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 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 linked to the geometry of the underlying algebra su(n). This ensures an intrinsic bound that is independent of the choice of parametrization.

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

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

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 reliable measure for informational completeness through collective encoding of quantum measurements, data and target states into grayscale images. By gradually accumulating measurements and data, these convolutional networks can efficiently certify a low-measurement-cost quantum-state characterization scheme. We confirm the potential of this machine-learning approach by presenting experimental results for both spatial-mode and multiphoton systems 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.

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

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

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

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

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 of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored and their potential applications 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 characterizes any rank-deficient completely-positive process with no a priori information about the process apart from the dimension of the system on which the process acts. It uses randomly-chosen input states and adaptive output von Neumann measurements. Both entangled and tensor-product configurations are flexibly employable in our scheme, the latter which naturally makes it especially compatible with many-body quantum computing. Two main features of this scheme are the certification protocol that verifies whether the accumulated data uniquely characterize the quantum process, and a compressive reconstruction method for the output states. We emulate multipartite scenarios with high-order electromagnetic transverse modes and optical fibers to positively demonstrate that, in terms of measurement resources, our assumption-free compressive strategy can reconstruct quantum processes almost equally efficiently using all types of input states and basis measurement operations, operations, independent of whether or not they are factorizable into tensor-product states.

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 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 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.

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, 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 single-photon level signals ten times smaller than their optical bandwidths with a ten-fold improvement in precision over the intensity-only Cramér-Rao 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 information processing. To deal with an infinite-dimensional Hilbert space, one in general has to handle large numbers of discretized measurements in tasks such as entanglement detection. Here we employ the continuous transverse spatial variables of photon pairs to experimentally demonstrate novel entanglement criteria based on a periodic structure of coarse-grained measurements. The periodization of the measurements allows for an efficient evaluation of entanglement using spatial masks acting as mode analyzers over the entire transverse field distribution of the photons and without the need to reconstruct the probability densities of the conjugate continuous variables. Our experimental results demonstrate the utility of the derived criteria with a success rate in entanglement detection of $\sim60\%$ relative to $7344$ studied cases.

Mutually unbiased coarse-grained measurements of two or more phase-space variables

E.C. Paul, S.P. Walborn, D.S. Tasca, Łukasz Rudnicki

Mutual unbiasedness of the eigenstates of phase-space operators-such as position and momentum, or their standard coarse-grained versions-exists only in the limiting case of infinite squeezing. In Phys. Rev. Lett. 120, 040403 (2018), it was shown that mutual unbiasedness can be recovered for periodic coarse graining of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two nonparallel phase-space operators. We illustrate these results through optics experiments, using the fractional Fourier transform to prepare and measure mutually unbiased phase-space variables. The differences between two and three mutually unbiased measurements is discussed. Our results contribute to bridging the gap between continuous and discrete quantum mechanics, and they could be useful in quantum-information protocols.

Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons

Frédéric Bouchard, Khabat Heshami, Duncan England, Robert Fickler, Robert W. Boyd, Berthold-Georg Englert, Luis Sanchez-Soto, Ebrahim Karimi

Quantum key distribution is on the verge of real world applications, where perfectly secure information can be distributed among multiple parties. Several quantum cryptographic protocols have been theoretically proposed and independently realized in different experimental conditions. Here, we develop an experimental platform based on high-dimensional orbital angular momentum states of single photons that enables implementation of multiple quantum key distribution protocols with a single experimental apparatus. Our versatile approach allows us to experimentally survey different classes of quantum key distribution techniques, such as the 1984 Bennett & Brassard (BB84), tomographic protocols including the six-state and the Singapore protocol, and to investigate, for the first time, a recently introduced differential phase shift (Chau15) protocol using twisted photons. This enables us to experimentally compare the performance of these techniques and discuss their benefits and deficiencies in terms of noise tolerance in different dimensions.

Tempering Rayleigh’s curse with PSF shaping

Martin Paúr, Bohumil Stoklasa, Jai Grover, Andrej Krzic, Luis Sanchez-Soto, Zdeněk Hradil, Jaroslav Řeháček

It has been argued that, for a spatially invariant imaging system, the information one can gain about the separation of two incoherent point sources decays quadratically to zero with decreasing separation. The effect is termed Rayleighx2019;s curse. Contrary to this belief, we identify a class of point-spread functions (PSFs) with a linear information decrease. Moreover, we show that any well-behaved symmetric PSF can be converted into such a form with a simple nonabsorbing signum filter. We experimentally demonstrate significant superresolution capabilities based on this idea.

Optimal measurements for quantum spatial superresolution

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

We construct optimal measurements, achieving the ultimate precision predicted by quantum theory, for the simultaneous estimation of centroid, separation, and relative intensities of two incoherent point sources using a linear optical system. We discuss the physical feasibility of the scheme, which could pave the way for future practical implementations of quantum-inspired imaging.

