Experimental Implementation of Device-Independent Dimension Witnesses

The authors in their laboratory in Stockholm last week. From left to right: Johan Ahrens, Adán Cabello, Piotr Badziag, Mohamed Bourennane.

Authors: Johan Ahrens¹, Piotr Badziag¹, Adán Cabello^1,2 and Mohamed Bourennane¹

Affiliation:
¹Physics Department, Stockholm University, Sweden
²Departamento de Fı´sica Aplicada II, Universidad de Sevilla, Spain

The concept of “dimension” is ubiquitous in physics. When we say that a physical system “has” dimension 2, or 3, or infinite, what do we mean? Why we say that a light switch has dimension 2 while a particle that can be anywhere within a line have infinite dimension? If we are provided by a black box emitting particles, how can we actually measure the dimension of the particles without knowing how the box works?

This month, Nature Physics publishes two papers [1, 2] describing experiments to determine the (minimum) dimension of particles emitted by a black box. These experiments measure some so-called “dimension witnesses”. Our experiment has been performed at Stockholm University (Sweden) in collaboration with the University of Seville (Spain) [1]; the other experiment was performed at the Institute of Photonic Sciences in Barcelona (Spain) in collaboration with the University of Bristol (UK) [2, also see last week's 2Physics article]. Both are based on a proposal of the Barcelona-Bristol group [3]. Here we explain what is a dimension witness and why is interesting to measure it in a black box scenario.

The dimension of a physical system on which a set of measurements can be carried out is the maximum number of perfectly distinguishable states using these measurements. This means that, among these measurements, there is at least one which allows us to distinguish between any two states.

A fundamental difference between classical and quantum physics is that, in classical physics, all the states are perfectly distinguishable, while this is not the case in quantum physics. For example, a quantum system of dimension 2 (or qubit) is a system in which the maximum number of perfectly distinguishable states is 2, but this does not mean, as in classical physics, that only 2 states are possible: there are infinite states, but it is only possible to distinguish 2.

Consider the following problem: We receive a black box with 3 buttons P1, P2 and P3, so every time we press one button, the box emits one particle. On this particle we can perform one measurement chosen between two, called M1 y M2 and represented by two buttons in a second box: whenever we press the button M1 (M2) we measure M1 (M2). Each of these measurements has two possible results that we denote as -1 and +1. The whole experiment is schematically illustrated in the following figure:

What can we say about the dimension of the particles emitted by the preparator? To answer that, we repeat the experiment many times, pressing all possible pairs of buttons Pi (i=1, 2, 3) and Mj (j=1, 2), and recording the frequencies of the different results.

A dimension witness is nothing but a linear combination of probabilities P(+1|Pi,Mj) of obtaining result +1 when preparing Pi and measuring Mj, such that its experimental value provides a lower bound to the dimension of the prepared systems. For example, the following combination T is a dimension witness:

T=P(+1|P1,M1)+P(+1|P1,M2)+P(+1|P2,M1)+P(-1|P2,M2)+P(-1|P3,M1).

Since probabilities cannot be higher than 1, then the maximum value for T is 5. Let us suppose we obtain T=5. This means that P(+1|P1,M2)=1 and P(-1|P2,M2)=1, which implies that M2 distinguishes P1 from P2. In addition, P(+1|P1,M1)=1 and P(-1|P3,M1)=1, indicating that M1 distinguishes P1 from P3. Finally, P(+1|P2,M1)=1 and P(-1|P3,M1)=1, thus M1 also distinguishes P2 from P3. Conclusion: If T=5, then dimension D is (at least) 3. However, if D=2, then T cannot be 5. Therefore, the experimental value of T allows us to have a lower bound for D.

Some dimension witnesses also allow us to distinguish between classical and quantum systems of the same dimension (e.g., between bits and qubits, or between trits and qutrits). For example, it can be proven that for classical systems of D=2 the maximum value of T is 4. However, for quantum systems of D=2 the maximum value is 4.414.

Specifically, in our experiment [1], we encode classical or quantum information in the polarization and spatial modes of individual photons and shown how the experimental value of two dimensional witnesses allow us to test whether the photons act as bits, or qubits, or trits, or qutrits.

The importance of such a tool is easy to understand if we notice that the physical system’s dimension determines its capacity to store, process and communicate information.

