# Hot Schrödinger Cat States

Hot Schrödinger Cat States
Ian Yang1,2,†, Thomas Agrenius2,3,†, Vasilisa Usova1,2, Oriol Romero-
Isart2,3,∗, Gerhard Kirchmair1,2,∗∗
1Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck,
Austria
2Institute for Quantum Optics and Quantum Information, Austrian Academy of
Sciences, 6020 Innsbruck, Austria
3Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck,
Austria

∗Present address: ICFO - Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Castelldefels (Barcelona) 08860, Spain
and ICREA, Passeig Lluis Companys 23, 08010, Barcelona, Spain
∗∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
> The observation of quantum phenomena often necessitates sufficiently pure
> states, a requirement that can be challenging to achieve. In this study, our
> goal is to prepare a non-classical state originating from a mixed state,
> utilizing dynamics that preserve the initial low purity of the state. We
> generate a quantum superposition of displaced thermal states within a
> microwave cavity using only unitary interactions with a transmon qubit. We
> measure the Wigner functions of these “hot” Schrödinger cat states for an
> initial purity as low as 0.06. This corresponds to a cavity mode temperature
> of up to 1.8 Kelvin, sixty times hotter than the cavity’s physical
> environment. Our realization of highly mixed quantum superposition states
> could be implemented with other continuous-variable systems e.g.
> nanomechanical oscillators, for which ground-state cooling remains
> challenging.

The quantum superposition principle allows us to prepare a system in a
superposition of two arbitrary states. The paradigmatic example is the
superposition of two coherent states, which are pure states with Heisenberg-
limited quantum fluctuations [1]. While the superposition of coherent states
is typically called a Schrödinger cat state, in Schrödinger’s original thought
experiment, the cat — a body-temperature and out-of-equilibrium system — is
prepared in a superposition of two mixed states dominated by classical
fluctuations [2].

Experimental demonstrations of Schrödinger cat states typically focus on a
continuous-variable degree of freedom, such as the one-dimensional motion of a
particle in a harmonic potential [3] or a single electromagnetic field mode in
a cavity [4]. The cat states are usually prepared by applying a given “cat
state protocol” to an initial vacuum Fock state
$\hat{\rho}_{0}=|{0}\rangle\langle{0}|$, created by cooling the system to its
ground state. This results in a pure quantum state which can be written as
$|{\alpha}\rangle+\mathrm{e}^{\mathrm{i}\phi}|{-\alpha}\rangle$, where
$|{\pm\alpha}\rangle$ are coherent states of the continuous-variable system
with complex amplitude $\alpha$ and relative phase $\phi$. We hereafter call
this a “cold” Schrödinger cat state. It has an emblematic Wigner function
exhibiting an interference pattern, see Figure 1A.

One could ask what type of state would be prepared if the same “cat state
protocol” is applied to an initial thermal state $\hat{\rho}_{\mathrm{T}}$
with a finite average thermal excitation number $n_{\mathrm{th}}$ [5, 6, 7, 8,
9, 10]. Would these “hot” Schrödinger cat states exhibit quantum features
given that (i) the purity of the initial state
$\mathcal{P}=\mathrm{tr}[\hat{\rho}_{\mathrm{T}}^{2}]=1/(2n_{\mathrm{th}}+1)$
is significantly less than one for large $n_{\mathrm{th}}$, and (ii) we
consider “cat state protocols” that do not remove entropy from the system? In
other words, can a highly mixed state exhibit unambiguous quantum features?

In this article, we experimentally show that indeed, these “hot” Schrödinger
cat states exhibit quantum features despite being highly mixed. More
precisely, we implement two unitary protocols, previously used to prepare cold
cats [11, 12], on initial states with a nonzero $n_{\mathrm{th}}$. We vary
$n_{\mathrm{th}}$ of the initial states up to $7.6(2)$ (corresponding to
$\mathcal{P}=0.062(2)$) and perform direct Wigner function measurements on the
final states. The created “hot” Schrödinger cat states show Wigner-negative
interference patterns for all investigated values of $n_{\mathrm{th}}$ in the
initial state (Figure 1B-G).

Our experimental platform is a circuit quantum electrodynamics (cQED) setup
[13]. The “hot” cat states are prepared in a microwave cavity mode, which is
well-described as a quantum harmonic oscillator. The cavity mode is coupled to
a two-level system which is used to prepare the cat states. The setup is
placed inside a dilution refrigerator and cooled to a temperature of $30$ mK
(see Figure S1 for the full experimental schematic). The cavity is a high-
coherence $\lambda/4$ post cavity, made up of high-purity niobium, with a
resonance frequency $\omega_{c}/2\pi=4.545$ GHz [14], and a relaxation time
$T_{1,c}=110(2)$ µs. For the two-level system, we use a transmon qubit with a
resonance frequency $\omega_{q}/2\pi=5.735$ GHz, qubit lifetime
$T_{1}=31.0(4)$ µs, and coherence time $T_{2}^{*}=12.5(4)$ µs. The cavity-
qubit interaction is dispersive with the dominant Hamiltonian term
$\hat{H}=-\hbar\chi_{\mathrm{qc}}\hat{c}^{\dagger}\hat{c}|{e}\rangle\langle{e}|$.
Here, $\hat{c}^{\dagger}$ and $\hat{c}$ are the cavity mode creation and
annihilation operators, $|{e}\rangle$ is the excited state of the qubit,
$\chi_{\mathrm{qc}}/2\pi=1.499(3)$ MHz is the dispersive shift, and $\hbar$ is
the reduced Planck’s constant. Qubit state measurements are performed through
an additional dispersively coupled microstrip resonator with frequency
$\omega_{r}/2\pi=7.534$ GHz. This setup allows for direct measurements of the
cavity Wigner function $W(\beta)\equiv
2\,\mathrm{Tr}\,[\hat{D}(\beta)\hat{\varPi}\hat{D}^{\dagger}(\beta)]/\pi$
[15]. Here $\beta$ is a complex parameter, $\hat{D}(\beta)$ is the cavity
displacement operator, and $\hat{\varPi}$ is the cavity parity operator [16].
We calibrate the Wigner function measurement by preparing a single photon Fock
state in the cavity (Figure S2).

The thermal initial state of the cavity mode, hotter than its environment, is
created by equilibrating the cavity mode with a heat bath in the form of
filtered and amplified Johnson-Nyquist noise from a 50 $\Omega$ resistor. The
heat bath is then disconnected (to prevent it from causing additional
decoherence) and the cat state preparation commences immediately [17]. The
state preparation and measurement protocols take up to $1.9$ µs which is much
faster than the cavity relaxation time $T_{1,c}$. Thus, there is neither
cooling nor heating of the cavity mode during the protocols. To verify that
the produced initial state is thermal, we characterize the photon statistics
of the cavity state with the added noise via qubit spectroscopy [18], which
also allows us to relate the noise power to $n_{\mathrm{th}}$ (Figure S4).

The two protocols used to prepare the hot cat states from the thermal initial
state are adaptations of two protocols known in the cQED community as the
echoed conditional displacement (ECD) [11] and qcMAP [12] protocols. The
quantum circuit diagrams for the protocols we use are shown in Figure 2A. To
illustrate the state generation, we decompose the initial thermal state in a
basis of cavity coherent states $|{\gamma}\rangle=\hat{D}(\gamma)|{0}\rangle$
with $\gamma$ a complex parameter [1]. We can then discuss the action of the
protocols on the state $|{\gamma}\rangle|{g}\rangle$, where $|{g}\rangle$ is
the qubit ground state. The first operation sequence prepares the qubit in the
superposition state $|{g}\rangle+\mathrm{e}^{\mathrm{i}\phi}|{e}\rangle$,
where $\phi$ is a controllable phase shift, and the cavity in the displaced
state $|{\gamma+\alpha}\rangle$ [19]. Next, the cavity-qubit state becomes
entangled through time evolution under the dispersive interaction $\hat{H}$,
as illustrated in Figure 2B and C for ECD and qcMAP respectively. The qcMAP
protocol has an uninterrupted time evolution, while the ECD protocol has
additional displacements and a qubit echo pulse inserted at half the evolution
time. At the end of the time evolution, we have created the state
$\mathrm{e}^{\mathrm{i}\phi}|{\mathrm{i}\gamma-\alpha}\rangle|{g}\rangle+|{\mathrm{i}\gamma+\alpha}\rangle|{e}\rangle$
with the ECD protocol and the state
$|{\gamma+\alpha}\rangle|{g}\rangle+\mathrm{e}^{\mathrm{i}\phi}|{-\gamma-\alpha}\rangle|{e}\rangle$
with the qcMAP protocol. The final three operations in Figure 2A disentangle
the qubit from the cavity. We center the $|{e}\rangle$ branch with a
displacement $-\alpha$ for ECD and $\alpha$ for qcMAP. Next, we apply a qubit
$\pi$-pulse selective to the $|{e}\rangle$ branch only. Finally, we invert the
previous displacement. The selective $\pi$-pulse has a Gaussian envelope with
standard deviation in time $\sigma_{t}$. This induces a Fock number-dependent
qubit flip probability $P_{g\leftrightarrow
e}(n)=\exp\\{-(\chi_{\mathrm{qc}}\sigma_{t}n)^{2}\\}$ [20, 21]. We choose
$\sigma_{t}$ so that this probability is large for the non-displaced thermal
state but small for the displaced thermal state (Figure 2D). This requires
that $\alpha$ is chosen large enough so that the Fock number distributions of
the initial thermal state and the thermal state displaced by $2\alpha$ do not
overlap. In a phase-space picture, we flip the qubit only for phase-space
points within a finite radius of the origin (Figure 2E).

