In the standard quantum state discrimination task^48⇓-50, Alice picks up a state [Math Processing Error]ρs with probability [Math Processing Error]ps from a state set denoted by $\mathcal{S}=\left\{{\rho }_s\right\}$ and then sends it to Bob, while Bob tries to distinguish these states by using a proper measurement strategy. Specifically, if Bob chooses a measurement modeled by a positive-operator-valued measure (POVM) $\mathcal{M}=\left\{M_s\right\}$ with positive elements satisfying ${\sum }_s{M}_s=\mathbb{I}$, then the probability of guessing the correct state is given by

Here each measurement outcome s associated with $M_s$ can be regarded as the correct guess for the state ${\rho }_s$. It is evident that given an arbitrary set of nonorthogonal quantum states, P is strictly smaller than 1 even for Bob’s optimal measurement strategy, implying that Bob can never perfectly discriminate these states by using a single measurement. We remark that in the above task Bob does not know the specific state ${\rho }_s$ in each round, whereas he knows the whole state set $\mathcal{S}$.

We then consider a modified state discrimination task in which Bob is provided with more intermediate information about the state set. Suppose that the state set $\mathcal{S}$ can be decomposed into two disjoint subsets, i.e., $\mathcal{S}={\mathcal{S}}_1\cup {\mathcal{S}}_2$ and ${\mathcal{S}}_1\cap {\mathcal{S}}_2=⌀$. If the information about the subset of which each state is sent by Alice to Bob prior to his measurement, then he can choose two measurement strategies ${\mathcal{M}}_1=\left\{M_{s_1}\right\}$ and ${\mathcal{M}}_2=\left\{M_{s_2}\right\}$ on each state subset. Consequently, the probability of correctly guessing these states yields

Here p denotes the probability of picking the state from the set ${\mathcal{S}}_1$, and is assumed to be 1/2 in the following sections. Instead, if the same information is sent to Bob, then he can still post-process the measurement statistics based on the given information and obtain the guessing probability ^22,51:

where $\mathcal{M}=\left\{M_{s_1,s_2}\right\}$ and there are ${\sum }_{s_2}{M}_{s_1,s_2}=M_{s_1}^c$ and ${\sum }_{s_1}{M}_{s_1,s_2}=M_{s_2}^c$ for two measurements ${\mathcal{M}}_1^c$ and ${\mathcal{M}}_2^c$. It is worth noting that the optimization in Eq. (3) is over all pairs of compatible measurements. It immediately follows from the above equations that Bob can use the prior state information to strictly improve his probability of correctly guessing the state only if two measurements ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are incompatible.

Incompatibility witnesses and robustness of incompatibility

It naturally follows from the above modified state discrimination task that the problem of whether two measurements ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are incompatible or not can be determined by comparing the two guessing probabilities based on the state information. In particular, following ref.²², we can introduce

Here W is nonpositive for a pair of compatible measurements and positive for two incompatible measurements performed on a well-chosen state set $\mathcal{S}$. Thus, it provides an experimentally friendly way to witness quantum incompatibility because the modified state discrimination task is easy to realize and there are also many freedoms to prepare the tested state set. Furthermore, by optimizing over all possible state sets, we are able to obtain a state-independent witness

It is proven that $W\in [0,1]$ and W=0 if and only if two measurements are compatible²².

Particularly, given two d-dimensional measurements described by mutually unbiased bases $\mathcal{M}\equiv \{M_i=|u_i〉{〈u_i\left|\right\}}_d$ and $\mathcal{N}\equiv \{N_j=|v_j〉{〈v_j\left|\right\}}_d$ with $\text{Tr}\left[M_i{N}_j\right]=1/d$ for i,j=1,…,d, we can choose the state subsets ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ as the eigenstates of the elements of the above two MUBs, respectively. Generally, denote ${\mathcal{S}}_1=\{{\rho }_i=u|u_i〉〈u_i|+(1-u)\mathbb{I}/d\}$ and ${\mathcal{S}}_2=\{{\rho }_j=v|v_j〉〈v_j|+(1-v)\mathbb{I}/d\}$, where $u,v\in [1/(1-d),1]$ models the white noise strengths in the state preparation process and $|u_i〉$ and $|v_j〉$ are the eigenstates of $M_i$ and $N_j$ associated with the nonzero eigenvalues. If each state is picked up at random from the state set with equal probability, then it immediately leads to

To optimize the postmeasurement guessing probability $P_{\text{guess}}^{\text{post}}$, one can construct a standard QSDT with a new state ensemble and modified measurements. In our case, the post-measurement guessing probability is:

for the case where [Math Processing Error]d=2 or [Math Processing Error]u>0 or [Math Processing Error]v>0. Otherwise, it simplifies to:

Note that it is straightforward to obtain the prior guessing probability, Eq. (6), by applying the measurement $M_i$ ($N_j$) on the corresponding states $|u_i〉$ ($|v_j〉$), while it is tedious to derive Eqs. (7) and (8) which essentially requires finding the optimal measurement to correctly guess the state mixtures $m{\rho }_i+n{\rho }_j$ with proper parameters m and n for all noisy states ${\rho }_i$ and ${\rho }_j$. The detailed derivation can be found in the seminal work²². Then, substituting Eqs. (6) and (7) into the incompatibility witness Eq. (4) immediately yields

Finally, the RoI for two MUBs is operationally defined as the critical point $\eta$ such that the measurement bases subjected to white noise with uniform strength $\eta$, in the form of ${\tilde{M}}_i=\eta M_i+(1-\eta )\mathbb{I}/d$ and ${\tilde{N}}_j=\eta N_j+(1-\eta )\mathbb{I}/d$, become compatible. This implies $W\left(\eta \right)\equiv W(\mathcal{S};\tilde{\mathcal{M}}\left(\eta \right),\tilde{\mathcal{N}}\left(\eta \right))\le 0$ for those values of $\eta$ for which the measurements are compatible. If the state preparation process is perfect, i.e., $u=v=1$, then it follows from $W\left(\eta \right)=0$ that the RoI is

This exactly recovers the result derived in the seminal work⁵². It is remarked that quantum incompatibility of the three MUBs can also be witnessed and quantified in a similar way. Taking the d=2 system as an example, if the whole state set is composed of three mutual bases, such as the eigenbases of the Pauli operators X,Y,Z, then the noise robustness can be derived as $\text{RoI}=1/\sqrt{3}$ (see METHODS section for details). We also point out that our results go beyond ref.⁴⁵, in the sense that our work demonstrates that the higher dimension and/or the more number MUBs are, the more incompatible they are, while the previous work⁴⁵ only studied the incompatibility of MUBs of dimensionality 3.

We consider the experimental verification of the quantum incompatibility of MUBs, namely the computational basis $\left\{\right|u_i〉\}{}_{i=0,...,d-1}$ and its Fourier conjugate basis $\left\{\right|v_j〉\}{}_{j=0,...,d-1}$ with dimension d = 2,3,4 in a high-dimensional quantum optics setup⁵³; the measurement elements formally read,

with $|u_i〉=|i〉$, $|v_j〉=\frac{1}{\sqrt{d}}{\sum }_{k=1}^d\exp (i2\pi kj/d)|k〉$, and $|v_j〉=\frac{1}{\sqrt{d}}{\sum }_{k=1}^d\exp (i2\pi kj/d)|k〉$ is the white noise proportion vector. To achieve this goal, we construct a QSDT game wherein the measurement settings are selected from the above MUBs according to the partial classical information in both prior and post scenarios. Correspondingly, the optimal choice of the state ensemble is the collection of the eigenstates of each MUB element with equal probability, i.e., $p=1/2d$. Under these settings, the optimal guessing probabilities in both scenarios can be calculated and the incompatibility witness in Eq. (9) is immediately obtained. For a Hilbert space with some certain dimension, there exist more than two MUBs and the incompatibility exhibits different properties for the case when more than two MUBs are involved. To investigate these properties, we consider the simplest scenario when d = 2. Specifically, we add a third MUB $\left\{\right|{\chi }_{1,2}〉=\frac{1}{\sqrt{2}}\left(\right|0〉\pm i|1〉)\}$, along with the two MUBs in Eq. (11), i.e. $\left\{\right|0〉,|1〉\}$ and $\left\{\right|\pm 〉\}$, to construct a QSDT with three MUB settings. We implement this protocol using the experimental setup in Fig. 1 and detailed information is shown in the METHODS section.

