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Abstract
Dimensionality plays an important role in various information-theoretic tasks and can be seen as a key resource for quantum information processing. In this work, for the first time to the best of our knowledge, we report an experimental test of classical and quantum dimensions in a prepare-and-measure scenario through a nonlinear dimension witness (NDW), where the preparer and the measurer have no shared randomness. We propose a theoretical method to certify whether the generated states can be decomposed as products of lower dimensional states with the NDW, certify the dimension of the high-dimensional system, and identify whether the quantum system is irreducible in a unified picture in our experiment. To demonstrate the practicability of this method, we experimentally witness the generation of classical and quantum systems encoded in the high-dimensional orbital angular momentum (OAM) state under the effects of atmospheric turbulence. Our results show that this nonlinear dimension witness is highly robust and can be used in the presence of noise and low detection efficiency; hence, we estimate the dimension in our experiment in a detection-loophole-free manner through the NDW. We believe our work opens new avenues for dimension witnesses, especially for device-independent estimation of quantum systems and quantum communications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
With the increasing complexity of quantum systems, there is a fast-growing requirement for certification and verification of their performance. The self-testing of quantum states and measurements has especially attracted a lot of attention [1–4]. Dimensionality is a fundamental property of physical systems and is widely regarded as a valuable resource in quantum computation and quantum communications [5], so is it possible to estimate the dimension of an uncharacterized physical system (loosely speaking, the number of relevant degrees of freedom) in a device-independent manner? Specifically, can lower bounds on the dimension of an unknown system be obtained from measurement data alone without making assumptions about the devices used to perform the experiment? This question has received much attention in recent years and has been known as device-independent dimension witness (DIDW) [6–9]. Estimating the dimension of an unknown quantum system is also relevant from the perspective of quantum information science [10–16] and ideas from DIDW have allowed us to prove the security of certain cryptographic schemes [17,18] and can also be used for randomness certification [19].
The dimension witness approach was later generalized to the prepare-and-measure (PM) scenario [20], which is simpler and makes it easier to implement experimentally [21–25]. These dimension witnesses allowed the devices to be correlated via shared randomness and generated the linear dimension witnesses. Later, the concept of an irreducible dimension witness was investigated to provide information about the system composition [26,27]. However, the previous dimension witnesses suffer from two drawbacks that restrict their applicability. First, the dimension witnesses do not always give a general lower bound for the dimension of the underlying quantum system; for example, the lower bound of a specific dimension for the irreducible dimension witness must be calculated numerically. Second, the detection loophole imposes a serious limitation when measuring high-dimensional systems. As shown in [28], the required efficiency of the lower bounds tends to one as the dimension increases; hence, the detection loophole plays an important role when dealing with high-dimensional systems.
Note that previous works on linear dimension witnesses allowed the devices to be correlated via shared randomness. However, shared randomness can be viewed as a resource [29] and, in a practical setup, it might be more natural to assume that all devices are independent. Therefore, conditions for a dimension witness with uncorrelated devices have received much attention, where one needs to use the nonlinear dimension witness (NDW) [30–32]. In our work, we overcome these challenges with the help of the NDW. We show a general lower bound in the NDW and propose, for what we believe is the first time, a general lower bound for the irreducible dimension witness.
So far, only linear dimension witnesses have been realized experimentally [21,25,27]. The aim of this work is to certify the dimension of the high-dimensional system and identify whether the quantum system is irreducible in a unified picture through the NDW. We propose a theoretical method to certify whether the generated states can be decomposed as products of lower dimensional states with the NDW. To demonstrate the practicability of the NDW, we also experimentally test the classical and quantum systems encoded in the high-dimensional OAM state under the effects of atmospheric turbulence. Our results show that this NDW is highly robust and can be used in the presence of noise and low detection efficiency; hence, we can estimate the dimension in a detection-loophole-free manner in our experiment.
2. THEORETICAL IDEA
The considered scenario consists of two devices (Fig. 1): the state preparator and the measurement device. When pressing button $x$, the box emits state ${\rho _x}$. When button $y$ is pressed, the device performs measurement ${M_y}$ on the incoming state and the measurement produces an outcome $b \in \{- 1, + 1\}$.
Fig. 1. Testing the dimension of an unknown system and the irreducible high-dimensional quantum systems in a prepare-and-measure scenario.