We run our experiment with an initial thermal excitation number of
$n_{\mathrm{th}}=0.75(1)$, $1.44(2)$, $1.84(3)$, $3.48(7)$, $7.6(2)$,
corresponding respectively to purities of $\mathcal{P}=0.400(3)$, $0.258(3)$,
$0.214(3)$, $0.126(2)$, $0.062(2)$. We use $\alpha=3$, set $\phi=0$, and use a
disentanglement pulse width of $\sigma_{t}=20$ ns. Figure 1B,C show Wigner
function measurements on a grid of phase-space points $\beta$ for
$n_{\mathrm{th}}=3.48$. Figure 1D-G show Wigner function measurements along
the $\mathrm{Re}\,\\{\beta\\}$ and $\mathrm{Im}\,\\{\beta\\}$ axes as
$n_{\mathrm{th}}$ is varied. While the ECD and qcMAP protocols are known to
prepare equivalent cold cats, we observe that they lead to distinct outcomes
when applied to thermal initial states; compare panels B and C in Figure 1. In
both cases, the Wigner functions have two separated Gaussians centered at
$\alpha$ and $-\alpha$, associated with the classical probability distribution
of the displaced thermal initial states. Centered between these is an
interference pattern of oscillations with negative values, which produces an
interference pattern in the marginal distribution along
$\mathrm{Im}\,\\{\beta\\}$. In the ECD state, the envelope of the interference
pattern grows in radius and decreases in amplitude with $n_{\mathrm{th}}$
similar to the displaced thermal states themselves (Figure 1D,E). In the qcMAP
state, the envelope shrinks with increasing $n_{\mathrm{th}}$, but its
amplitude decreases more slowly than the displaced thermal states (Figure
1F,G). The data for all prepared states show clear negative values in the
interference pattern regardless of their $n_{\mathrm{th}}$.

These results can be understood as follows. We show in [22] that, under ideal
conditions, the ECD and qcMAP protocols are equivalent to the application of
two different quantum operators, namely
$\hat{S}_{1}\equiv[\hat{D}(\alpha)+\mathrm{e}^{\mathrm{i}\phi}\hat{D}(-\alpha)]/\sqrt{2}$
(ECD) and
$\hat{S}_{2}\equiv[1+\mathrm{e}^{\mathrm{i}\phi}\hat{\varPi}]\hat{D}(\alpha)/\sqrt{2}$
(qcMAP), to the initial state $\hat{\rho}_{T}$ [23]. The Wigner functions
prepared from $\hat{\rho}_{T}$ by the operators $\hat{S}_{1,2}$ are
$W_{1,2}(\beta)\equiv\frac{1}{2}\left[W_{T}(\beta-\alpha)+W_{T}(\beta+\alpha)+C_{1,2}(\beta)\right]$
where $W_{T}(\beta\pm\alpha)$ are the Wigner functions of the left- and right-
displaced thermal states, and the third term, which represents the quantum
superposition property of the state, is different for the two preparations:
$C_{1,2}(\beta)=2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi\right)f_{1,2}(\beta)$
[24]. For ECD, $f_{1}(\beta)\equiv
W_{T}(\beta)=2\mathcal{P}\mathrm{e}^{-2\mathcal{P}|\beta|^{2}}/\pi$ is the
thermal initial state. For qcMAP, $f_{2}(\beta)\equiv
2\mathrm{e}^{-2|\beta|^{2}/\mathcal{P}}/\pi$ is related to the characteristic
function [25] of the initial state. We plot examples of $W_{1,2}(\beta)$ in
Figure 3A. As illustrated there, when $n_{\mathrm{th}}\to 0$ ($\mathcal{P}\to
1$), $f_{1,2}(\beta)$ become identical and $W_{1,2}(\beta)$ both become equal
to the cold cat Wigner function. When $n_{\mathrm{th}}>0$, the phase-space
radius of $f_{1}(\beta)$ grows at the same rate as the phase-space radii of
$W_{T}(\beta\pm\alpha)$, while the phase-space radius of $f_{2}(\beta)$
shrinks. For both states, the $\mathrm{Im}\,\\{\beta\\}$ marginal probability
distribution contains interference fringes with period $\pi/2\alpha$ and full
contrast independently of $n_{\mathrm{th}}$. We remark that for $\phi=n\pi$
with $n$ integer, $W_{2}(0)$ is always saturated to the Wigner function
upper/lower bounds $\pm 2/\pi$, corresponding to parity values
$\langle\hat{\varPi}\rangle=\pm 1$, independently of $n_{\mathrm{th}}$.
Realizing this saturation of parity could be useful for hardware-efficient
encoding in bosonic qubit states in the presence of finite mode temperature
[26, 27].

We gain further understanding of the hot cat states by using the ideal Wigner
functions $W_{1,2}(\beta)$ to compute the coherence function
$g(x_{1},x_{2})\equiv|\langle{x_{1}}|\hat{\rho}|{x_{2}}\rangle|/\sqrt{\langle{x_{1}}|\hat{\rho}|{x_{1}}\rangle\langle{x_{2}}|\hat{\rho}|{x_{2}}\rangle}$
of the hot cat states. Here, $x_{1,2}$ are eigenvalues and $|{x_{1,2}}\rangle$
eigenkets of the dimensionless quadrature operator
$\hat{x}\equiv(\hat{c}+\hat{c}^{\dagger})/\sqrt{2}$. The coherence function is
upper-bounded as $g(x_{1},x_{2})\leq 1$, where saturation of the bound implies
that full-contrast quantum interference could be observed [28]. The thermal
state coherence function is a Gaussian
$\mathrm{e}^{-|x_{1}-x_{2}|^{2}/2\xi_{\mathrm{th}}^{2}}$ where the scale is
the coherence length
$\xi_{\mathrm{th}}=\sqrt{2\mathcal{P}/(1-\mathcal{P}^{2})}\approx
1/\sqrt{n_{\mathrm{th}}}$. As $n_{\mathrm{th}}$ increases, the coherence
length shrinks and the coherence function decays faster with $|x_{1}-x_{2}|$
(Figure 3B). The cat state coherence functions computed from $W_{1,2}(\beta)$
can be written $\mathrm{e}^{-(|x_{1}|-|x_{2}|)^{2}/2\xi_{\mathrm{th}}^{2}}$
along the one-dimensional section shown in Figure 3C. Compared to the thermal
state, the cat state coherence function has an additional peak at
$x_{1}=-x_{2}$ along this section, contributed by the term $C_{1,2}(\beta)$ in
$W_{1,2}(\beta)$. The value of $n_{\mathrm{th}}$ affects only how narrow this
extra peak is, but not its maximum value, which is always saturated to $1$ in
the ideal case. We present the full two-dimensional coherence functions of the
cat states in Figure S5.

We emphasize that the additional peak in the cat state coherence function
comes from the action of the cat creation operators $\hat{S}_{1,2}$. These
operators thus generate coherence not present in the initial state. The
additional coherence function peak is also generated in experiments which
observe interference patterns from superpositions of thermal clouds of atoms
prepared using Bragg diffraction [29] or Stern-Gerlach interferometry [30]. In
contrast, the additional peak is not present in the state obtained by sending
a thermal state through a double slit grating (see e.g. [31]), nor in a
completely dephased cold cat state, namely
$\hat{\rho}_{\mathrm{dephased}}=\frac{1}{2}(|{\alpha}\rangle\langle{\alpha}|+|{-\alpha}\rangle\langle{-\alpha}|)$
(see e.g. [32]). Note that $\hat{\rho}_{\mathrm{dephased}}$ has purity 1/2 but
a completely positive Wigner function ($W_{1,2}(\beta)$ with
$C_{1,2}(\beta)=0$), whereas we experimentally prepare states with negative
Wigner functions down to purities of 0.06.

The measured Wigner functions (Figure 1) deviate from the ideal Wigner
functions $W_{1,2}(\beta)$ (Figure 3A) mainly due to qubit operation
imperfections and perturbative Hamiltonian nonlinearities. We construct an _ab
initio_ model of the state preparation protocols that includes the
nonlinearities, the shape and timings of the qubit pulses, and decoherence
(see Figure S6 and Table S1 for the experimental characterisation). We account
for the effects of residual cavity-qubit entanglement (due to pulse
imperfections) in the modelling of the Wigner function measurement. We find
that the model reproduces the imperfections seen in the data (Figures S7-S8).

By turning the model features on and off, we attribute each imperfection to a
cause (Figures S9-S11). Here, we summarize the conclusions and further details
are given in [22]. The leading perturbative Hamiltonian terms
$\hat{H}^{\prime}=-\frac{1}{2}(K_{\mathrm{c}}+\chi^{\prime}_{\mathrm{qc}}|{e}\rangle\langle{e}|)\hat{c}^{\dagger}\hat{c}^{\dagger}\hat{c}\hat{c}$
($K_{\mathrm{c}}/2\pi=4.9(1)$ kHz, $\chi^{\prime}_{\mathrm{qc}}/2\pi=12.8(9)$
kHz) cause a smearing and bending distortion of the Wigner functions which is
similar to that observed in previous experiments [33]. The 20 ns width of the
disentanglement pulse causes left-right asymmetries of the displaced thermal
states as well as bending distortions of the fringes. In particular, the
nonlinearities and finite pulse widths together cause the qcMAP linecuts
(Figure 2D) to deviate from those predicted by $W_{2}(\beta)$, with an
$n_{\mathrm{th}}$-dependent reduction of the maximum of $C_{2}(\beta)$. The
extra fringes in the ECD state as compared to theory are caused by
instrumentation limitations to the minimum qubit pulse width that we can
achieve (Gaussian standard deviation of 6 ns).

We have demonstrated and characterized the preparation of quantum
superposition states directly from thermally excited initial states using only
unitary dynamics. Preparing “hotter” (larger initial thermal occupation
number) cat states requires using larger cavity displacements, where
limitations eventually appear due to the finite coherence time and
perturbative nonlinearities in our experiment. State-of-the-art setups capable
of cold cat states with $\alpha=32$ were recently reported [34]. Under ideal
conditions (including ideal measurement precision), standard quantum-
mechanical theory predicts no upper limit on and no loss of contrast due to
the thermal occupation number of a hot cat state [5, 6, 7, 8, 9, 10].

