---
abstract: 'Wave functions and density matrices represent our knowledge about a quantum system and give probabilities for the outcomes of measurements. If the combined dynamics and measurements on a system lead to a density matrix $\rho(t)$ with only diagonal elements in a given basis $\{|n\rangle\}$, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states $|n\rangle$ with probabilities $\rho_{nn}(t)$. Equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states $|n\rangle$ may improve our ability to retrodict what was the outcome of a projective state measurement at time $t$. Here, we show with experiments on a superconducting qubit that the smoothed probabilities do not, in the same way as the diagonal elements of $\rho(t)$, permit a classical mixture interpretation of the state of the system at the past time $t$.'
author:
- 'D. Tan'
- 'M. Naghiloo'
- 'K. Mølmer'
- 'K. W. Murch'
---

The quantum mechanical wavefunction, $\psi(x)$, yields the probability for detection of a particle at location $x$, but most textbooks carefully emphasize that this does not imply that, prior to detection, the particle *was* at the location $x$ with that probability. In contrast, a density matrix $\rho$ is often attributed a mixed interpretation as a classical random mixture of quantum states, i.e., the system is said to populate one out of several candidate states. A density matrix $\rho$ which is diagonal in a particular basis $|n\rangle$, indeed, leads to the same predictions about projective measurements in that basis, $P(n)=\rho_{nn}$, as if states had been Moreover, for any general measurement, described by a positive operator valued measure (POVM) [@Nielsenbook] with operators $\Omega_m$ that fulfill $\sum_m  \Omega_m^\dagger \Omega_m = I$ (the identity operator), the outcome probabilities $P(m) =$Tr($\Omega_m \rho{\textcolor{black}{(t)}} \Omega_m^\dagger$) equal the weighted mean of the probabilities over states $|n\rangle$, $$\begin{aligned}
 \label{eq:weighted}
P^{cm}(m) = \sum_n P(n) \textrm{Tr}\bigl(\Omega_m \rho_n \Omega_m^\dagger\bigr),\end{aligned}$$ where $\rho_n=|n\rangle \langle n|$.

$$\label{eq:pqsprob}
P_P(m) = \frac{\textrm{Tr}(\Omega_m \rho(t) \Omega_m^\dagger E(t))}{\sum_{m'} \textrm{Tr}(\Omega_{m'} \rho(t) \Omega_{m'}^\dagger E(t))}.$$

When applied to measurements on quantum systems, the PQS expression Eq. reveals unique features such as anomalous weak values [@Ahar88] arising from the pre- and postselection process [@Hatr13; @murc13traj; @Groe13; @Lang14] and quantum coherence [@dres15; @tan15]. Smoothed predictions for the outcomes of measurements on quantum systems have been tested in a variety of experimental systems [@tan15; @Ryba15; @whea10] and they have been used in the interpretation of temporal signal correlation functions [@camp13; @chan15; @xu15; @Foro16]. $$\label{eq:qfb}
P_P(n)=\frac{\rho_{nn}{\textcolor{black}{(t)}}E_{nn}{\textcolor{black}{(t)}}}{\sum_{n'}\rho_{n'n'}{\textcolor{black}{(t)}}E_{n'n'}{\textcolor{black}{(t)}}}$$ $$\label{eq:cmprob}
P_P^{cm}(m)= \sum_{{\textcolor{black}{n}}} P_P(n) \textrm{Tr}\bigl(\Omega_m|n\rangle \langle n|\Omega_m^\dagger\bigr),$$

So far, we merely observed an inconsistency between different theoretical predictions for experiments. We shall now present experiments on a superconducting qubit, where projective test measurements in bases different from the density matrix eigenbasis will illustrate and confirm Eq. while rejecting the classical mixture leading to Eq. .