Consider the situation of classical systems of dimension $d$. Given the choice of preparation $x$, the preparator sends a classical message $m = 0,1,\ldots,d - 1$ to the measurement device. If the internal randomness of the preparation is ${\lambda _1}$ and the internal randomness of the measurement is ${\lambda _2}$, then the behavior observed in the experiment is given by
For classical systems of dimension $d$, one has
and one can achieve ${W_k} \geqslant 1$ for $d \gt k$.For quantum systems of dimension $d$, we get
while ${W_k} \gt 0$ is possible whenever $d \gt \sqrt k$ (details in Supplement 1).We study the concept of irreducible dimension using the NDW and investigate whether the statistics of our experiments could have come from composite systems of a smaller dimension. For example, imagine that a quantum system of dimension $d$ is composed of $N$ independent quantum systems each of dimension ${d_i}$, so that it has Hilbert space dimension $d = \prod\limits_{i = 1}^N {d_i}$. Since the number of real parameters in a density matrix of dimension ${d_i} \times {d_i}$ is $d_i^2 - 1$, the composite quantum system can be parametrized in a real space of dimension $\sum\limits_{i = 1}^N (d_i^2 - 1)$. For simplicity, suppose the $N$ independent quantum systems have the same dimension ${d_0}$. Then the number of real parameters needed to describe the composite quantum system is $N(d_0^2 - 1)$, and it then follows that
For example, the observation ${W_7} \gt 0$ for a quantum system of dimension 4 implies that the system cannot be composed of two product qubit degrees of freedom.
In our work, we perform the experiment in a more practical way and imagine we don’t know the system dimension. To witness the system dimension, we implement the NDW of the family ${W_k}$ and randomly choose the initial states and measurements. The basic idea (see the flow chart shown in Fig. 2) is to measure ${W_k}$ for increasingly larger values of $k$ until one finds ${W_k} = 0$, in which case one has a lower bound on the dimension via Eq. (3) or Eq. (4). For example, in the classical case, when one arrives at $k$ such that ${W_k} = 0$, one necessarily has that ${W_{k - 1}} \ne 0$; therefore, from Eq. (3) $d \ge k$. For quantum systems we have a similar process, although to certify integer values of dimension we have a slightly more complicated update rule for $k$ due to the quadratic relation between $k$ and the certified dimension in Eq. (4). In the experiment, if we find that ${W_k}$ is 0, we must reselect the preparations and the measurements and perform the experiment again to determine whether ${W_k}$ is 0. We also show that the probability of ${W_{\sqrt k \lt d}} = 0$ is 0 when you randomly choose the initial states and measurements for the quantum systems, which means you can get the system dimension in a more practical way (details in Supplement 1). Essentially, the quadratic separation between the classical and quantum bounds is a result of the ability of quantum systems to have off-diagonal terms (i.e., coherence) in the density matrix. Such a large separation makes the method particularly convenient to distinguish between quantum and classical systems.
3. EXPERIMENTAL SETUP AND RESULTS
The experimental setup of the dimension witness is shown in Fig. 3. In our experiment, a type-2 beta–barium–borate (BBO, $9.0 \times 7.0 \times 1.0\;{\rm mm^3}$, $\theta = 41.44^ \circ$) crystal is pumped by a frequency-doubled femtosecond pulse (400 nm, 76 MHz repetition rate) from a mode-locked Ti:sapphire laser to generate the degenerate photon pairs. After passing through the interference filter (IF, $\Delta \lambda = 3\;{\rm nm}$, $\lambda = 800\;{\rm nm}$), the photon pairs generated in the spontaneous parametric down-conversion (SPDC) process are coupled into single-mode fibers separately. A single-photon state is prepared by triggering on one of the two photons. The coincidence counting rate collected by the avalanche photodiodes (APDs) is about $1.8 \times {10^5}$ in 60 s, and the measurement time for each experiment was 7 s.
Fig. 3. Experimental setup for testing classical and quantum dimensions. We creatively use the Faraday rotator to decrease the angle between the incident light and the reflected light when the laser beam is reflected from the spatial light modulator. To avoid the Gouy phase-shift effect, an imaging $4f$ system is implemented between the screens of the two spatial light modulators. Legend: PH, pinhole; HWP, half-wave plate; FC, fiber coupler; PBS, polarizing beam splitter; SLM, spatial light modulator; FR, Faraday rotator.