Hot Schrödinger cat states are in principle realizable in any continuous-
variable quantum system. This is particularly relevant for systems where long
coherence times have been achieved but ground-state cooling is not (yet)
available. Specific examples include nanomechanical systems such as carbon
nanotubes [35], and levitated magnetic [36, 37] and electrostatically trapped
dielectric particles [38, 39, 40, 41].

## Acknowledgments

† IY and TA contributed equally to this work.

TA and ORI wish to thank Lukas Neumeier for useful discussions. IY, VU and GK
wish to thank Johannes Fink for the etching of our Niobium cavities.

Funding: IY, VU and GK were funded in part by the Austrian Science Fund (FWF)
DOI 10.55776/F71 and 10.55776/I4395 QuantERA grant QuCOS. For the purpose of
open access, the author has applied a CC BY public copyright licence to any
Author Accepted Manuscript version arising from this submission. TA and ORI
were supported by the European Research Council (ERC) under Grant Agreement
No. [951234] (Q-Xtreme ERC-2020-SyG).

Author contributions: IY carried out the experiment with support from VU. TA
carried out the theoretical analysis with support from the other authors. The
numerical simulations were done by IY, TA, and VU. ORI and GK conceived of and
supervised the project. All authors were involved in the writing and the
editing of the paper.

Competing interests: The authors declare no competing interests.

Data availability: The data that supports the plots within this paper and the
supplementary material will be available via a permanent, public repository,
Zenodo.

## References

  * [1] Roy J. Glauber  “Coherent and Incoherent States of the Radiation Field”  In _Physical Review_ 131.6 American Physical Society, 1963, pp. 2766–2788  DOI: 10.1103/PhysRev.131.2766
  * [2] E. Schrödinger  “Die gegenwärtige Situation in der Quantenmechanik”  In _Naturwissenschaften_ 23.48, 1935, pp. 807–812  DOI: 10.1007/BF01491891
  * [3] C. Monroe, D.. Meekhof, B.. King and D.. Wineland  “A “Schrödinger Cat” Superposition State of an Atom”  In _Science_ 272.5265, 1996, pp. 1131–1136  DOI: 10.1126/science.272.5265.1131
  * [4] M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J.. Raimond and S. Haroche  “Observing the Progressive Decoherence of the “Meter” in a Quantum Measurement”  In _Physical Review Letters_ 77.24 American Physical Society, 1996, pp. 4887–4890  DOI: 10.1103/PhysRevLett.77.4887
  * [5] Kaicheng Zhu, Huiqin Tang, Cuiliang Li, Duzhi Huang, Xinguang Li and Xiangru Li  “The Quantum Statistical Properties of Superposition Coherent States with Thermal Noise”  In _Journal of Modern Optics_ 43.2 Taylor & Francis, 1996, pp. 323–336  DOI: 10.1080/09500349608232745
  * [6] Guillaume Huyet, Sonja Franke-Arnold and Stephen M. Barnett  “Superposition States at Finite Temperature”  In _Physical Review A_ 63.4 American Physical Society, 2001, pp. 043812  DOI: 10.1103/PhysRevA.63.043812
  * [7] Hyunseok Jeong and Timothy C. Ralph  “Transfer of Nonclassical Properties from a Microscopic Superposition to Macroscopic Thermal States in the High Temperature Limit”  In _Physical Review Letters_ 97.10 American Physical Society, 2006, pp. 100401  DOI: 10.1103/PhysRevLett.97.100401
  * [8] Shi-Biao Zheng  “Macroscopic Superposition and Entanglement for Displaced Thermal Fields Induced by a Single Atom”  In _Physical Review A_ 75.3 American Physical Society, 2007, pp. 032114  DOI: 10.1103/PhysRevA.75.032114
  * [9] Hyunseok Jeong and Timothy C. Ralph  “Quantum Superpositions and Entanglement of Thermal States at High Temperatures and Their Applications to Quantum-Information Processing”  In _Physical Review A_ 76.4 American Physical Society, 2007, pp. 042103  DOI: 10.1103/PhysRevA.76.042103
  * [10] Fernando Nicacio, Raphael N.. Maia, Fabricio Toscano and Raúl O. Vallejos  “Phase Space Structure of Generalized Gaussian Cat States”  In _Physics Letters A_ 374.43, 2010, pp. 4385–4392  DOI: 10.1016/j.physleta.2010.08.076
  * [11] Alec Eickbusch, Volodymyr Sivak, Andy Z. Ding, Salvatore S. Elder, Shantanu R. Jha, Jayameenakshi Venkatraman, Baptiste Royer, S.. Girvin, Robert J. Schoelkopf and Michel H. Devoret  “Fast universal control of an oscillator with weak dispersive coupling to a qubit”  In _Nature Physics_ 18.12, 2022, pp. 1464–1469  DOI: 10.1038/s41567-022-01776-9
  * [12] Zaki Leghtas, Gerhard Kirchmair, Brian Vlastakis, Michel H. Devoret, Robert J. Schoelkopf and Mazyar Mirrahimi  “Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity”  In _Physical Review A_ 87.4, 2013, pp. 042315  DOI: 10.1103/PhysRevA.87.042315
  * [13] Matthew Reagor, Wolfgang Pfaff, Christopher Axline, Reinier W. Heeres, Nissim Ofek, Katrina Sliwa, Eric Holland, Chen Wang, Jacob Blumoff, Kevin Chou, Michael J. Hatridge, Luigi Frunzio, Michel H. Devoret, Liang Jiang and Robert J. Schoelkopf  “Quantum memory with millisecond coherence in circuit QED”  In _Physical Review B_ 94.1 American Physical Society, 2016, pp. 014506  DOI: 10.1103/PhysRevB.94.014506
  * [14] P. Heidler, C… Schneider, K. Kustura, C. Gonzalez-Ballestero, O. Romero-Isart and G. Kirchmair  “Non-Markovian Effects of Two-Level Systems in a Niobium Coaxial Resonator with a Single-Photon Lifetime of 10 milliseconds”  In _Physical Review Applied_ 16.3, 2021, pp. 034024  DOI: 10/gnxb57
  * [15] Brian Vlastakis, Gerhard Kirchmair, Zaki Leghtas, Simon E. Nigg, Luigi Frunzio, S.. Girvin, Mazyar Mirrahimi, M.. Devoret and R.. Schoelkopf  “Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States”  In _Science_ 342.6158, 2013, pp. 607–610  DOI: 10.1126/science.1243289
  * [16] Antoine Royer  “Wigner Function as the Expectation Value of a Parity Operator”  In _Physical Review A_ 15.2 American Physical Society, 1977, pp. 449–450  DOI: 10.1103/PhysRevA.15.449
  * [17] In Figure S3, we additionally report results from running the experiment with the heat bath left connected during the entire protocol.
  * [18] D.. Schuster, A.. Houck, J.. Schreier, A. Wallraff, J.. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M.. Devoret, S.. Girvin and R.. Schoelkopf  “Resolving photon number states in a superconducting circuit”  In _Nature_ 445.7127, 2007, pp. 515–518  DOI: 10.1038/nature05461
  * [19] For simplicity of presentation, we ignore normalization, global phases, and we absorb all relative phase shifts into $\phi$. In particular, we absorb a geometric phase shift $2|\alpha|^{2}$ which accumulates during the ECD displacement operations. In the experiment, the geometric phase shift is cancelled via $\phi$.
  * [20] Gerald F. Thomas  “Validity of the Rosen-Zener Conjecture for Gaussian-modulated Pulses”  In _Physical Review A_ 27.5 American Physical Society, 1983, pp. 2744–2746  DOI: 10.1103/PhysRevA.27.2744
  * [21] Ivo S. Mihov and Nikolay V. Vitanov  “Pulse Shape Effects in Qubit Dynamics Demonstrated on an IBM Quantum Computer”, 2023  DOI: 10.48550/arXiv.2301.10004
  * [22] See the Supplemental Materials to this article.
  * [23] The ECD protocol is more generally equivalent to the operator $\hat{S}_{1}^{\prime}\equiv\left.\hat{S}_{1}\right|_{\phi\to\phi+2|\alpha|^{2}}\mathrm{i}^{\hat{c}^{\dagger}\hat{c}}$, but these differences vanish when the initial state is thermal and the geometric phase is cancelled by $\phi$.
  * [24] The expressions for $W_{1,2}(\beta)$ are normalized under the no-overlap assumption made for $\alpha$, which also implies that $C_{1,2}(\beta)$ integrate to $0$.
  * [25] Stephen M. Barnett and Paul M. Radmore  “Methods in Theoretical Quantum Optics”  Clarendon Press, 2002
  * [26] A. Grimm, N.. Frattini, S. Puri, S.. Mundhada, S. Touzard, M. Mirrahimi, S.. Girvin, S. Shankar and M.. Devoret  “Stabilization and operation of a Kerr-cat qubit”  In _Nature_ 584.7820, 2020, pp. 205–209  DOI: 10.1038/s41586-020-2587-z
  * [27] Raphaël Lescanne, Marius Villiers, Théau Peronnin, Alain Sarlette, Matthieu Delbecq, Benjamin Huard, Takis Kontos, Mazyar Mirrahimi and Zaki Leghtas  “Exponential suppression of bit-flips in a qubit encoded in an oscillator”  In _Nature Physics_ 16.5, 2020, pp. 509–513  DOI: 10.1038/s41567-020-0824-x
  * [28] Roy J. Glauber  “The Quantum Theory of Optical Coherence”  In _Physical Review_ 130.6 American Physical Society, 1963, pp. 2529–2539  DOI: 10.1103/PhysRev.130.2529
  * [29] D.. Miller, J.. Anglin, J.. Abo-Shaeer, K. Xu, J.. Chin and W. Ketterle  “High-Contrast Interference in a Thermal Cloud of Atoms”  In _Physical Review A_ 71.4 American Physical Society, 2005, pp. 043615  DOI: 10.1103/PhysRevA.71.043615
  * [30] Yair Margalit, Zhifan Zhou, Shimon Machluf, Yonathan Japha, Samuel Moukouri and Ron Folman  “Analysis of a High-Stability Stern–Gerlach Spatial Fringe Interferometer”  In _New Journal of Physics_ 21.7, 2019, pp. 073040  DOI: 10.1088/1367-2630/ab2fdc
  * [31] I. Bloch, T.. Hänsch and T. Esslinger  “Measurement of the Spatial Coherence of a Trapped Bose Gas at the Phase Transition”  In _Nature_ 403.6766 Nature Publishing Group, 2000, pp. 166–170  DOI: 10.1038/35003132
  * [32] Samuel Deléglise, Igor Dotsenko, Clément Sayrin, Julien Bernu, Michel Brune, Jean-Michel Raimond and Serge Haroche  “Reconstruction of Non-Classical Cavity Field States with Snapshots of Their Decoherence”  In _Nature_ 455.7212 Nature Publishing Group, 2008, pp. 510–514  DOI: 10.1038/nature07288
  * [33] Gerhard Kirchmair, Brian Vlastakis, Zaki Leghtas, Simon E. Nigg, Hanhee Paik, Eran Ginossar, Mazyar Mirrahimi, Luigi Frunzio, S.. Girvin and R.. Schoelkopf  “Observation of quantum state collapse and revival due to the single-photon Kerr effect”  In _Nature_ 495.7440, 2013, pp. 205–209  DOI: 10.1038/nature11902
  * [34] Ofir Milul, Barkay Guttel, Uri Goldblatt, Sergey Hazanov, Lalit M. Joshi, Daniel Chausovsky, Nitzan Kahn, Engin Çiftyürek, Fabien Lafont and Serge Rosenblum  “Superconducting Cavity Qubit with Tens of Milliseconds Single-Photon Coherence Time”  In _PRX Quantum_ 4.3 American Physical Society, 2023, pp. 030336  DOI: 10.1103/PRXQuantum.4.030336
  * [35] C. Samanta, S.. De Bonis, C.. Møller, R. Tormo-Queralt, W. Yang, C. Urgell, B. Stamenic, B. Thibeault, Y. Jin, D.. Czaplewski, F. Pistolesi and A. Bachtold  “Nonlinear Nanomechanical Resonators Approaching the Quantum Ground State”  In _Nature Physics_ 19.9 Nature Publishing Group, 2023, pp. 1340–1344  DOI: 10.1038/s41567-023-02065-9
  * [36] Martí Gutierrez Latorre, Gerard Higgins, Achintya Paradkar, Thilo Bauch and Witlef Wieczorek  “Superconducting Microsphere Magnetically Levitated in an Anharmonic Potential with Integrated Magnetic Readout”  In _Physical Review Applied_ 19.5 American Physical Society, 2023, pp. 054047  DOI: 10.1103/PhysRevApplied.19.054047
  * [37] J. Hofer, R. Gross, G. Higgins, H. Huebl, O.. Kieler, R. Kleiner, D. Koelle, P. Schmidt, J.. Slater, M. Trupke, K. Uhl, T. Weimann, W. Wieczorek and M. Aspelmeyer  “High-Q Magnetic Levitation and Control of Superconducting Microspheres at Millikelvin Temperatures”  In _Physical Review Letters_ 131.4 American Physical Society, 2023, pp. 043603  DOI: 10.1103/PhysRevLett.131.043603
  * [38] J. Millen, P… Fonseca, T. Mavrogordatos, T.. Monteiro and P.. Barker  “Cavity Cooling a Single Charged Levitated Nanosphere”  In _Physical Review Letters_ 114.12 American Physical Society, 2015, pp. 123602  DOI: 10.1103/PhysRevLett.114.123602
  * [39] T. Delord, P. Huillery, L. Nicolas and G. Hétet  “Spin-Cooling of the Motion of a Trapped Diamond”  In _Nature_ 580.7801 Nature Publishing Group, 2020, pp. 56–59  DOI: 10.1038/s41586-020-2133-z
  * [40] Gerard P. Conangla, Raúl A. Rica and Romain Quidant  “Extending Vacuum Trapping to Absorbing Objects with Hybrid Paul-Optical Traps”  In _Nano Letters_ 20.8 American Chemical Society, 2020, pp. 6018–6023  DOI: 10.1021/acs.nanolett.0c02025
  * [41] Lorenzo Dania, Dmitry S. Bykov, Florian Goschin, Markus Teller, Abderrahmane Kassid and Tracy E. Northup  “Ultrahigh Quality Factor of a Levitated Nanomechanical Oscillator”  In _Physical Review Letters_ 132.13 American Physical Society, 2024, pp. 133602  DOI: 10.1103/PhysRevLett.132.133602
  * [42] Paul Horowitz and Winfield Hill  “The art of electronics”  Cambridge University Press, 1989
  * [43] F. Diedrich, J.. Bergquist, Wayne M. Itano and D.. Wineland  “Laser Cooling to the Zero-Point Energy of Motion”  In _Physical Review Letters_ 62.4 American Physical Society, 1989, pp. 403–406  DOI: 10.1103/PhysRevLett.62.403
  * [44] S.. Hamann, D.. Haycock, G. Klose, P.. Pax, I.. Deutsch and P.. Jessen  “Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice”  In _Physical Review Letters_ 80.19 American Physical Society, 1998, pp. 4149–4152  DOI: 10.1103/PhysRevLett.80.4149
  * [45] J.. Johansson, P.. Nation and Franco Nori  “QuTiP 2: A Python Framework for the Dynamics of Open Quantum Systems”  In _Computer Physics Communications_ 184.4, 2013, pp. 1234–1240  DOI: 10.1016/j.cpc.2012.11.019

