive F-4-codes of length n and minimum weight d from circulant graphs. The quantum codes with these parameters are constructed for the first time. (C) 2016 Elsevier B.V. All rights reserved.

Multimode theory of single-photon subtraction

V. Averchenko, C. Jacquard, V. Thiel, C. Fabre, N. Treps

Achieving the ultimate optical resolution

Martin Paur, Bohumil Stoklasa, Zdenek Hradil, Luis L. Sanchez-Soto, Jaroslav Rehacek

Concatenated Codes for Amplitude Damping

Tyler Jackson, Markus Grassl, Bei Zeng

Implementing a neutral-atom controlled-phase gate with a single Rydberg pulse

Rui Han, Hui Khoon Ng, Berthold-Georg Englert

One can implement fast two-qubit entangling gates by exploiting the Rydberg blockade. Although various theoretical schemes have been proposed, experimenters have not yet been able to demonstrate two-atom gates of high fidelity due to experimental constraints. We propose a novel scheme, which only uses a single Rydberg pulse illuminating both atoms, for the construction of neutral-atom controlled-phase gates. In contrast to the existing schemes, our approach is simpler to implement and requires neither individual addressing of atoms nor adiabatic procedures. With parameters estimated based on actual experimental scenarios, a gate fidelity higher than 0.99 is achievable. Copyright (C) EPLA, 2016

Optical resolution from Fisher information

L. Motka, B. Stoklasa, M. D'Angelo, P. Facchi, A. Garuccio, Z. Hradil, S. Pascazio, F. V. Pepe, Y. S. Teo, et al.

Codeword Stabilized Quantum Codes for Asymmetric Channels

Tyler Jackson, Markus Grassl, Bei Zeng

Quantum technology: from research to application

Wolfgang P. Schleich, Kedar S. Ranade, Christian Anton, Markus Arndt, Markus Aspelmeyer, Manfred Bayer, Gunnar Berg, Tommaso Calarco, Harald Fuchs, et al.

The term quantum physics refers to the phenomena and characteristics of atomic and subatomic systems which cannot be explained by classical physics. Quantum physics has had a long tradition in Germany, going back nearly 100 years. Quantum physics is the foundation of many modern technologies. The first generation of quantum technology provides the basis for key areas such as semiconductor and laser technology. The "new" quantum technology, based on influencing individual quantum systems, has been the subject of research for about the last 20 years. Quantum technology has great economic potential due to its extensive research programs conducted in specialized quantum technology centres throughout the world. To be a viable and active participant in the economic potential of this field, the research infrastructure in Germany should be improved to facilitate more investigations in quantum technology research.

The many facets of the Fabry-Perot

Luis L. Sanchez-Soto, Juan J. Monzon, Gerd Leuchs

Parsing polarization squeezing into Fock layers

Christian R. Mueller, Lars S. Madsen, Andrei B. Klimov, Luis L. Sanchez-Soto, Gerd Leuchs, Christoph Marquardt, Ulrik L. Andersen

We investigate polarization squeezing in squeezed coherent states with varying coherent amplitudes. In contrast to the traditional characterization based on the full Stokes parameters, we experimentally determine the Stokes vector of each excitation subspace separately. Only for states with a fixed photon number do the methods coincide; when the photon number is indefinite, we parse the state in Fock layers, finding that substantially higher squeezing can be observed in some of the single layers. By capitalizing on the properties of the Husimi Q function, we map this notion onto the Poincare space, providing a full account of the measured squeezing.

Applying Grover's Algorithm to AES: Quantum Resource Estimates

Markus Grassl, Brandon Langenberg, Martin Roetteler, Rainer Steinwandt

We present quantum circuits to implement an exhaustive key search for the Advanced Encryption Standard (AES) and analyze the quantum resources required to carry out such an attack. We consider the overall circuit size, the number of qubits, and the circuit depth as measures for the cost of the presented quantum algorithms. Throughout, we focus on Clifford+T gates as the underlying fault-tolerant logical quantum gate set. In particular, for all three variants of AES (key size 128, 192, and 256 bit) that are standardized in FIPS-PUB 197, we establish precise bounds for the number of qubits and the number of elementary logical quantum gates that are needed to implement Grover's quantum algorithm to extract the key from a small number of AES plaintext-ciphertext pairs.