Our experiment, depicted in Figure \[fig:rho\]a, consists of a superconducting transmon circuit that is dispersively coupled to aluminum cavity [@EXP]. The dispersive interaction between the qubit and cavity is given by an interaction Hamiltonian $H_\mathrm{int.} = -\hbar \chi \sigma_z a^\dagger a$, $a^\dagger (a)$ are the creation (annihilation) operators for a photon in the cavity mode and $\chi$ . This interaction allows quantum non-demolition (QND) measurements of the qubit in the $\sigma_z$ basis through probing of the qubit-state-dependent cavity resonance. , represented by the projection operators $\Pi_{\pm,z}$. achieve measurement fidelities in excess of $95\%$ with the predominant sources of infidelity arising from qubit transitions [@slic12; @sank16] that occur during the finite duration of the measurement [@Evan14; @Rist12; @John12; @Mack15; @Wall05; @sank16; @chen16].

We can make more general projective measurements by combining measurements in the $\sigma_z$ basis with arbitrary rotations ($R_x^\theta,\ R_y^\theta$) about the $x$ and $y$ axes of the qubit. For example, a projective measurement along the axis that forms an angle $\theta$ with the $z$ axis and azimuthal angle $\phi = 0$ can be performed through the , $\Pi_{ \pm,\theta} = R^{-\theta}_y\ \Pi_{\pm,z}\ R^{\theta}_y$ (Fig. \[fig:rho\]b). In the following these projective measurements will constitute the POVMs, $\Omega_{ \pm,\theta} = \Omega_{ \pm,\theta}^\dagger \equiv \Pi_{ \pm,\theta}$, for which we will test the predictions, Eqs. (\[eq:cmprob\], \[eq:pqsprob\]). If the qubit is described by a diagonal density matrix $\rho{\textcolor{black}{(t)}}$, the probability of obtaining eigenvalue $+1$ from such a measurement is given by, $$\begin{aligned}
P_\rho(+,\theta)=\rho_{00}{\textcolor{black}{(t)}} \cos^2\bigg(\frac{\theta}{2}\bigg)
+ \rho_{11}{\textcolor{black}{(t)}} \sin^2\bigg(\frac{\theta}{2}\bigg).\label{Ptheta} $$

In Figure \[fig:rho\]c we test the predictions given by Eq. (\[Ptheta\]) for different values of $\rho{\textcolor{black}{(t)}}$. To prepare different mixed states, we apply a qubit rotation pulse $R_y^{\varphi}$ followed by a projective measurement $\Pi_{\pm, z}$. When the result of this measurement is ignored, the projective measurement decoheres the system and prepares the qubit in a diagonal mixed state in the qubit basis eigenstates $|0(1)\rangle \equiv | +(-)z\rangle$ with $\rho_{00}{\textcolor{black}{(t)}}$ and $\rho_{11}{\textcolor{black}{(t)}}$ determined by the initial rotation angle $\varphi$ and $T_1$ decay during the first measurement. Following this preparation, we make projective measurements at different angles $\theta$ to determine $\tilde{P}(+,\theta) {\textcolor{black}{\equiv}} N_+/(N_++N_-)$ from the number of positive (negative) eigenvalue results $N_+$ ($N_-$). The projective measurements $\Pi_{\pm, \theta}$ are subject to infidelities originating predominantly from $T_1$ decay during the $t_\mathrm{m} = 400$ ns projective measurement. This results in a $\theta$-dependent measurement fidelity that is given by the overlap of the $\Pi_{\pm, \theta}$ eigenstates and the qubit excited state, $\mathcal{F}_\theta = 0.99-\sin^2 (\theta/2) (1-e^{-t_\mathrm{m}/T_1})$ and ranges from $0.945$ when $\theta = \pi$ to $0.99$ when $\theta = 0$. The maximum readout fidelity of $0.99$ arises from residual overlap of the measurement distributions. After correcting for the measurement fidelity, the predictions given by $\rho{\textcolor{black}{(t)}}$ are in good agreement with the measured probabilities as shown in Figure \[fig:rho\]c.