Fig. 1. Wigner function measurement results. (A) Cold Schrödinger cat prepared
using the qcMAP protocol with the heat bath disconnected
($n_{\mathrm{th}}=0.03$, $\mathcal{P}=0.94$). (B,C) ‘Hot’ Schrödinger cat
states prepared from an initial thermal state with $n_{\mathrm{th}}=3.48(7)$
using the ECD (B) and qcMAP (C) protocols. Also displayed in A-C are the
marginal distributions obtained by summation along the
$\mathrm{Re}\,\\{\beta\\}$ axis. To increase the visibility of small parity
values, the color brightness changes nonlinearly across the colorbar. (D-G)
Linecuts of the Wigner function along the coordinate axes in the ECD (D,E) and
qcMAP (F,G) protocols with $n_{\mathrm{th}}$ of the initial state as indicated
in the legend.

Fig. 2. Hot cat state generation protocol. (A) Quantum circuit diagram of hot
cat state generation sequence. For ECD (qcMAP), $j=1$ ($j=2$), and the
displacements use the lower (upper) sign. $\hat{X}(\pi/2,\phi)$ is a qubit
$\pi/2$ pulse with phase $\phi$, and $\hat{X}(\pi,\sigma_{t})$ is the
disentanglement pulse. (B) Definition of $\hat{V}_{1}$ and visualization of
its action in the joint cavity-qubit phase space. $\hat{T}(t)$ denotes free
evolution for time $t$, $\hat{Y}(\pi)$ is a qubit $\pi$ pulse, and
$\zeta=-(1+i)\alpha/2$. Cavity states are entangled with the qubit states
whose marker they touch. The arrows illustrate how the total state evolves
under the indicated operator. (C) Definition and visualization of
$\hat{V}_{2}$. (D) Qubit-conditional cavity Fock state distributions
$P_{q}(n)=\langle{n}|\langle{q}|\hat{\rho}|{q}\rangle|{n}\rangle$,
($q\in\\{g,e\\}$) of the total state before the $\hat{X}(\pi,\sigma_{t})$
operation. Here $\alpha=3$ and $n_{\mathrm{th}}=3.5$. The dashed line shows
$P_{g\leftrightarrow e}(n)$ (defined in the main text) with
$\sigma_{\mathrm{t}}=20$ ns. (E) The choice of $\sigma_{t}$ corresponds to the
choice of a radius in the cavity-qubit phase space within which the qubit
state is flipped with a certain probability. At this stage of the protocol,
the $|{g}\rangle$ branch is displaced by $2\alpha$.

Fig. 3. Hot Schrödinger Cat States in Theory. (A) Left: Plots of the thermal
state Wigner function $W_{T}(\beta)$ with $n_{\mathrm{th}}=0$ (top),
$n_{\mathrm{th}}=2$ (middle). To increase the visibility of the hotter state,
the color brightness changes nonlinearly across the colorbar (bottom). Center:
Cat state Wigner functions $W_{1,2}(\beta)$ which result from applying the
operators $\hat{S}_{1,2}$ with $\alpha=3.5$ and $\phi=\pi$ to the initial
states on the left according to the arrows. Right: Marginal probability
distributions obtained from the cat state Wigner functions by integrating
along the $\mathrm{Re}\,\\{\beta\\}$ axis. (B) Sections of the coherence
function $g(x_{1},x_{2})$ for the thermal states. The number in the legend
indicates the $n_{\mathrm{th}}$ of the state. (C) Sections of the coherence
function $g(x_{1},x_{2})$ for the cat states (lower panel) along the line from
$(x_{1},x_{2})=\sqrt{2}\alpha(-1,-1)$ to $(x_{1},x_{2})=\sqrt{2}\alpha(3,-1)$.
The plots are made with $\alpha=3.5$.