References:
[1] Johan Ahrens, Piotr Badziacedilg, Adán Cabello, Mohamed Bourennane, “Experimental device-independent tests of classical and quantum dimensions”. Nature Physics, 8, 592–595(2012). Abstract.
[2] Martin Hendrych, Rodrigo Gallego, Michal Mičuda, Nicolas Brunner, Antonio Acín, Juan P. Torres, “Experimental estimation of the dimension of classical and quantum systems”. Nature Physics, 8, 588–591(2012). Abstract. 2Physics article.
[3] Rodrigo Gallego, Nicolas Brunner, Christopher Hadley, and Antonio Acín, “Device-independent tests of classical and quantum dimensions”. Physical Review Letters 105, 230501 (2010). Abstract.

Testing the Dimension of Classical and Quantum Systems

Group leaders: Antonio Acin (left) and Juan Pérez Torres (right)

Authors: Martin Hendrych¹, Rodrigo Gallego¹, Michal Mičuda^1,2, Nicolas Brunner³, Antonio Acín^1,4, Juan P. Torres^1,5

Affiliations:
¹ICFO-Institut de Ciencies Fotoniques, Barcelona, Spain
²Department of Optics, Palacký University, Czech Republic
³H.H. Wills Physics Laboratory, University of Bristol, UK
⁴ICREA-Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
⁵Department of Signal Theory and Communications, Universitat Politècnica de Catalunya, Barcelona, Spain

The main goal of any scientific theory is to predict and explain the results of experiments. In doing so, the theory makes some assumptions about the experiment under consideration, based, for instance, on some a priori knowledge, or symmetries of the setup. Based on these assumptions, a model -- possibly with some free parameters -- is constructed. The model is satisfactory whenever it is able to reproduce the observed results for reasonable values of the free parameters.

Consider for instance a quantum experiment involving measurements on several interacting particles. The quantum postulates tell us that the state, interactions and measurements in the setup should be described by operators acting on a Hilbert space of a given dimension. A standard practice is to assume that the dimension of the Hilbert space, that is, the number of independent parameters necessary to describe the setup is known. The theoretical model should then provide the operators in this space that reproduce the observed measurement statistics.

However one may ask whether this initial assumption on the dimension of the Hilbert space is in fact unavoidable to describe the experimental data or, on the contrary, if it is possible to estimate the dimension of a completely unknown quantum system only from the statistics of measurements performed on it. The concept of dimension witnesses gives a positive answer to the last question, as it provides lower bounds on the dimension of an unknown system only from the collected measurement results and without making any assumptions on the physical system under consideration. Clearly, without any assumption, the best one can hope for is to get lower bounds on the minimal dimension needed to describe an unknown system. In fact, one can never exclude the existence of further degrees of freedom in the system that are not seen in the present setup but can be accessed using a more refined experimental arrangement.

Our recent work [1] and a similar and independent experiment [2] represent the first experimental demonstrations of a dimension witness. Dimension witnesses were introduced in Reference[3] in the context of Bell inequalities. Later, alternative techniques for bounding the dimension of unknown systems were proposed based on random access codes [4] or the time evolution of a quantum observable [5]. In our experimental demonstration, we followed the approach presented in Reference[6], which applies to a “prepare and measurement” scenario as the one depicted in Figure 1. In this scenario there are two devices, the state preparator and the measurement device. These devices are seen as black boxes, as no assumptions are made on their internal working. At the state preparator, a quantum state ρ_x is prepared, out of N possible states. The state is then sent to the measuring device. There, a measurement y is performed, among M possible measurements, which produces a result b that can take K different values. The whole experiment is thus described by the probability distribution p(b|x,y), giving the probability of obtaining outcome b when measurement y is performed on the prepared state x. The goal is to estimate the minimal dimension of the mediating quantum particle between the two devices needed to describe the observed statistics.

Figure 1: Prepare-and-measure scenario for dimension witnesses. At the state preparator one can choose to prepare one out of N possible quantum states. The prepared state is denoted by ρ_x. The quantum state is then sent to the measuring device, where a measurement y is performed among M possibilities. The measurement result is denoted by b and can take K different values. For instance, the figure shows a scenario with four preparations and three measurements.

A dimension witness for a system of dimension d is simply a function of the observed probabilities that is bounded by a given value for all systems of dimension not larger than d. If in a given experiment the observed value of the dimension witness exceeds this bound, the system must necessarily have dimension larger than d. In our experiment, we observed the violation of one of the dimension witnesses introduced in [6], denoted by I₄. The maximum values this witness can take for classical or quantum systems of dimension up to four are given in Table 1. Note that if the system dimension is assumed to be bounded, the witness also allows distinguishing between classical and quantum systems.

	Bit	Qubit	Trit	Qutrit	Quart/Ququart
I4	5	6	7	7.97	9

Table 1: Classical and quantum bounds for the dimension witness I₄. The witness I₄ can be used to discriminate ensembles of classical and quantum states of dimension up to 4. Note that for some values of the dimension a gap appears between classical and quantum systems. Thus, if one assumes a bound on the dimension of the system, the witness can be used to certify its quantum nature.