The experimental setup is shown in Fig. 1. A cw violet laser (1 mW, 404 nm) was used to pump a type-II cut ppKTP crystal and to generate photon pairs with a degenerate wavelength of 808 nm. The idler photon was used as a herald and the target photon was used to implement the above QSDT. The whole process can be divided into two parts: the state preparation (part a) and MUB measurement (part b for 3-dimensional MUB and part c for 4-dimensional MUB). The key idea of state preparation is to encode the state from the ensemble into the polarization and path degree of freedom of a single photon that spans the Hilbert space of dimensionality d=2,3,4. In particular, the horizontal- and vertical-polarization photons of the upper path encode states $|0〉$ and $|1〉$, while the horizontal- and vertical-polarization photons of the lower path encode states $|2〉$ and $|3〉$. The apparatus in part a can be used to prepare an arbitrary pure state of the form $|\varphi 〉={\sum }_{i=1}^d{\alpha }_i|i〉$ with complex coefficients ${\alpha }_i$ satisfying ${\sum }_{i=1}^d{\left|{\alpha }_i\right|}^2=1$. In our experiment, we choose two MUBs M and N as in Eq. (11) to perform the modified QSDT. The elements of these two measurements are the computational basis of d-dimensional Hilbert space $|u_i〉=|i〉$ and the corresponding Fourier conjugate $|v_j〉=\frac{1}{\sqrt{d}}{\sum }_{k=1}^d\exp (i2\pi kj/d)|k〉$ with d=2,3,4. The goal of measuring the incompatibility and its robustness requires determining the $P_{\text{guess}}^{\text{prior}}$ and $P_{\text{guess}}^{\text{post}}$ in Eqs. (6) to (8). To determine $P_{\text{guess}}^{\text{post}}$, one needs to construct a new standard QSDT with modified state ensembles and measurements. As our goal here is to explore the incompatibility of certain MUBs, we use the theoretical value of $P_{\text{guess}}^{\text{post}}$. To determine $P_{\text{guess}}^{\text{prior}}$, we need to prepare all these basis states by properly setting the angles of the HWPs (h1 to h3) and Phasers (p1 to p3). Here a phaser consists of two QWPs and a HWP sandwiched between them and can be used to apply an arbitrary phase between the horizontal- and vertical-polarization photons. We list the settings for the 3− and 4− dimensional cases in Table 1. In the qubit case, we need to prepare only the eigenstates of the Pauli operators. To investigate the boundary feature of MUB incompatibility, we also need to simulate the case when the above states are subjected to white noise with varied proportions. We simulate the white noise by preparing all computational basis states with an equal probability. The noisy states are achieved by modifying the proportions of preparing the pure states and the white noise.

Parts b and c are used to implement 3- and 4- dimensional MUB measurements respectively. For the 3-dimensional case, the measurement outcomes of the two MUBs M and N admit a quantum-mechanical description with the state vectors

and

The three elements in Eqs. (12) and (13) can be realized simultaneously by placing 5 HWPs, a QWP, 2 beam displacers (BDs), a polarisation beam splitter (PBS), and 3 single photon detectors sequentially. For the four dimensional case, the measurement outcomes of the two MUBs M and N are

and

Similarly, we can use basic optical elements, including 12 HWPs, a QWP, 4 BDs, a PBS, and 4 single-photon detectors to simulate these two measurement settings. All necessary parameters related to the optical elements for the 3- and 4- dimensional MUBs are given in Tables 1 and 2. To simulate noisy MUB measurements, we adopt the method in refs.^53,57 by adding two independent light sources to the measurement apparatus (not shown in the figure). The light sources are composed of two variable-intensity LEDs and are set before two fiber couplers to realize the independent adjustment of the noise proportions entering each single photon detector. This is done by changing the brightness of the LEDs to control the total number of photons scattered into the detectors.

The incompatibility witness, as well as the corresponding RoI of more than two MUBs, can also be constructed by means of the modified state discrimination task. In this section, we take the case of 3-qubit Pauli operators as an example. To be consistent with the case of two MUBs, the three Pauli operators are denoted by $\mathcal{M}\equiv \{M_i=|u_i〉{〈u_i\left|\right\}}_2$, $\mathcal{N}\equiv \{N_j=|v_j〉{〈v_j\left|\right\}}_2$, and $\mathcal{O}\equiv \{O_k=|w_k〉{〈w_k\left|\right\}}_2$ with $|u_i〉$s, $|v_j〉$s, and $|w_k〉$s being eigenstates of the three Pauli operators respectively. In the discrimination game, we can choose three state subsets S1, S2, and S3 that are composed of the elements of the above three MUBs respectively. Generally, denote ${\mathcal{S}}_1=\{{\rho }_i=u|u_i〉〈u_i|+(1-u)\mathbb{I}/2\}$, ${\mathcal{S}}_2=\{{\rho }_j=v|v_j〉〈v_j|+(1-v)\mathbb{I}/2\}$, and ${\mathcal{S}}_3=\{{\rho }_k=w|w_k〉〈w_k|+(1-w)\mathbb{I}/2\}$, where $u,v,w\in [-1,1]$ models the white noise strengths in the state preparation process. When each state is picked up at random with equal probability, then

The postmeasurement guessing probability can be calculated by the guessing probability of an auxiliary state ensemble in a standard state discrimination task. We refer the interested reader to ref.⁵¹ for details. In our case, the postmeasurement guessing probability is

Substituting Eqs. (16) and (17) into the incompatibility witness Eq. (4) immediately yields

Analogous to the two MUB case, the RoI of three MUBs can be operationally defined as the critical point η such that the measurement bases subjected to white noise with uniform strength η, in the form of ${\tilde{M}}_i=\eta M_i+(1-\eta )\mathbb{I}/2$, ${\tilde{N}}_j=\eta N_j+(1-\eta )\mathbb{I}/2$, and ${\tilde{O}}_k=\eta O_k+(1-\eta )\mathbb{I}/2$ become compatible. In the case of perfect state preparation, i.e., u=v=w=1, it follows from $W\left(\eta \right)=0$ that the RoI is $1/\sqrt{3}$.