We now address how the subsequent continuous probing of the qubit in the $\sigma_z$ basis, as depicted in Figure \[fig2\]a, yields our smoothed predictions for the outcomes of the projective measurements $\Pi_{\pm, \theta}$. After the dispersive interaction, the phase of the coherent probe field depends on the qubit state, and the time integral $\xi$ of the measured $Q$-quadrature is Gaussian distributed with opposite mean values for the states $|0(1)\rangle$. In Fig \[fig2\]b, we show the experimentally obtained distributions $P(\xi|0)$ and $P(\xi|1)$, where we have normalized the integrated signal to have mean values $\pm 1$ for the two qubit states. The Gaussian widths are significant for short probing times and become much narrower when the system is probed for longer. For a given measured signal $\xi$, we can extract the values $P(\xi|0)$ and $P(\xi|1)$, i.e., the probability of the measured signal conditioned on the state. By Bayes’ rule, these are precisely the factors multiplying the prior probabilities $\rho_{nn}{\textcolor{black}{(t)}}$ to yield the classical smoothing theory. *I.e*., if we disregard the effect of qubit decay during the probing, they yield the values of $E_{00}{\textcolor{black}{(t)}}$ and $E_{11}{\textcolor{black}{(t)}}$ in Eq.(2), $$\begin{aligned}
\label{Emap}
E_{00}{\textcolor{black}{(t)}}=\frac{P(\xi|0)}{P(\xi|1)+P(\xi|0)}, \
E_{11}{\textcolor{black}{(t)}} =\frac{P(\xi|1)}{P(\xi|1)+P(\xi|0)},\end{aligned}$$ where we have applied a common normalization factor, leading to Tr($E$)$=1$. Fig \[fig2\]c shows how the inferred normalized value of $E_{00}$ ($E_{11}=1-E_{00}$) depends on the measured signal $\xi$. The continuous probing constitutes a QND measurement of the qubit state, and the accumulated back-action on the qubit state populations in the forward propagation of $\rho$ [@koro11] amounts to the same factors—which confirms that the evolution of $E$ is, indeed equivalent to the evolution of $\rho$ (the QND back-action is equal to its adjoint).

For a projective measurement in the qubit basis $(\theta = 0)$ at time $t$, $\rho(t)$ leads to the prediction $P_{\rho}(0) = \rho_{00}(t)$, while the pair of matrices $\bigl(\rho(t),E(t)\bigr)$ implies $${\textcolor{black}{P_P(0)\equiv P_P(+,0) =}} \frac{\rho_{00}(t)E_{00}(t)}{\rho_{00}(t)E_{00}(t)+\rho_{11}(t)E_{11}(t)}.$$ If the values of $P_P(0)$ and could be interpreted as refined populations of a classical mixture of the two qubit states at time $t$, the projective measurement, corresponding to $\Pi_{+,\theta}$ would have the probability $$\begin{aligned}
\label{Pcmtheta}
P_P^{cm}(+,\theta)=P_P(0) \cos^2\left(\frac{\theta}{2}\right)
+ P_P(1) \sin^2\bigg(\frac{\theta}{2}\bigg),\end{aligned}$$ while insertion of the projection operators $\Pi_{\pm,\theta}$ for $\Omega_m$ in yields the expression $$\label{eq:Ppthetared}
P_P(+,\theta) = \frac{P_\rho(+,\theta) P_E(+,\theta)}{P_\rho(+,\theta) P_E(+,\theta) + P_\rho(-,\theta)P_E(-,\theta)},$$ where $P_\rho(+,\theta)$ is given in , and we have introduced the formally similar $P_E(+,\theta) = E_{00} \cos^2\left(\frac{\theta}{2}\right) + E_{11} \sin^2\left(\frac{\theta}{2}\right)$ and $P_\rho(-,\theta)=1-P_\rho(+,\theta)$, $P_E(-,\theta)=1-P_E(+,\theta)$.