Obviously, to demonstrate the dimension witness, we need to construct quantum states of different dimensions. Fortunately, photons have a rich structure: they have polarization, frequency and spatial shape. Moreover, pairs of photons can be entangled [7]. In our experiment we take advantage of all these features. First of all, multidimensional spaces of up to dimension 4 are created by generating photons in a superposition of two orthogonal polarization states (two dimensions) embedded into one out of two specific spatial modes (two more dimensions).

The state preparation works as follows. By means of spontaneous parametric down conversion, namely the generation of two lower frequency photons when a second order nonlinear crystal is pumped by an intense higher frequency optical beam, we generate photon pairs entangled in the polarization and spatial degrees of freedom. The detection of one of the photons in a tailored state effectively prepares (projects) the second photon in the desired quantum state. In our experiment, we are also interested in demonstrating the separation between quantum and classical systems of the same dimension. We achieve this at the preparation by exploiting the frequency degree of freedom, which is used to change the superposition that occurs in polarization from coherent (quantum) to incoherent (classical). The prepared photon is finally sent to the measuring device, where it is detected using optical tools very similar to those used in its generation: spatial light modulators, polarizers, optical fibers and single-photon counting modules.

Figure 2: (click on the image to view higher resolution version) Experimental results. The experiment probes the dimension witness I₄ using systems of different nature, classical or quantum, and dimension (bit-qubit, trit-qutrit and quart). In the case of dimension 4 (quart), the dimension witness is insensitive to the quantum/classical transition (see also Table 1). For all dimensions, a violation of the corresponding bound is observed, certifying the dimension of the system.

To conclude, we have demonstrated that the dimension of classical and quantum systems can be bounded only from the measurement statistics without any extra assumption on the devices used in the experiment. Dimension witnesses represent an example of a device-independent estimation technique, in which relevant information about an unknown system is obtained only from the measurement data. Our work demonstrates how the device-independent approach can be employed to experimentally estimate the dimension of an unknown system. Beyond the fundamental motivation, the estimation of the dimension of unknown quantum systems is also relevant from a quantum information perspective, where the Hilbert space dimension is a resource that enables more powerful quantum information protocols. In fact, the quantum/classical distinction provided by dimension witnesses when the system dimension is bounded has recently been used for constructing protocols for secure key distribution [8] and randomness generation [9].

References
[1] Martin Hendrych, Rodrigo Gallego, Michal Mičuda, Nicolas Brunner, Antonio Acín, Juan P. Torres, "Experimental estimation of the dimension of classical and quantum systems", Nature Physics 8, 588–591 (2012). Abstract.
[2] Johan Ahrens, Piotr Badziacedilg, Adán Cabello, Mohamed Bourennane, "Experimental device-independent tests of classical and quantum dimensions", Nature Physics, 8 592–595 (2012). Abstract.
[3] Nicolas Brunner, Stefano Pironio, Antonio Acin, Nicolas Gisin, André Allan Méthot, and Valerio Scarani, "Testing the Dimension of Hilbert Spaces", Physical Review Letters, 100, 210503 (2008). Abstract.
[4] Stephanie Wehner, Matthias Christandl, and Andrew C. Doherty, "Lower bound on the dimension of a quantum system given measured data", Physical Review A 78, 062112 (2008). Abstract.
[5] Michael M. Wolf and David Perez-Garcia, "Assessing Quantum Dimensionality from Observable Dynamics", Physical Review Letters, 102, 190504 (2009). Abstract.
[6] Rodrigo Gallego, Nicolas Brunner, Christopher Hadley, and Antonio Acín, "Device-Independent Tests of Classical and Quantum Dimensions", Physical Review Letters, 105, 230501 (2010). Abstract.
[7] Juan P. Torres, K. Banaszek and I. A. Walmsley, "Engineering Nonlinear Optic Sources of Photonic Entanglement", Progress in Optics 56, Chapter V, 227-331 (2011). Abstract.
[8] Marcin Pawłowski and Nicolas Brunner, "Semi-device-independent security of one-way quantum key distribution", Physical Review A 84, 010302(R) (2011). Abstract.
[9] Hong-Wei Li, Marcin Pawłowski, Zhen-Qiang Yin, Guang-Can Guo, and Zheng-Fu Han, "Semi-device-independent randomness certification using n→1 quantum random access codes", Physical Review A 85, 052308 (2012). Abstract.