Supplemental Material for Hot Schrödinger Cat States
Ian Yang1,2 , Thomas Agrenius2,3, Vasilisa Usova1,2, Oriol Romero-Isart2,3,∗,
Gerhard Kirchmair1,2,∗∗
1Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck,
Austria
2Institute for Quantum Optics and Quantum Information, Austrian Academy of
Sciences, 6020 Innsbruck, Austria
3Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck,
Austria

∗Present address: ICFO - Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Castelldefels (Barcelona) 08860, Spain
and ICREA, Passeig Lluis Companys 23, 08010, Barcelona, Spain
∗∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
###### Contents

  1. S1 Experiment
    1. S1.1 Experimental Setup
    2. S1.2 Calibration and Scaling of Wigner Function
    3. S1.3 Additional Hot Cat Wigner Function Measurements
    4. S1.4 Characterizing the Initial Thermal State
    5. S1.5 Characterization of Hamiltonian
  2. S2 Theory
    1. S2.1 Theoretical analysis of the qcMAP and ECD protocols
    2. S2.2 Derivation of the Wigner functions $W_{1,2}(\beta)$ from the operators $\hat{S}_{1,2}$
    3. S2.3 Hot cat state coherence functions
  3. S3 Numerical model
    1. S3.1 Method
    2. S3.2 Comparison of Simulation and Data
  4. S4 Analysis of Imperfections
    1. S4.1 Residual cavity-qubit entanglement
    2. S4.2 Comparison of simulations with different parameters
    3. S4.3 Free evolution timing errors

##  S1 Experiment

###  S1.1 Experimental Setup

The schematic of the experimental setup is shown in Figure S1. The high
coherence cavity has a post length of $14.8$ mm, inner radius of $2$ mm and
outer radius of $6.4$ mm. This geometry gives a bare cavity frequency of
approximately $4.5$ GHz. The tunnel for the qubit chip has a diameter of $4$
mm, which is a compromise between cavity mode leakage into the tunnel and
qubit capacitance to ground. The cavity was made from high-purity niobium at
the Institute for Quantum Optics and Quantum Information Innsbruck mechanical
workshop. The manufacturing process used electro-discharge machining with a
tungsten alloy electrode. The cavity was then etched with our collaborators at
the Institute of Science and Technology, Vienna with the group of Prof.
Johannes Fink. This process used a buffer chemical polishing etching solution
of 1:1:1 hydrofluoric, nitric and phosphoric acid for one hour at 5 ∘C.
Phosphoric acid was then slowly added to reach a ratio of 1:1:2 for another
hour of polishing. Afterwards, the niobium cavity was rinsed heavily with
deionized (DI) water. In total, this process removes approximately $150$ µm of
material.

The transmon qubit and readout resonator were patterned by electron-beam
lithography (Raith eLINE Pllus $30$ kV) on a bi-layer resist ($1$ µm MMA (8.5)
EL13 and $0.3$ µm of 950 PMMA A4). The substrate started from a 2-inch
sapphire wafer that was first piranha-cleaned before processing. To prevent
charging of the substrate, a thin gold layer was sputtered on top of the PMMA.
After lithography, this gold layer was etched in a solution of Lugol (5 %
potassium iodide) and DI water in a ratio of 1:15, before being washed in DI
water and developed in a 3:1 solution of isopropyl alcohol and water. In the
next step, two layers of aluminum ($25$ nm and $50$ nm) were evaporated onto
the sample using a Plassys MEB550S electron-beam evaporator. A controlled
oxidation step ($5$ mbar for $5.5$ min) was carried out in between the
deposition of the two aluminum layers. Subsequently, the qubit chip was laser-
diced, and the resist layer was lifted off. The sample chips were thermalized
by a copper clamp. An additional aluminum sheet was used to cover the copper
clamp to reduce losses due to the presence of the copper material.

The measurements were conducted in a Triton DU7-200 Cryofree dilution
refrigerator system. The input coaxial cables were attenuated by $20$ dB at
the $4$ K plate and $10$ dB at the still plate. Finally, at the mixing chamber
plate, the input signal was filtered by a K&L DC-12 GHz low pass filter and
then attenuated by a $20$ dB directional coupler followed by a thermalized
cryogenic $20$ dB attenuator and filtered by microtronics $4-8$ GHz bandpass
filter. The experiment was done in reflection with a Quinstar double junction
$4-8$ GHz circulator. Before and after the sample, the input and output
signals passed through a home-built eccosorb filter. The input signal for the
high coherence cavity was attenuated and filtered similarly, except at the
base plate where a $10$ dB thermalized cryogenic attenuator was used instead.

The output signal was filtered via a microtronics $4-8$ GHz bandpass filter,
before passing through a quantum-limited parametric amplifier. Finally, the
output signal was filtered by a K&L filter which was connected to two Quinstar
isolators giving $40$ dB isolation. The signal was amplified at the $4$ K
plate by high electron mobility transistor (HEMT) amplifiers and again with
room temperature amplifiers outside of the refrigerator.

Control of the thermal noise was done by amplifying and filtering the noise
from a 50 $\Omega$ resistor. The added noise has a frequency spectrum shown in
Figure S3A. The added noise power level was controlled by a digital
attenuator. A fast, home-built microwave switch was used to disconnect the
cavity mode from this added noise. To initialize the cavity state, the cavity
mode was allowed to come into thermal equilibrium with the controlled noise
environment for 1 ms. Afterwards, the microwave switch was opened and the
state preparation and measurement started.

Leaving the microwave switch opened resulted in a thermal state in equilibrium
with the residual thermal excitations of the setup
($n_{\mathrm{th}}=0.0338(7)$), which was the coldest initial state we could
achieve with this setup.

The samples were placed in a $\mu$-metal shield which sat in a superconducting
shield to protect the experiment against stray magnetic fields. The shield was
filled with eccosorb foam for the absorption of any stray infrared photons.

The pulses for the high coherence cavity and readout resonator were generated
by an arbitrary waveform generator (AWG), specifically the Operator X from
Quantum Machines. These pulses were up-mixed with a local oscillator (LO)
using a Marki microwave IQ mixer. The qubit pulses, on the other hand, were
up-mixed through a double-super-heterodyne [42] setup employing two LOs and
two single side-band mixers. These pulse generation setups also incorporated
various amplifiers, filters, alternators, and fast microwave switches to
achieve effective suppression of unwanted mixing products and to minimize
leakage of LO signals. The signal from the refrigerator was down-mixed using
the same readout LO and further amplified before being digitized by Operator X
from Quantum Machines.

Fig. S1. Schematic of Experimental Wiring. The cavity and readout resonator
were driven by an IQ mixing setup while the qubit tones were up-converted via
a double-super-heterodyne setup. The qubit-resonator line had a total of $70$
dB input line attenuation while the cavity had $60$ dB. Cavity noise was added
via amplifying Johnson-Nyquist noise at room temperature and filtered with a
MiniCircuits filter. The noise level was reduced by a digital attenuator. The
readout tone was first amplified with a parametric amplifier before reaching
the HEMTs. Noise from the output and amplifier pump lines were attenuated with
isolators. Additional microwave and eccosorb filters were used to remove
unwanted radiation from reaching the experiment. The setup was placed in
superconducting and µ-metal shields and was surrounded by eccosorb foam.

###  S1.2 Calibration and Scaling of Wigner Function

The Wigner function measurement was calibrated by the measurement of a single
photon Fock state. The single photon Fock state was prepared by using a blue
sideband transition. This technique is similar to that used in ion traps [43]
or atomic arrays [44]. The measurement data, which has arbitrary units, was
collected in the pulse quadrature variables $I,Q$, which also have arbitrary
units (Figure S2A). To calibrate the measurement, we seek a linear map from
the data space into phase space. We find this map by fitting the function
$\chi_{W}^{-1}W_{|{1}\rangle}(\chi_{I}I+\mathrm{i}\chi_{Q}Q)$ to the Fock
state measurement data, where $\chi_{W}$, $\chi_{I}$, and $\chi_{Q}$ are
scaling constants, and $W_{|{1}\rangle}(\beta)$ is the Wigner function of the
first Fock state (Figure S2B,C). Using the fitted scaling constants, we then
map all other Wigner function measurement data into phase-space by
constructing $\beta=\chi_{I}I+\mathrm{i}\chi_{Q}Q$ and
$W(\beta)=\chi_{W}D(I,Q)$, where $D(I,Q)$ is the measurement data.

Fig. S2. Wigner Measurement and Fit of a Single Photon Fock State. (A)
Measured data and (B) analytical fit of a $|{1}\rangle$ cavity Fock state in
data space. (C) Residuals of fit. We use the parameters of the fit to
construct a linear map from data space into phase space (Section S1.2).

###  S1.3 Additional Hot Cat Wigner Function Measurements

In Figure S3, we report Wigner function measurements on states prepared by the
qcMAP and ECD protocols in an earlier experimental setup. In this setup, we
had $\chi_{\mathrm{qc}}/2\pi=1.272$ MHz, $K_{\mathrm{c}}/2\pi=2.33$ kHz,
$\chi_{\mathrm{qc}}^{\prime}/2\pi=7.1$ kHz, cavity lifetime
$T_{1,\mathrm{\mathrm{c}}}=128$ µs, qubit lifetime $T_{1}=6.3$ µs, and qubit
coherence time $T_{2}^{*}=2.4$ µs. In these experimental runs, the heat bath
used to prepare the initial state was kept connected throughout the
preparation and measurement protocol. Note that, in our setup, the cavity
lifetime is limited by the external coupling via the coupling pin to the
environment. Thus, the cavity coupling rate to the heat bath is the same as
the cavity photon loss rate. The experiment was run with $\alpha=2.5$ and
$n_{\mathrm{th}}=2.07$.

Fig. S3. Hot cat state measurements from a previous experimental setup with
the heat bath left connected. (A) qcMAP protocol (B) ECD protocol. Note that
this plot uses a linear scaling of the color bar. See section S1.3 for the
parameters of this setup.

###  S1.4 Characterizing the Initial Thermal State

To determine the initial thermal state of the cavity mode, number-split qubit
spectroscopy was performed (Figure S4). This allowed for the measurement of
the cavity photon distribution. The steps involved are shown in Figure S4B.
First, the cavity mode was equilibrated with the thermal bath as described in
Section S1.1. Next, a cavity photon number selective qubit $\pi$-pulse was
applied and the qubit state was measured. By repeating the measurement to get
an ensemble average, the probability to excite the qubit at a certain
frequency was determined.