New self-dual additive F4-codes constructed from circulant graphs

Markus Grassl, Masaaki Harada

In order to construct quantum [[n,0,d]] codes for (n,d)=(56,15), (57,15), (58,16), (63,16), (67,17), (70,18), (71,18), (79,19), (83,20), (87,20), (89,21), (95,20), we construct self-dual additive F4-codes of length n and minimum weight d from circulant graphs. The quantum codes with these parameters are constructed for the first time.<br>

Stars of the quantum Universe: extremal constellations on the Poincare sphere

Gunnar Bjork, Markus Grassl, Pablo de la Hoz, Gerd Leuchs, Luis L. Sanchez-Soto

The characterization of the polarization properties of a quantum state requires the knowledge of the joint probability distribution of the Stokes variables. This amounts to assessing all the moments of these variables, which are aptly encoded in a multipole expansion of the density matrix. The cumulative distribution of these multipoles encapsulates in a handy manner the polarization content of the state. We work out the extremal states for that distribution, finding that SU(2) coherent states are maximal to any order, so they are the most polarized allowed by quantum theory. The converse case of pure states minimizing that distribution, which can be seen as the most quantum ones, is investigated for a diverse range of number of photons. Exploiting the Majorana representation, the problem appears to be closely related to distributing a number of points uniformly over the surface of the Poincare sphere.

Extremal quantum states and their Majorana constellations

G. Bjork, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, L. L. Sanchez-Soto

The characterization of quantum polarization of light requires knowledge of all the moments of the Stokes variables, which are appropriately encoded in the multipole expansion of the density matrix. We look into the cumulative distribution of those multipoles and work out the corresponding extremal pure states. We find that SU(2) coherent states are maximal to any order whereas the converse case of minimal states (which can be seen as the most quantum ones) is investigated for a diverse range of the number of photons. Taking advantage of the Majorana representation, we recast the problem as that of distributing a number of points uniformly over the surface of the Poincare sphere.

New Constructions of Codes for Asymmetric Channels via Concatenation

Markus Grassl, Peter W. Shor, Graeme Smith, John Smolin, Bei Zeng

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors.

Least-bias state estimation with incomplete unbiased measurements

Jaroslav Rehacek, Zdenek Hradil, Yong Siah Teo, Luis L. Sanchez-Soto, Hui Khoon Ng, Jing Hao Chai, Berthold-Georg Englert

Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimation strategies. We propose to use instead the physical state estimators that maximize the Shannon entropy of the unmeasured outcomes, which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions.

Classical polarization multipoles: paraxial versus nonparaxial

P. de la Hoz, G. Bjork, H. de Guise, A. B. Klimov, G. Leuchs, L. L. Sanchez-Soto

We discuss the polarization of paraxial and nonparaxial classical light fields by resorting to a multipole expansion of the corresponding polarization matrix. It turns out that only a dipolar term contributes when one considers SU(2) (paraxial) or SU(3) (nonparaxial) as fundamental symmetries. In this latter case, one can alternatively expand in SU(2) multipoles, and then both a dipolar and a quadrupolar component contribute, which explains the richer structure of this nonparaxial instance. These multipoles uniquely determine Wigner functions, in terms of which we examine some intriguing hallmarks arising in this classical scenario.

Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sanchez-Soto, E. Karimi, R. W. Boyd

Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81-0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.

Extremal states for photon number and quadratures as gauges for nonclassicality

Z. Hradil, J. Rehacek, P. de la Hoz, G. Leuchs, L. L. Sanchez-Soto

Rotated quadratures carry the phase-dependent information of the electromagnetic field, so they are somehow conjugate to the photon number. We analyze this noncanonical pair, finding an exact uncertainty relation, as well as a couple of weaker inequalities obtained by relaxing some restrictions of the problem. We also find the intelligent states saturating that relation and complete their characterization by considering extra constraints on the second-order moments of the variables involved. Using these moments, we construct performance measures tailored to diagnose photon-added and Schrodinger-cat-like states, among others.

Practical implementation of mutually unbiased bases using quantum circuits

U. Seyfarth, L. L. Sanchez-Soto, G. Leuchs

The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits. We also analyze the quantum circuits implementing the various entanglement classes.

Experimental violation of a Bell-like inequality with optical vortex beams

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sanchez-Soto, G. S. Agarwal

Optical beams with topological singularities have a Schmidt decomposition. Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement. This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions. We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum. Our experimental scheme, which is designed to give directly the parity of the Wigner function, yields negativity at the origin for LG(10) beams, whereas for LG(20) we always get a positive value.