A Newtonian Approach to Negative Index Metamaterials

Hosang Yoon (left) and Donhee Ham (right)














Authors: Hosang Yoon and Donhee Ham 
Affiliation:
School of Engineering and Applied Sciences, Harvard University, USA

Link to Donhee Ham Research Group >>

Negative index metamaterials have been celebrated due to their unusual ability to manipulate electromagnetic waves, such as bending light in the ‘wrong’ direction [1] and focusing light below the diffraction limit [2], which may prove technologically useful. To achieve negative refraction, a variety of material systems with engineered electric and/or magnetic properties have been developed. Reporting in Nature [3], we have demonstrated a ‘kinetic’ route to negative refraction, where exploitation of acceleration of electrons in a two-dimensional (2D) conductor leads to extraordinarily strong negative refraction with refractive index as large as -700.

Electrons in a conductor subjected to an electric field collectively accelerate according to Newton’s 2nd law of motion, creating a current that lags the electric field by 90°, as in a magnetic inductor. The conductor may then be considered as an inductor, but with its inductance being of kinetic origin [4]. This kinetic inductance is usually not much appreciable in ordinary three-dimensional conductors, but in a 2D conductor, it is orders-of-magnitude larger than magnetic inductance, which is what we exploit to create the dramatically strong negative refraction.

The acceleration of electrons is continually interrupted by their collisions with vibrations, impurities, or defects of the crystal lattice. Imagine a time-varying sinusoidal electric field at a given frequency, which accelerates/decelerates electrons into an oscillation motion. If the mean scattering rate is too high to accommodate an appreciable fraction of an oscillation period, the acceleration effect is hard to observe, and the conductor acts as an Ohmic resistor. This can be avoided in two ways: one is to cool the conductor to lower the scattering rate as in our work [3]; the other is to increase the frequency at room temperature, as currently pursued in our lab.

Figure 1. Optical micrograph (left) and schematic (right) of the metamaterial. The 2DEG strip array is connected to electromagnetic waveguides to the left and right. (Image reproduced from the paper published in Nature [3])

We employ a GaAs/AlGaAs 2D electron gas (2DEG) as a demonstrational 2D conductor. Our metamaterial designed in the GHz frequency range is an array of 2DEG strips (Fig. 1). Signal (S) lines flanked by ground (G) lines on the left and right of the strip array are coplanar waveguides (CPWs) that guide electromagnetic waves to and from the metamaterial. As an electromagnetic wave arrives from the left CPW, its electric field between the S and G lines accelerates electrons in/along the leftmost few strips. This inductive movement of electrons is capacitively coupled to the strip on the right, accelerating electrons there. This dynamics repeats to propagate an ‘electro-kinetic’ wave from left to right, perpendicular to the strips acting as kinetic inductors. It is this electro-kinetic wave that is negatively refracting. The negative refractive index is large due to the large kinetic inductance of the GaAs/AlGaAs 2DEG strip.

Figure 2. (Click on the image to see high resolution version) Dispersion relation (left) and refractive index (middle) of a 13-strip metamaterial with strip length 112 μm and periodicity 1.25 μm measured at different cryogenic temperatures. (right) Refractive index measured for metamaterials with various strip length (l) and periodicity (a). (Image reproduced from the paper published in Nature[3]) 

Microwave scattering experiments over 1~50 GHz confirms negative refraction; the measured dispersion of the electro-kinetic wave (Fig. 2, left) shows opposite signs for the phase and group velocities above a cutoff frequency. The corresponding negative refractive index is hundreds in magnitude, two orders of magnitude larger than typical negative refractive indices (Fig. 2, middle). With varying geometric parameters, an index as large as -700 is obtained (Fig. 2, right).

The very large negative index brings the science of negative refraction into drastically miniaturized scale, enabling ultra-subwavelength manipulation of electromagnetic waves. We expect to see the similar result at room temperatures by increasing the frequency towards the THz range, which is the direction our lab is now taking.

References:
[1] V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Soviet Physics Uspekhi, 10, 509–514 (1968). Abstract.
[2] J. B. Pendry, "Negative refraction makes a perfect lens," Physical Review Letters, 85, 3966–3969 (2000). Abstract.
[3] H. Yoon, K. Y. M. Yeung, V. Umansky, and D. Ham, "A Newtonian approach to extraordinarily strong negative refraction," Nature, 488, 65–69 (2012). Abstract.
[4] R. Meservey, "Measurements of the kinetic inductance of superconducting linear structures," Journal of Applied Physics, 40, 2028–2034 (1969). Abstract.