Due to the cavity photon distribution and the dispersive coupling between the
cavity and the qubit, the qubit resonance frequency is split into a spectrum
of multiple frequencies where the relative resonance peak height depends on
the cavity photon distribution (Figure S4C). By measuring the probability to
excite the qubit across the frequency spectrum, we also directly measure the
cavity photon number distribution.

For a thermal state with average photon occupation number $n_{\mathrm{th}}$,
the probability of measuring $n$ photons is

$P_{n_{\mathrm{th}}}(n)=\frac{n_{\mathrm{th}}^{n}}{(1+n_{\mathrm{th}})^{n+1}}.$
(S1)

By fitting this to the spectral qubit excitation probability (Figure S4C), we
determined $n_{\mathrm{th}}$ of the cavity state. By varying the added photon
noise power, we found a relationship between the attenuator setting and
$n_{\mathrm{th}}$ (Figure S4D).

Fig. S4. Thermal State Measurement Technique (A) An illustration of frequency
spectrum of the added noise. The thermal noise is only added at the cavity
frequency while qubit frequencies are filtered out. The noise level is
controlled by a digital attenuator. (B) Experimental Pulse sequence. (C) Qubit
spectroscopy measurement result for a thermal state. Here,
$n_{\mathrm{th}}=3.3(1)$. (D) Thermal population measurements for different
attenuation settings, corresponding to noise powers. The added noise power
with the mean thermal photon is fitted with a straight line.

###  S1.5 Characterization of Hamiltonian

The full system Hamiltonian, including perturbative terms due to higher
excitation levels of the qubit and cavity nonlinearities, is

$\hat{H}/\hbar=\omega_{c}\hat{c}^{{\dagger}}\hat{c}-\frac{K_{c}}{2}\hat{c}^{{\dagger}}\hat{c}^{{\dagger}}\hat{c}\hat{c}+\omega_{q}\hat{q}^{{\dagger}}\hat{q}-\frac{K_{q}}{2}\hat{q}^{{\dagger}}\hat{q}^{{\dagger}}\hat{q}\hat{q}-\chi_{qc}\hat{c}^{{\dagger}}\hat{c}\hat{q}^{{\dagger}}\hat{q}-\frac{K^{\prime}_{c}}{6}\hat{c}^{{\dagger}}\hat{c}^{{\dagger}}\hat{c}^{{\dagger}}\hat{c}\hat{c}\hat{c}-\frac{\chi^{\prime}_{qc}}{2}\hat{c}^{{\dagger}}\hat{c}^{{\dagger}}\hat{c}\hat{c}\hat{q}^{{\dagger}}\hat{q}$
(S2)

Here $\hat{q}^{{\dagger}}$ and $\hat{c}^{{\dagger}}$ are the creation
operators for the qubit and cavity mode respectively, and $\hat{q},\hat{c}$
are the corresponding annihilation operators. The values of the Hamiltonian
parameters were measured experimentally and are reported in Table S1.

To characterize the Hamiltonian of our system, we employed a measurement
method to accurately determine the cavity frequency as a function of the
cavity photon number and qubit initial state (Figure S6A). A similar technique
was reported in [11]. First, the cavity was displaced by $\beta$ and allowed
to evolve for a time delay $t$, with the qubit in the ground state. The delay
time was varied up to a maximum delay time, $T$. Subsequently, a second
displacement with displacement parameter $\beta\exp\\{-i\phi(t)\\}$ was
applied, where $\phi(t)=2\pi\times 5t/T$. Finally, a cavity ground state
selective $\pi$-pulse (described by the operator $\hat{X}(\pi,\sigma_{t})$
with $\sigma_{t}=300$ ns) was applied to the qubit, and the qubit state was
measured.

The principle of the measurement is illustrated in Figure S6B. For weak Kerr
effects and the qubit in the ground state, the displaced state approximately
evolves as $|{\beta\mathrm{e}^{\mathrm{i}\omega(\beta)t}}\rangle$, where

$\omega(\beta)=\Delta-|\beta|^{2}\frac{K_{\mathrm{c}}}{2}-|\beta|^{4}\frac{K_{c}^{\prime}}{6},$
(S3)

and the cavity frequency detuning
$\Delta=\omega_{\mathrm{c}}-\omega_{\mathrm{drive}}$ from the drive frequency
$\omega_{\mathrm{drive}}$. The ground-state selective $\pi$-pulse will flip
the qubit only if $\phi(t)$ in the second displacement pulse matches
$\omega(\beta)$, i.e. $\phi(t)=\omega(\beta)$. Thus, the probability of
measuring the cavity in the ground state, or equivalently, qubit in the
excited state, is expected to be

$P(|{e}\rangle)=\lvert{\langle
0|\beta(t)\rangle}\rvert^{2}=e^{-2|\beta|^{2}[1-\cos{(\omega(\beta)t)}]-t/T_{1,\mathrm{c}}},$
(S4)

The exponential decay comes from the finite cavity lifetime.

We measure $P(|{e}\rangle)$ as a function of $t$ and $\beta$ (Figure S6C). We
then fit Eq. (S4) as a function of $t$ for the different values of $\beta$
used in the measurement (Figure S6D). From the fit, we extract $\omega(\beta)$
for the given value of $\beta$. We then fit Eq. (S3) to the measured
$\omega(\beta)$ to extract $\Delta$, $K_{c}$, and $K_{c}^{\prime}$ (Figure
S6E). Finally, we repeat the procedure with the qubit initially in the excited
state, which allows us to determine $\chi_{\mathrm{qc}}$ and
$\chi_{\mathrm{qc}}^{\prime}$.

Fig. S6. Hamiltonian Measurement Technique (A) Experimental pulse sequence.
(B) Phase space evolution of the cavity during the experiment. (C) Measurement
data of the qubit excited state probability for different values of the delay
time and the initial displacement. Here, the qubit is initialized in the
ground state. (D) Measurement of the qubit excited state probability for a
fixed value of $\beta$. The solid line is a fit of Eq. (S4). (E) The photon-
number dependent cavity frequency $\omega(\beta)$ as a function of the initial
average photon number $|\beta|^{2}$ with a fit of Eq. (S3).

Parameter Symbol Value Qubit Frequency $\omega_{\mathrm{q}}/2\pi$ $5.676001$
GHz Qubit Anharmonicity $K_{\mathrm{q}}/2\pi$ $189.9\pm 0.4$ MHz Qubit
Lifetime $T_{1}$ $31.0\pm 0.4$ µs Qubit Coherence Time $T_{2}^{*}$ $12.5\pm
0.4$ µs Qubit Hann Echo Time $T_{2}^{E}$ $16.9\pm 0.5$ µs High Q Cavity
Frequency $\omega_{\mathrm{c}}/2\pi$ $4.544939$ GHz High Q Cavity Self-Kerr
$K_{\mathrm{c}}/2\pi$ $4.9\pm 0.1$ kHz High Q Cavity Second Order Self-Kerr
$K^{\prime}_{\mathrm{c}}/2\pi$ $14\pm 8$ Hz High Q Cavity Lifetime
$T_{1,\mathrm{c}}$ $110\pm 2$ µs High Q Cavity - Qubit Dispersive Shift
$\chi_{\mathrm{qc}}/2\pi$ $1.499\pm 0.003$ MHz High Q Cavity - Qubit Second
Order Dispersive shift $\chi^{\prime}_{\mathrm{qc}}/2\pi$ $12.8\pm 0.9$ kHz
High Q Cavity Residual Mean Thermal Photon Number $n_{\mathrm{th,\ residual}}$
$0.0338\pm 0.007$ High Q Cavity Residual Mode Temperature $T_{\mathrm{cavity,\
residual}}$ $63.7\pm 0.4$ mK Qubit Residual Mode Temperature
$T_{\mathrm{qubit}}$ $68\pm 1$ mK Readout Resonator Frequency
$\omega_{\mathrm{r}}/2\pi$ $7.528852$ GHz Readout Resonator External Coupling
$\kappa_{\mathrm{c\ ext}}/2\pi$ $1.33$ MHz Readout Resonator - Qubit
Dispersive Shift $\chi_{\mathrm{rq}}/2\pi$ $1.61$ MHz Readout Resonator - High
Q Cavity Dispersive Shift (calculated) $\chi_{\mathrm{rc,cal}}/2\pi$ $3.4$ kHz

Table. S1. Hamiltonian Parameters.  The high Q cavity lifetime is limited by
its external coupling. The reported cross Kerr $\chi_{\mathrm{readout-
cavity,cal}}$ is a lower bound calculated via
$\chi_{\mathrm{rc}}=\chi_{\mathrm{qc}}\chi_{\mathrm{qr}}(\frac{1}{\Delta_{\mathrm{qc}}}+\frac{1}{\Delta_{\mathrm{qr}}})$.

##  S2 Theory

We show that the qcMAP and ECD protocols are described by the operators
$\hat{S}_{1,2}$ under ideal conditions, derive the hot cat Wigner functions
$W_{1,2}(\beta)$, and analyze the coherence functions of the hot cats.