In our experiment, the signal related to $E{\textcolor{black}{(t)}}$ is obtained from measurement $\Pi_{\pm, \theta}$. $E{\textcolor{black}{(t)}}$ is given by the Eq. ($\ref{Emap}$) and depicted in Figure \[fig2\]c. In Figure \[fig2\]a, we display our experimental results that test the prediction of Eq. ($\ref{eq:Ppthetared}$) for three different combinations of $\rho{\textcolor{black}{(t)}}$ and $E{\textcolor{black}{(t)}}$ [@eref]. The experimental and theoretical curves show good agreement and highlight how information before and after the projective measurement contribute to the smoothed prediction.

Figure $\ref{fig3}$ summarizes our experimental results, showing the measured $\tilde{P}(+,\theta)$ as a function of the angle $\theta$ and the post-selected value of $E_{00}{\textcolor{black}{(t)}}$ (the corresponding values of the integrated signal $\xi$ are given on the right hand axis in the figure). Results are shown for three different density matrices $\rho{\textcolor{black}{(t)}}$ prior to the projective measurement along the direction $\theta$. For $\theta=\pi/2$ both conventional and smoothed predictions assign unbiased probabilities $0.5$ to the outcomes $\pm,\theta$. For any $\theta$ and for all three values of $\rho{\textcolor{black}{(t)}}$, a certain value of the probing signal after the projective measurements results in an unbiased smoothed prediction $P_P(+,\theta)=0.5$. This amounts to an increased uncertainty about the outcome and it happens because the subsequent probing of the system is at loggerheads with the prior state $\rho{\textcolor{black}{(t)}}$ (e.g., $\rho_{00}{\textcolor{black}{(t)}} = 0.91$, $E_{00}{\textcolor{black}{(t)}} = 0.25$, cf., Fig. 2c). Conversely, when the $\rho{\textcolor{black}{(t)}}$ and $E{\textcolor{black}{(t)}}$ are similar (e.g., $\rho_{00}{\textcolor{black}{(t)}} = 0.91$, $E_{00}{\textcolor{black}{(t)}} = 0.94$, cf., Fig. 2c), the later probing “confirms” the prediction by $\rho{\textcolor{black}{(t)}}$, and thus enhances the probability of the most likely outcome of the projective measurement. These trends are most clearly observed in Figure \[fig4\], where we compare the measurement probabilities $\tilde{P}(+,\theta)$ to the smoothed prediction $P_P(+,\theta)$ and the classical mixture interpretation $P_P^\mathrm{cm} (+,\theta)$. Notably, the figure shows a clear disagreement of the experimental data with the classical mixture interpretation.

In conclusion, we have presented a description of a quantum system, evolving without developing coherences, and hence, both prior and posterior information about the system are represented by diagonal matrices. While the theory of smoothing yields probabilities in better agreement with predictions for the outcome of measurements in the eigenstate basis, we have shown that these probabilities do not permit a classical mixture interpretation of the (past) quantum state.

At a more foundational level, our work dismisses simple “hidden variable theories” that equate eigenstates of incoherent ensembles with hidden “true” states of the system, and it offers an illustration of the problematic character of macrorealism [@legg85] which separates the evolution of quantum states and the measurements performed. Rather than demonstrating an explicit statistical violation of the Bell [@Bell65], CSCH [@CSCH81], or Leggett-Garg [@Groe13; @pala10; @will08; @gogg11; @whit15] inequalities, we have merely shown the failure of the simplest preconceived probabilistic classical mixture interpretation of the quantum description, and we have shown that the pair of matrices $\rho(t)$ and $E(t)$ offers a satisfactory account of the outcomes of past measurements on a quantum system.

We acknowledge P. Harrington and N. Foroozani for discussions and assistance with the manuscript and G. Zhao, L. Xu, and L. Yang for fabrication assistance. This research was supported in part by the John Templeton Foundation and the Sloan Foundation and used facilities at the Institute of Materials Science and Engineering at Washington University. K.M. acknowledges support from the Villum Foundation.

Correspondence and requests for materials should be addressed to K.W.M. (murch@physics.wustl.edu)

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