In our experiment, the prepared states are encoded in the OAM of single photons [33]. OAM-carrying photons are used to enable high-capacity optical communications [34,35] and versatile optical tweezers [36]. As shown in Fig. 3, the signal photon is projected on the Gaussian state by means of a single-mode fiber. After expanding the laser beam with two lens, the OAM state of signal photons is manipulated with the first spatial light modulator (SLM) to prepare desired superposition state. Many good methods have been developed to encode high-dimensional states with a single phase-only SLM [37–39]. We adopt the method [39] that modulates the wavefront according to computer-generated holograms specifically calculated to maximize the state fidelity. Projective measurements are performed by combining the second SLM with a single-mode fiber and a single-photon detector. If the incoming photon carried the OAM mode corresponding to a projection of the measurement, the phase of the mode is flattened and the photon will couple to the single-mode fiber. We can achieve different measurements by changing the holograms displayed on the SLM using the same experimental apparatus without intermediate adjustments. Figure 4(a) shows some intensity profiles of the high-dimensional OAM quantum states used in our experiment. We also give the theoretically calculated intensity profiles of the OAM quantum states, which show that the intensity profiles produced in our experiment are very good. We also show the crosstalk matrices of OAM states used in our experiment [Figs. 4(b) and 4(c)], measure the power of the reflected light in the first order of diffraction for each SLM, and normalize the projection outcomes by the reflection efficiencies of each SLM (details in Supplement 1), so the cross-talk of the OAM states in our experiment is very small.
Fig. 4. Experimental analysis of different OAM qudit photonic states. (a) Theoretically calculated intensity profiles and experimentally produced intensity profiles of the quantum states used in our experiment. (b) and (c) Cross-talk matrix between the different OAM modes.
The experimental results of the dimension witnesses for the classical systems are shown in Fig. 5(a). We have performed dimension witnesses for different dimensional classical systems and the experimental results agree well with the corresponding theoretical prediction. To perform the dimension witnesses for the classical systems of dimension 10, we start the measurement with $k = 1$, which means we need two initial states and one binary measurement. Then we find that the value of the dimension witness ${W_1}$ is $0.999 \pm 0.00185$, which is larger than zero, so we need to continue the experiment and increase the value of $k$ each time. When the value of $k$ is 10, we find that the value of the dimension witness ${W_{10}}$ is $0.00413 \pm 0.00397$. After checking the result of ${W_{10}} = 0$ again and the value of ${W_9}$ is not zero, we can make sure that the dimension of the classical system is at least 10. We also show the dimension witnesses for classical systems of dimension 2, 4, 6, and 8, whose results can also be seen in Fig. 5(a) (details in Supplement 1).
Fig. 5. Experimental results of the dimension witnesses. (a) Dimension witnesses of classical systems for dimension 2, 4, 6, 8, and 10. (b) Dimension witnesses of quantum systems for dimension 2, 4, and 6.
The experimental results of the dimension witnesses for the quantum systems are shown in Fig. 5(b). The dimension witnesses for quantum systems differ from classical systems since the value of ${W_k}$ is 0 only when $d \le \sqrt k$. Here we show the dimension witness for quantum systems of dimension 6. We start the experiment with $k = 2$. The value of the dimension witness ${W_2}$ was found to be $0.459 \pm 0.0646$, which means $d \gt \sqrt 2$. We therefore need to repeat the experiment with a higher value of $k$. Since ${W_k} \gt 0 \Rightarrow d \gt \sqrt k$, we focus only on the cases where $\sqrt k$ is an integer. For the dimension witness ${W_4}$ we obtained $0.583 \pm 0.084$, which means $d \gt \sqrt 4$. We then continued this process for $k = 9$, $k = 16$, $k = 25$, and $k = 36$. When the value of $k$ is 36, we find that ${W_{36}}$ is $0.000147 \pm 0.00149$, which is very close to zero. Since the value of ${W_{25}}$ is not zero, we can make sure that the dimension of the quantum system is at least 6. We also show the dimension witnesses of quantum systems for dimension 2 and 4. The corresponding results can be seen in Fig. 5(b).
In addition to certifying classical or quantum dimension, one can use the NDW to compare classical and quantum systems of the same dimension. For example, when the dimension of a classical or quantum system is 4, one has ${W_4} = 0$ for quarts and ${W_4} \gt 0$ for ququarts; therefore, ququarts outperform classical systems of the same dimension. In our experiment, the value of the dimension witness ${W_4}$ for the classical system is $0.000197 \pm 0.00324$ and the value of the dimension witness ${W_4}$ for the quantum system is $0.583 \pm 0.084$. Hence, our results also show a difference between classical and quantum systems of the same dimension. In the dimension witness, quantum system means that the density matrix of the states has off-diagonal terms; on the other hand, the off-diagonal terms vanish for all states in classical systems and the coherence is lost.