###  S2.1 Theoretical analysis of the qcMAP and ECD protocols

In this section, we show that the ECD and qcMAP pulse sequences displayed in
the quantum circuit diagrams in Figure 2A-C are respectively equivalent to
applying the operators $\hat{S}_{1}^{\prime}$ and $\hat{S}_{2}$ to the initial
cavity state $\hat{\rho}_{0}$. A necessary condition is that $\hat{\rho}_{0}$
has no overlap with itself when displaced by $2\alpha$. We repeat the operator
definitions from the main text for convenience. The equivalent operator for
the ECD protocol is

$\hat{S}_{1}^{\prime}\equiv\frac{1}{\sqrt{2}}\left[\hat{D}(\alpha)-\mathrm{e}^{\mathrm{i}(\phi+2|\alpha|^{2})}\hat{D}(-\alpha)\right]\mathrm{i}^{\hat{n}},$
(S5)

which in turn is equivalent to the operator

$\hat{S}_{1}\equiv\frac{1}{\sqrt{2}}\left[\hat{D}(\alpha)+\mathrm{e}^{\mathrm{i}\phi}\hat{D}(-\alpha)\right]$
(S6)

when the initial state is thermal and $\phi$ is experimentally controllable.
The equivalent operator for the qcMAP protocol is

$\hat{S}_{2}\equiv\frac{1}{\sqrt{2}}\left[1-\mathrm{e}^{\mathrm{i}\phi}\hat{\varPi}\right]\hat{D}(\alpha).$
(S7)

From Figure 2A-C, we read that the ECD and qcMAP protocols first prepare the
initial thermal state $\hat{\rho}_{0}$ of the cavity and then apply the
unitary operators

$\hat{U}_{1}\equiv\hat{D}(\alpha)\hat{X}(\pi,\sigma_{t})\hat{D}(-\alpha)\hat{T}(\pi/2\chi_{\mathrm{qc}})\hat{D}[-\alpha\tfrac{(1+\mathrm{i})}{2}]\hat{Y}(\pi)\hat{D}[-\alpha\tfrac{(1+\mathrm{i})}{2}]\hat{T}(\pi/2\chi_{\mathrm{qc}})\hat{D}(\alpha)\hat{X}(\pi/2,\phi)$
(S8)

for ECD, and

$\hat{U}_{2}\equiv\hat{D}(-\alpha)\hat{X}(\pi,\sigma_{t})\hat{D}(\alpha)\hat{T}(\pi/\chi_{\mathrm{qc}})\hat{D}(\alpha)\hat{X}(\pi/2,\phi)$
(S9)

for qcMAP, to the cavity-qubit initial state
$\hat{\rho}_{0}|{g}\rangle\langle{g}|$ (we define all operators appearing in
these expressions in the next paragraph). Under ideal conditions, the final
state of the protocols is

$\hat{U}_{i}\hat{\rho}_{0}|{g}\rangle\langle{g}|\hat{U}_{i}^{\dagger}=\hat{\rho}_{i}|{g}\rangle\langle{g}|$
(S10)

($i\in\\{1,2\\}$), with

$\hat{\rho}_{i}\equiv\hat{S}_{gg,i}\hat{\rho}_{0}\hat{S}_{gg,i}^{\dagger},$
(S11) $\hat{S}_{gg,i}\equiv\langle{g}|\hat{U}_{i}|{g}\rangle.$ (S12)

It is straightforward to show that the final cavity-qubit state Eq. (S10) is a
product state, with the qubit in the $|{g}\rangle$ state, if and only if

$\mathrm{Tr}\,\\{\hat{S}_{gg,i}\hat{\rho}_{0}\hat{S}_{gg,i}^{\dagger}\\}=1.$
(S13)

Consequently, the ECD and qcMAP protocols, given exactly by $\hat{U}_{1,2}$,
are equivalent to the operators $\hat{S}_{gg,\\{1,2\\}}$ for all initial
states $\hat{\rho}_{0}$ which satisfy Eq. (S13).

The operators appearing in Eqs. (S8) and (S9) are: I) The cavity displacement
operator

$\hat{D}(\beta)\equiv\exp{\left\\{\beta\hat{c}^{\dagger}-\beta^{*}\hat{c}\right\\}}$
(S14)

where $\hat{c},\hat{c}^{\dagger}$ are the cavity annihilation and creation
operators and $\beta$ is a complex-valued argument. II) The time-evolution
operator under the dispersive interaction Hamiltonian $\hat{T}(t)$. The
interaction Hamiltonian is
$\hat{H}=-\hbar\chi_{\mathrm{qc}}\hat{n}|{e}\rangle\langle{e}|$, where
$\hat{n}=\hat{c}^{\dagger}\hat{c}$, so the time-evolution operator is

$\hat{T}(t)=\exp{\left\\{\mathrm{i}t\chi_{\mathrm{qc}}\hat{n}|{e}\rangle\langle{e}|\right\\}}=|{g}\rangle\langle{g}|+\exp{\left\\{\mathrm{i}t\chi_{\mathrm{qc}}\hat{n}\right\\}}|{e}\rangle\langle{e}|.$
(S15)

In particular,

$\hat{T}(\pi/\chi_{\mathrm{qc}})=|{g}\rangle\langle{g}|+\hat{\varPi}|{e}\rangle\langle{e}|$
(S16)

where $\hat{\varPi}=(-1)^{\hat{n}}$ is the parity operator, and

$\hat{T}(\pi/2\chi_{\mathrm{qc}})=|{g}\rangle\langle{g}|+\mathrm{i}^{\hat{n}}|{e}\rangle\langle{e}|.$
(S17)

In phase space, the operator $\mathrm{i}^{\hat{n}}$ leads to a
counterclockwise rotation by $\pi/2$ around the phase-space origin. III) The
qubit rotation operators acting only on the qubit Hilbert space

$\hat{X}(\pi/2,\phi)\equiv\frac{1}{\sqrt{2}}\left[\left(|{g}\rangle+\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}|{e}\rangle\right)\langle{g}|+\left(|{e}\rangle+\mathrm{i}\mathrm{e}^{-\mathrm{i}\phi}|{g}\rangle\right)\langle{e}|\right],$
(S18)

where $\phi$ is an experimentally controllable phase,

$\hat{X}(\pi)\equiv\mathrm{i}\left(|{g}\rangle\langle{e}|+|{e}\rangle\langle{g}|\right),$
(S19)

and

$\hat{Y}(\pi)\equiv\hat{R}(\pi,\pi/2)=|{g}\rangle\langle{e}|-|{e}\rangle\langle{g}|.$
(S20)

These operators are special cases of the general qubit rotation operator

$\hat{R}(a,{\boldsymbol{u}})\equiv\exp\left\\{\frac{\mathrm{i}a}{2}{\boldsymbol{u}}\cdot\hat{{\boldsymbol{\sigma}}}\right\\}=\hat{1}\cos\frac{a}{2}+\mathrm{i}{\boldsymbol{u}}\cdot\hat{{\boldsymbol{\sigma}}}\sin\frac{a}{2}$
(S21)

with $a$ a real variable, ${\boldsymbol{u}}$ a unit vector on the
3-dimensional unit sphere, and
$\hat{{\boldsymbol{\sigma}}}=\hat{\sigma}_{x}{\boldsymbol{e}}_{x}+\hat{\sigma}_{y}{\boldsymbol{e}}_{y}+\hat{\sigma}_{z}{\boldsymbol{e}}_{z}$
(${\boldsymbol{e}}_{j}$ are the coordinate unit vectors and $\hat{\sigma}_{j}$
the Pauli matrices
$\hat{\sigma}_{x}=|{g}\rangle\langle{e}|+|{e}\rangle\langle{g}|$,
$\hat{\sigma}_{y}=\mathrm{i}(-|{g}\rangle\langle{e}|+|{e}\rangle\langle{g}|)$,
$\hat{\sigma}_{z}=|{g}\rangle\langle{g}|-|{e}\rangle\langle{e}|$). We have
$\hat{X}(\pi/2,\phi)=\hat{R}(\pi/2,\cos\phi\,{\boldsymbol{e}}_{x}+\sin\phi\,{\boldsymbol{e}}_{y})$,
$\hat{X}(\pi)=\hat{R}(\pi,{\boldsymbol{e}}_{x})$,
$\hat{Y}(\pi)=\hat{R}(\pi,{\boldsymbol{e}}_{y})$. IV) The cavity-selective
qubit rotation operator $\hat{X}(\pi,\sigma_{t})$. We assume that it takes the
form

$\hat{X}(\pi,\sigma_{t})=\sum_{n=0}^{\infty}|{n}\rangle\langle{n}|\hat{R}(a_{n},{\boldsymbol{u}}_{n}),$
(S22)

i.e. that it is defined by specifying a sequence of qubit rotation operators
on the Fock states $|{n}\rangle$. We further assume that the parameters
$\\{a_{n}\\}_{n=0}^{\infty}$ and $\\{{\boldsymbol{u}}_{n}\\}_{n=0}^{\infty}$
are such that for $n\leq N$, where $N$ is a number that depends on
$\sigma_{t}$, $a_{n}=\pi$ and ${\boldsymbol{u}}_{n}={\boldsymbol{e}}_{x}$.
Additionally, the sequence of parameters $\\{a_{n}\\}_{n=0}^{\infty}$ decays
so that there is a number $M$ (also determined by $\sigma_{t}$) such that
$a_{n}=0$ when $n>M$. In this case, we can write

$\hat{X}(\pi,\sigma_{t})=\mathrm{i}\hat{P}_{\leq
N}\hat{\sigma}_{x}+\hat{P}_{>M}+\hat{Q}_{NM},$ (S23)

where we have introduced the operators

$\hat{P}_{\leq N}\equiv\sum_{n=0}^{N}|{n}\rangle\langle{n}|,$ (S24)
$\hat{P}_{>M}\equiv\sum_{n=M+1}^{\infty}|{n}\rangle\langle{n}|,$ (S25)
$\hat{Q}_{NM}\equiv\sum_{n=N+1}^{M}|{n}\rangle\langle{n}|\hat{R}(a_{n},{\boldsymbol{u}}_{n}).$
(S26)

It is possible (by using a Magnus approximation [20]) to find an explicit
expression for the operator $\hat{X}(\pi,\sigma_{t})$ that results when a
Gaussian qubit pulse is applied to our dispersively coupled cavity-qubit
system. That expression is indeed well-described by Eq. (S23). However, for
the present discussion, we do not need to specify $\hat{X}(\pi,\sigma_{t})$
beyond the description already made.

