### 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 Ahrens**

^{1}, Piotr Badziag^{1}, Adán Cabello^{1,2}and Mohamed Bourennane^{1}**Affiliation:**

^{1}Physics Department, Stockholm University, Sweden

^{2}Departamento de Fı´sica Aplicada II, Universidad de Sevilla, SpainThe 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.

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