Residual and Destroyed Accessible Information after Measurements

Rui Han, Gerd Leuchs, Markus Grassl

When quantum states are used to send classical information, the receiver performs a measurement on the signal states. The amount of information extracted is often not optimal due to the receiver’s measurement scheme and experimental apparatus. For quantum nondemolition measurements, there is potentially some residual information in the postmeasurement state, while part of the information has been extracted and the rest is destroyed. Here, we propose a framework to characterize a quantum measurement by how much information it extracts and destroys, and how much information it leaves in the residual postmeasurement state. The concept is illustrated for several receivers discriminating coherent states.

Polarimetric purity and the concept of degree of polarization

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

The concept of degree of polarization for electromagnetic waves, in its general three-dimensional version, is revisited in the light of the implications of the recent findings on the structure of polarimetric purity and of the existence of nonregular states of polarization [J. J. Gil et al., Phys Rev. A 95, 053856 (2017)]. From the analysis of the characteristic decomposition of a polarization matrix R into an incoherent convex combination of (1) a pure state Rp, (2) a middle state Rm given by an equiprobable mixture of two eigenstates of R, and (3) a fully unpolarized state Ru−3D, it is found that, in general, Rm exhibits nonzero circular and linear degrees of polarization. Therefore, the degrees of linear and circular polarization of R cannot always be assigned to the single totally polarized component Rp. It is shown that the parameter P3D proposed formerly by Samson [J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973)] takes into account, in a proper and objective form, all the contributions to polarimetric purity, namely, the contributions to the linear and circular degrees of polarization of R as well as to the stability of the plane containing its polarization ellipse. Consequently, P3D constitutes a natural representative of the degree of polarimetric purity. Some implications for the common convention for the concept of two-dimensional degree of polarization are also analyzed and discussed.

Uncertainty-reality complementarity and entropic uncertainty relations

Łukasz Rudnicki

Reality of quantum observables, a feature of long-standing interest within foundations of quantum mechanics, has recently been quantified and deeply studied by means of entropic measures (Dieguez and Angelo 2018 Phys. Rev. A 97 022107). However, there is no state-independent 'reality trade-off' between non-commuting observables, as in certain systems all observables are real (Bilobran and Angelo 2015 Europhys. Lett. 112 40005). We show that the entropic uncertainty relation in the presence of quantum memory (Berta et al 2010 Nat. Phys. 6 659) perfectly supplements the discussed notion of reality, rendering trade-offs between reality and quantum uncertainty. State-independent complementarity inequalities involving entropic measures of both, uncertainty and reality, for two observables are presented.

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 constitutes a remarkable example of this quantum-enhanced precision. The recently introduced Kings of Quantumness are especially germane for this task 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 experimental realization of these states by generating up to 21-dimensional orbital angular momentum states of single photons, and confirm their high metrological abilities.

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 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 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.

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

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 phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In 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.

Towards optimal quantum tomography with unbalanced homodyning

Yong Siah Teo, Hyunseok Jeong, Luis L. Sanchez-Soto

Balanced homodyning, heterodyning and unbalanced homodyning are the three well-known sampling techniques used in quantum optics to characterize all possible photonic sources in continuous-variable quantum information theory. We show that for all quantum states and all observable-parameter tomography schemes, which includes the reconstructions of arbitrary operator moments and phase-space quasi-distributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently known to often give more accurate parameter reconstructions than conventional marginalized sampling with balanced homodyning. This result also holds for realistic photodetectors with subunit efficiency. With examples from first- through fourth-moment tomography, we demonstrate that unbalanced homodyning can outperform balanced homodyning when heterodyning fails to do so. This new benchmark takes us one step towards optimal continuous-variable tomography with conventional photodetectors and minimal experimental components.

Joint measurement of complementary observables in moment tomography

Yong Siah Teo, Christian R. Mueller, Hyunseok Jeong, Zdenek Hradil, Jaroslav Rehacek, Luis L. Sanchez-Soto

Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning (HOM) and heterodyning (HET) that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that HET can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.

Dimensionality of random light fields

Andreas Norrman, Ari T. Friberg, Jose J. Gil, Tero Setala

Background: The spectral polarization state and dimensionality of random light are important concepts in modern optical physics and photonics. Methods: By use of space-frequency domain coherence theory, we establish a rigorous classification for the electricfield vector to oscillate in one, two, or three spatial dimensions. Results: We also introduce a new measure, the polarimetric dimension, to quantify the dimensional character of light. The formalism is utilized to show that polarized three-dimensional light does not exist, while an evanescent wave generated in total internal reflection generally is a genuine three-dimensional light field. Conclusions: The framework we construct advances the polarization theory of random light and it could be beneficial for near-field optics and polarization-sensitive applications involving complex-structured light fields.

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.

Multimode theory of single-photon subtraction

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

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

Codeword Stabilized Quantum Codes for Asymmetric Channels

Tyler Jackson, Markus Grassl, Bei Zeng

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.

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.