We also study the concept of irreducible dimension using the NDW. Dimension witnesses allow for the device-independent certification of the minimum dimension required to describe a given physical system; however, these tests do not provide information about the system composition. This point has been recently investigated in the Bell scenario [26] and in the PM scenario [27]. We take a similar approach to investigate whether the statistics of our experiments could have come from composite systems of a smaller dimension through the NDW. From Eq. (5), we know that the observation ${W_7} \gt 0$ for a quantum system of dimension 4 implies that the system cannot be composed of two product qubit degrees of freedom. We verified this assumption by testing the witness ${W_7}$ on systems of dimension 4 encoded in OAM, and systems of dimension 4 encoded using noncoupled different degrees of freedom of a photon (i.e., polarization and OAM). In our experiment [Fig. 6(a)], for the irreducible four-dimensional (4D) systems, the value of ${W_7}$ is $0.32 \pm 0.0624$, but for the 4D state that is encoded using polarization and OAM degrees of freedom, the value of ${W_7}$ is $0.0000881 \pm 0.0218$, which shows that we can investigate whether the statistics of our experiments could have come from composite systems of a smaller dimension.
Fig. 6. (a) Experimental results of the irreducible dimension witness with ${W_7}$. (b) Robustness of the NDW for the four-dimensional quantum systems.
Finally, we investigate the robustness of the NDW. As shown in [30], to show the robustness of the dimension witness, the effect of technical imperfections (such as background noise and detection efficiency) was investigated, then we can find that the nonlinear dimension witness can tolerate an arbitrary amount of background noise or an arbitrarily low efficiency in the theory. In our experiment, we study the effects of atmospheric turbulence on high-dimensional OAM states and the atmospheric turbulence is simulated by a single phase screen based on the Kolmogorov theory of turbulence [40,41]. The turbulence is implemented as a phase-only distortion on a single phase screen with the second spatial light modulator in our experiment. It is difficult to accurately simulate the OAM states under the effect of noise (such as atmospheric turbulence) purely numerically according to [42]. We considered values of $N = \frac{{{w_0}}}{{{r_0}}}$ [41], representing the scintillation strength of the random phase function, in the range 0 to 1.0. As shown in [41], negativity decreases gradually with increasing the scintillation strength of the random phase function, which proves that it is an effective method. In our experiment, to show the robustness of the NDW, we test the 4D quantum systems. ${W_{16}} = 0$ is the key condition that determines whether we can estimate the dimension of the system. Therefore, we need to test whether ${W_{16}}$ is always 0 under the effects of atmospheric turbulence, otherwise we will obtain an incorrect dimension. More importantly, we must make sure that ${W_9}$ will not become 0 from the effects of atmospheric turbulence at the same time. The results are shown in Fig. 6(b), the theoretical value of ${W_{16}}$ is 0, and one can see that the value of ${W_{16}}$ is robust to noise with increasing ${N}$. Our experimental results show that NDW is highly robust to technical imperfections and can be used in the presence of noise and low detection efficiency; hence, we can estimate the dimension in a detection-loophole-free manner in our experiment.
4. CONCLUSIONS
In conclusion, we have experimentally tested the dimension of quantum and classical systems (up to dimension 10) based on NDWs with independent devices, where the preparer and the measurer have no shared randomness. We can certify the dimension of classical and quantum systems in a more practical manner and identify whether the quantum system is irreducible in our experiment. We propose a theoretical method to show the general lower bound for the irreducible dimension witness for what we believe, to the best of our knowledge, is the first time. Our results show that the NDWs are highly robust to technical imperfections and can be used in the presence of noise and low detection efficiency; hence, we can estimate the dimension in our experiment in a detection-loophole-free manner through the NDW. A very good agreement between the experimental results and the theoretical predictions confirms that the method can be extended to more complex quantum information tasks.
Funding
National Key Research and Development Program of China (2017YFA0304100); National Natural Science Foundation of China (11325419, 11774335, 11821404, 61327901); Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-SLH003); Science Foundation of the CAS (ZDRW-XH-2019-1); Anhui Initiative in Quantum Information Technologies (AHY020100); Fundamental Research Funds for the Central Universities (WK2470000026); China Postdoctoral Science Foundation (2020M671862); AXA Chair in Quantum Information Science; Juan de la Cierva (formación) fellowship.
Acknowledgment
We are most grateful to Qi-Hang Lu and Xin Wu for their discussion on this paper.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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