Importance of Electron-Electron Interactions in Graphene

Michael Crommie [Photo by Roy Kaltschmidt, Courtesy: Lawrence Berkeley National Laboratory]

 Perhaps no other material is generating as much excitement in the electronics world as graphene, sheets of pure carbon just one atom thick through which electrons can race at nearly the speed of light – 100 times faster than they move through silicon. Superthin, superstrong, superflexible and superfast as an electrical conductor, graphene has been touted as a potential wonder material for a host of electronic applications, starting with ultrafast transistors. For the vast potential of graphene to be fully realized, however, scientists must first learn more about what makes graphene so super. The latest step in this direction has been taken by researchers with the U.S. Department of Energy (DOE)’s Lawrence Berkeley National Laboratory (Berkeley Lab) and the University of California (UC) Berkeley.

Michael Crommie, a physicist who holds joint appointments with Berkeley Lab’s Materials Sciences Division and UC Berkeley’s Physics Department, led a study in which the first direct observations at microscopic lengths were recorded of how electrons and holes respond to a charged impurity – a single Coulomb potential – placed on a gated graphene device. The results provide experimental support to the theory that interactions between electrons are critical to graphene’s extraordinary properties. This work has been published online on July 29th in the journal 'Nature Physics'[1].

“We’ve shown that electrons in graphene behave very differently around charged impurities than electrons in other materials,” Crommie says. “Some researchers have held that electron-electron interactions are not important to intrinsic graphene properties while others have argued they are. Our first-time-ever pictures of how ultra-relativistic electrons re-arrange themselves in response to a Coulomb potential come down on the side of electron-electron interactions being an important factor.”

Graphene sheets are composed of carbon atoms arranged in a two-dimensional hexagonally patterned lattice, like a honeycomb. Electrons moving through this honeycomb lattice perfectly mimic the behavior expected of highly relativistic charged particles with no mass: think of a ray of light that is electrically charged. Because this is the same behavior displayed by highly relativistic free electrons, charge-carriers in graphene are referred to as “Dirac quasiparticles,” after Paul Dirac, the scientist who first described the behavior of relativistic fermions in 1928.

“In graphene, electrons behave as massless Dirac fermions,” Crommie says. “As such, the response of these electrons to a Coulomb potential is predicted to differ significantly from how non-relativistic electrons behave in traditional atomic and impurity systems. However, until now, many key theoretical predictions for this ultra-relativistic system had not been tested.”

Image 1: This zoom-in STM topograph shows one of the cobalt trimers placed on graphene for the creation of Coulomb potentials – charged impurities – to which electrons and holes could respond. (Image courtesy of Crommie group)

Working with a specially equipped scanning tunneling microscope (STM)in ultra-high vacuum, Crommie and his colleagues probed gated devices consisting of a graphene layer deposited atop boron nitride flakes which were themselves placed on a silicon dioxide substrate, the most common of semiconductor substrates.

“The use of boron-nitride significantly reduced the charge inhomogeneity of graphene, thereby allowing us to probe the intrinsic graphene electronic response to individual charged impurities,” Crommie says. In this study, the charged impurities were cobalt trimers constructed on graphene by atomically manipulating cobalt monomers with the tip of an STM.”

Image 2: The response of ultrarelativistic electrons in graphene to Coulomb potentials created by cobalt trimers was observed to be signficantly different the response of non-relativistic electrons in traditional atomic and impurity systems. (Image courtesy of Crommie group)

The STM used to fabricate the cobalt trimers was also used to map (through spatial variation in the electronic structure of the graphene) the response of Dirac quasiparticles – both electron-like and hole-like – to the Coulomb potential created by the trimers. Comparing the observed electron–hole asymmetry to theoretical simulations allowed the research team to not only test theoretical predictions for how Dirac fermions behave near a Coulomb potential, but also to extract graphene’s dielectric constant.

“Theorists have predicted that compared with other materials, electrons in graphene are pulled into a positively-charged impurity either too weakly, the subcritical regime; or too strongly, the supercritical regime,” Crommie says. “In our study, we verified the predictions for the subcritical regime and found the value for the dielectric to be small enough to indicate that electron–electron interactions contribute significantly to graphene properties. This information is fundamental to our understanding of how electrons move through graphene.”

Reference:
[1] Yang Wang, Victor W. Brar, Andrey V. Shytov, Qiong Wu, William Regan, Hsin-Zon Tsai, Alex Zettl, Leonid S. Levitov, Michael F. Crommie, "Mapping Dirac quasiparticles near a single Coulomb impurity on graphene", Nature Physics, doi:10.1038/nphys2379 (Published online July 29, 2012). Abstract.

[This article is written by Lynn Yarris of Lawrence Berkeley National Laboratory]