Certification and Quantification of Multilevel Quantum Coherence

Quantum coherence, present whenever a quantum system exists in a superposition of multiple classically distinct states, marks one of the fundamental departures from classical physics. Quantum coherence has recently been investigated rigorously within a resource-theoretic formalism. However, the finer-grained notion of multilevel coherence, which explicitly takes into account the number of superposed classical states, has remained relatively unexplored. A comprehensive analysis of multilevel coherence, which acts as the single-party analogue to multipartite entanglement, is essential for understanding natural quantum processes as well as for gauging the performance of quantum technologies. Here, we develop the theoretical and experimental groundwork for characterizing and quantifying multilevel coherence. We prove that nontrivial levels of purity are required for multilevel coherence, as there is a ball of states around the maximally mixed state that do not exhibit multilevel coherence in any basis. We provide a simple, necessary, and sufficient analytical criterion to verify the presence of multilevel coherence, which leads to a complete classification of multilevel coherence for three-level systems. We present the robustness of multilevel coherence, a bona fide quantifier, which we show to be numerically computable via semidefinite programming and experimentally accessible via multilevel coherence witnesses, which we introduce and characterize. We further verify and lower bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, which is implemented experimentally with four-level quantum probes in a photonic setup. Our results contribute to understanding the operational relevance of genuine multilevel coherence, also by demonstrating the key role it plays in enhanced phase discrimination—a primitive for quantum communication and metrology—and suggest new ways to reliably and effectively test the quantum behavior of physical systems.

Popular Summary

Unlike everyday objects, quantum particles can, counterintuitively, be in a superposition of two locations at once. The feature behind this is quantum coherence, which underpins virtually every quantum phenomenon. Things quickly get complicated when the particle may be in three or more possible locations. Is the particle really in a superposition of all three locations—A, B, and C—at once, or is there some probability of a superposition A and B with some other probability of A and C? Here, we provide a comprehensive toolset to characterize, detect, quantify, and experimentally measure such multilevel coherence.

We start by developing a theoretical framework in which to characterize multilevel coherence, along with analytical criteria that together lead to a complete characterization of such systems. We demonstrate this toolset experimentally in a photonic four-level quantum system. We show that multilevel coherence obeys a strict hierarchy, where higher levels of coherence are more challenging to establish and maintain but also offer advantages in practical tasks.

Given the fundamental role played by quantum coherence, our results have far-reaching applicability for the foundations of physics, quantum technologies, and exciting fields such as quantum biology.

Figure 1

Hierarchy of multilevel coherence. (a) The set of states $\mathcal{D}(\mathcal{H})$ (outer circle) of a $d$-dimensional quantum system, which can be structured according to coherence number into the convex sets $C_k$ (orange shading) with $C_1\subset C_2\subset \dots \subset C_d=\mathcal{D}(\mathcal{H})$. Note that a different choice of classical basis leads to a different hierarchy of sets $C_k^{\prime }$ (green shading). However, as shown in the Supplemental Material [41], irrespective of the classical basis, there is a finite volume ball within $C_2$ (blue inner circle). This implies that, while almost all states exhibit some form of coherence, achieving genuine multilevel coherence is instead nontrivial and requires the state to be sufficiently far from the maximally mixed state. (b) Real part of the density matrix of two example three-dimensional quantum states with equal off-diagonal elements yet different multilevel-coherence properties. The upper state is a mixture of two level-coherent states of the form $\frac{1}{3}(|{\psi }_{0,1}⟩⟨{\psi }_{0,1}|+|{\psi }_{0,2}⟩⟨{\psi }_{0,2}|+|{\psi }_{1,2}⟩⟨{\psi }_{1,2}|)$, where $|{\psi }_{i,j}⟩=\frac{1}{\sqrt{2}}(|i⟩+|j⟩)$, and thus has coherence number $n_C(\rho )=2$. The lower state is a mixture of the maximally coherent state $(|0⟩+|1⟩+|2⟩)/\sqrt{3}$ with weight $1/2$ and of the incoherent-basis states $|0⟩$ and $|2⟩$, each with weight $1/4$. Every pure-state decomposition of the lower state must contain a superposition of all $|0⟩$, $|1⟩$, $|2⟩$, as it can be verified by numerically calculating the robustness of three-level coherence $R_{C_2}\approx 0.0361$ [see Eq. (2)] or by using the comparison matrix criterion of Sec. 2c. Hence, although both states exhibit the same off-diagonal elements, only the lower state has genuine multilevel coherence, requiring experimental control that is coherent across multiple levels. This exemplifies the fine-grained classification of coherence and of experimental capabilities that studying multilevel coherence provides.

Figure 2

Experimental setup for probing multilevel coherence in systems of dimension $d\le 4$. Pairs of single photons are created via spontaneous parametric down-conversion (SPDC) in a $\beta$-Barium-borate (BBO) crystal, pumped at a wavelength of 410 nm. The detection of one photon heralds the presence of the other, which is initialized in a horizontal polarization state by means of a Glan-Taylor polarizer (GT). A four-level quantum system is then prepared using polarization encoding in each of the two spatial modes created by a calcite beam displacer (BD). Three sets of half-wave plates (HWP) and quarter-wave plates (QWP) are used to control the amplitude and phase of the generated states. We prepare noisy maximally coherent states $\rho (p)$, Eq. (4), for several values of $p$ in dimension $d=4$, as detailed in (b). Arbitrary states can be prepared and measured in dimension $d\le 4$ by manipulating only the corresponding subspaces.

Figure 3

Multilevel coherence in four-dimensional noisy maximally coherent states. With varying parameter $p\in [0,1]$, the coherence number of a four-dimensional, noisy maximally coherent state ranges from $n_C\mathbf{(}\rho (p)\mathbf{)}=1$ for $p=0$, to $n_C\mathbf{(}\rho (p)\mathbf{)}=2$ for $p\in ]0,\frac{1}{3}]$ (blue region), $n_C\mathbf{(}\rho (p)\mathbf{)}=3$ for $p\in ]\frac{1}{3},\frac{2}{3}]$ (orange region), and $n_C\mathbf{(}\rho (p)\mathbf{)}=4$ for $p\in ]\frac{2}{3},1]$ (green region). We use this color scheme throughout the paper to represent the three nontrivial levels of coherence in a four-dimensional system. On the right, we show examples of ideal density matrices for $p=1$ (top), $p=1/2$ (middle), and $p=0$ (bottom).

Figure 4

Measuring multilevel coherence. The plot shows experimentally measured robustness of ($k+1$)-level coherence for a four-dimensional, noisy maximally coherent state $\rho (p)$ as a function of $p\in [0,1]$. The solid lines represent the theory predictions from Eq. (5), and the shaded areas indicate the regions where multilevel coherence for $k=1$ (blue), $k=2$ (orange), $k=3$ (green) can be observed. The open squares correspond to the robustness of ($k+1$)-level coherence estimated from SDP in Eq. (11) applied to the experimentally reconstructed density matrices. The $5\sigma$ statistical confidence regions obtained from Monte Carlo resampling are on the order of ${10}^{-3}$ for $p$ and on the order of ${10}^{-2}$ for the RMC. These are smaller than the symbol size and thus not shown. The data points with error bars correspond to the absolute values of the negative expectation values of $W_k({\psi }_4^{+})$ in Eq. (9), which provide a lower bound on the RMC.

Figure 5

Bounding multilevel coherence from arbitrary measurements. The blue, orange, and green solid lines correspond to the experimental lower bound on the robustness of multilevel coherence for $k=2$, 3, 4, respectively, for a maximally coherent state $|{\psi }_4^{+}⟩$, while the grey solid lines are the theory prediction. These bounds are obtained from the SDP in Eq. (S16) in the Supplemental Material [41] for an increasing number of randomly chosen projective measurements, taking into account $5\sigma$ statistical uncertainties. The colored dashed lines correspond to the lower bounds for the noisy maximally coherent state $\rho (0.8874\pm 0.0007)$ using the same observables, with the grey dashed line being the theory prediction for this state.

Figure 6

Semi-device-independent witnessing of multilevel coherence. (a) A $d$-dimensional probe state $\rho$ is sent into a black box, which imprints one of $n$ phases $\{{\varphi }_m{\}}_{m=1}^n$ onto the state at random according to the prior probability distribution $\{p_m{\}}_{m=1}^n$. To infer the index $m$ of the imprinted phase, the state is then subjected to a single generalized measurement with elements $\{M_m{\}}_{m=1}^n$, yielding outcome $m^{\prime }$. This strategy succeeds, i.e., $m^{\prime }=m$, with probability $p_{\mathrm{succ}}^{\mathrm{\Theta }}(\rho )$, given by Eq. (13), which exceeds the optimal classical success probability $p_{\max }≔\max \{p_m{\}}_{m=1}^n$ by a factor greater than $k$ only when ($k+1$)-level coherence is present in the initial state $\rho$. Since evaluating the probability of success can be done without any information about the measurement device, based only on the assumption that incoherent states are unchanged by the black box, this scheme provides a semi-device-independent method to witness and estimate the robustness of multilevel coherence in the probe. (b) The experimentally measured bounds on the robustness of ($k+1$)-level coherence from the performance of noisy maximally coherent states $\rho (p)$ in the phase discrimination task $\tilde{\mathrm{\Theta }}$, as a function of $p\in [0,1]$. Plot as in Fig. 4, where solid lines represent the theory predictions, and open squares are the measured RMC from quantum-state tomography for comparison. The filled circles (higher) correspond to the semi-device-independent witness as discussed in the text, under the assumption that the application of the phases leaves incoherent states invariant. An upside-down triangle (lower) corresponds to each filled circle, which represents the conservative estimate of multilevel coherence obtained from the phase discrimination task by taking into account experimental imperfections in the implementation of the unitaries (see Supplemental Material [41] for details). The gray lines connecting circles and the corresponding triangles serve as a visual aid. For all data, $5\sigma$ statistical confidence regions obtained from Monte Carlo resampling are on the order of ${10}^{-3}$ for $p$ and on the order of ${10}^{-2}$ for the RMC.