We now show the equivalence of $\hat{U}_{1,2}$ to $\hat{S}_{1,2}$ under
conditions which we identify during the analysis. We begin with the qcMAP
protocol $\hat{U}_{2}$. From the definitions made, $\hat{U}_{2}$ can be
rewritten as

$\hat{U}_{2}=\frac{1}{\sqrt{2}}\hat{D}(-\alpha)\hat{X}(\pi,\sigma_{t})\left[\hat{D}(2\alpha)|{g}\rangle\left(\langle{g}|+\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\langle{e}|\right)+\hat{\varPi}|{e}\rangle\left(\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\langle{g}|+\langle{e}|\right)\right]$
(S27)

from which we identify

$\hat{S}_{gg,2}=\frac{1}{\sqrt{2}}\hat{D}(-\alpha)\left[\langle{g}|\hat{X}(\pi,\sigma_{t})|{g}\rangle\hat{D}(2\alpha)+\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\hat{\varPi}\langle{g}|\hat{X}(\pi,\sigma_{t})|{e}\rangle\right]\\\
=\frac{1}{\sqrt{2}}\hat{D}(-\alpha)\left[\left(\hat{P}_{>M}+\langle{g}|\hat{Q}_{NM}|{g}\rangle\right)\hat{D}(2\alpha)+\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\hat{\varPi}\left(\mathrm{i}\hat{P}_{\leq
N}+\langle{g}|\hat{Q}_{NM}|{e}\rangle\right)\right]$ (S28)

where we inserted Eq. (S23) to go from the first to the second line. We now
wish to identify the initial states $\hat{\rho}_{0}$ for which
$\hat{S}_{gg,2}$ fulfils Eq. (S13). Consider first an initial state
$\hat{\rho}_{0}$ which has non-zero matrix elements only in the first $N$ Fock
states, i.e.

$\hat{\rho}_{0}=\sum_{k,l=0}^{N}\langle{k}|\hat{\rho}_{0}|{l}\rangle|{k}\rangle\langle{l}|.$
(S29)

For this $\hat{\rho}_{0}$, $\hat{P}_{\leq N}\hat{\rho}_{0}=\hat{\rho}_{0}$. If
we have additionally chosen $\alpha$ large enough so that the matrix element
$\langle{M}|\hat{D}(2\alpha)|{M}\rangle$ is negligible, then it also follows
(using the triangle inequality) that $\hat{P}_{\leq
M}\hat{D}(2\alpha)\hat{\rho}_{0}=0$. In a Wigner function picture, the
condition on $\alpha$ can be formulated as the $M$:th Fock state Wigner
function being negligible for arguments larger than $|\alpha|$, i.e.
$W_{|}{M}\rangle(|\beta|\geq|\alpha|)=0$. From $\hat{P}_{\leq
N}\hat{\rho}_{0}=\hat{\rho}_{0}$ and $\hat{P}_{\leq
M}\hat{D}(2\alpha)\hat{\rho}_{0}=0$ it follows that
$\hat{Q}_{NM}\hat{\rho}_{0}=0$,
$\hat{Q}_{NM}\hat{D}(2\alpha)\hat{\rho}_{0}=0$, and
$\hat{P}_{>M}\hat{D}(2\alpha)\hat{\rho}_{0}=(1-\hat{P}_{\leq
M})\hat{D}(2\alpha)\hat{\rho}_{0}=\hat{D}(2\alpha)\hat{\rho}_{0}$. We
therefore obtain

$\hat{S}_{gg,2}\hat{\rho}_{0}=\frac{1}{\sqrt{2}}\left[\hat{D}(\alpha)-\mathrm{e}^{\mathrm{i}\phi}\hat{D}(-\alpha)\hat{\varPi}\right]\hat{\rho}_{0}=\hat{S}_{2}\hat{\rho}_{0}.$
(S30)

The condition Eq. (S13) is also satisfied. Consequently,
$\hat{U}_{2}\hat{\rho}_{0}|{g}\rangle\langle{g}|\hat{U}_{2}^{\dagger}=\hat{S}_{2}\hat{\rho}_{0}\hat{S}_{2}^{\dagger}|{g}\rangle\langle{g}|$,
and $\hat{S}_{2}$ accurately describes the action of the qcMAP protocol, for
all $\hat{\rho}_{0}$ of the form Eq. (S29) when $\alpha$ is such that
$W_{|}{M}\rangle(|\beta|\geq|\alpha|)=0$.

Identical arguments show the equivalence of $\hat{U}_{1}$ to
$\hat{S}_{1}^{\prime}$ . From Eq. (S8), $\hat{U}_{1}$ can be written

$\hat{U}_{1}=\frac{1}{\sqrt{2}}\hat{D}(\alpha)\hat{X}(\pi,\sigma_{t})\left[\hat{D}(-2\alpha)\mathrm{e}^{\mathrm{i}|\alpha|^{2}}|{g}\rangle\left(\langle{g}|\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}+\langle{e}|\right)-\mathrm{e}^{-\mathrm{i}|\alpha|^{2}}|{e}\rangle\left(\langle{g}|+\mathrm{i}\mathrm{e}^{-\mathrm{i}\phi}\langle{e}|\right)\right]\mathrm{i}^{\hat{n}}.$
(S31)

We identify

$\hat{S}_{gg,1}=\frac{1}{\sqrt{2}}\hat{D}(\alpha)\left[\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\left(\hat{P}_{>M}+\langle{g}|\hat{Q}_{NM}|{g}\rangle\right)\hat{D}(-2\alpha)\mathrm{e}^{\mathrm{i}|\alpha|^{2}}-\mathrm{e}^{-\mathrm{i}|\alpha|^{2}}\left(\mathrm{i}\hat{P}_{\leq
N}+\langle{g}|\hat{Q}_{NM}|{e}\rangle\right)\right]\mathrm{i}^{\hat{n}}.$
(S32)

When this operator acts on a state $\hat{\rho}_{0}$ described by Eq. (S29),
and $\alpha$ is large enough so that $\hat{P}_{\leq
M}\hat{D}(2\alpha)\hat{\rho}_{0}=0$ as before, then

$\hat{S}_{gg,1}\hat{\rho}_{0}=\frac{-\mathrm{i}\mathrm{e}^{-\mathrm{i}|\alpha|^{2}}}{\sqrt{2}}\left[\hat{D}(\alpha)-\mathrm{e}^{\mathrm{i}(\phi+2|\alpha|^{2})}\hat{D}(-\alpha)\right]\mathrm{i}^{\hat{n}}\hat{\rho}_{0}=-\mathrm{i}\mathrm{e}^{-\mathrm{i}|\alpha|^{2}}\hat{S}_{1}^{\prime}\hat{\rho}_{0}.$
(S33)

Eq. (S13) is satisfied. The global phase vanishes when considering
$\hat{S}_{gg,1}\hat{\rho}_{0}\hat{S}_{gg,1}^{\dagger}$.

We now relax the assumption that $\hat{\rho}_{0}$ is of the form Eq. (S29). In
general, the state after the ECD and qcMAP protocols can be written

$\hat{U}_{i}\hat{\rho}_{0}|{g}\rangle\langle{g}|\hat{U}_{i}^{\dagger}=p_{g}\hat{\rho}_{g}|{g}\rangle\langle{g}|+p_{e}\hat{\rho}_{e}|{e}\rangle\langle{e}|+\hat{\psi}$
(S34)

where

$p_{g}=\mathrm{Tr}\,\\{\hat{S}_{gg,i}\hat{\rho}_{0}\hat{S}_{gg,i}^{\dagger}\\},$
(S35) $p_{e}=1-p_{g},$ (S36)

are the probabilities of finding the qubit in the $|{g}\rangle$ and
$|{e}\rangle$ states,

$\hat{\rho}_{g}\equiv\frac{\hat{S}_{gg,i}\hat{\rho}_{0}\hat{S}_{gg,i}^{\dagger}}{p_{g}},$
(S37)
$\hat{\rho}_{e}\equiv\frac{\hat{S}_{eg,i}\hat{\rho}_{0}\hat{S}_{eg,i}^{\dagger}}{p_{e}}$
(S38)

are the qubit-conditional cavity states (where
$\hat{S}_{eg,i}\equiv\langle{e}|\hat{U}_{i}|{g}\rangle$), and $\hat{\psi}$
represents off-diagonal terms in the qubit basis. The previous condition Eq.
(S13) is $p_{g}=1$. For initial states $\hat{\rho}_{0}$ which have non-
negligible matrix elements only for Fock numbers $\leq N$, i.e. are described
by Eq. (S29) plus a negligible part, one has $p_{g}\approx 1$, $p_{e}\approx
0$. More precisely, one computes

$p_{g}=\frac{1}{2}\left[\langle\hat{P}_{\leq
N}\rangle+\langle\hat{D}(-2\alpha)\hat{P}_{>M}\hat{D}(2\alpha)\rangle+q(\sigma_{t},\alpha,n_{\mathrm{th}})\right]$
(S39)

where the expectation value is with respect to $\hat{\rho}_{0}$, and $q$
represents terms related to expectation values of $\hat{Q}_{NM}$. If both the
first two terms are $1$, then $q(\sigma_{t},\alpha,n_{\mathrm{th}})=0$. If
$\langle\hat{P}_{\leq N}\rangle\approx 1$, it is always possible to choose
$\alpha$ such that also
$\langle\hat{D}(-2\alpha)\hat{P}_{>M}\hat{D}(2\alpha)\rangle\approx 1$ and
therefore $q(\sigma_{t},\alpha,n_{\mathrm{th}})\approx 0$. For these states,
the equivalence of $\hat{U}_{1,2}$ to $\hat{S}_{1,2}$ can therefore be
satisfied to arbitrary precision in principle, and it is a good approximation
to consider the ECD and qcMAP protocols equivalent to $\hat{S}_{1,2}$. The
thermal state

$\hat{\rho}_{T}\equiv\frac{1}{n_{\mathrm{th}}+1}\sum_{n=0}^{\infty}\left(\frac{n_{\mathrm{th}}}{n_{\mathrm{th}}+1}\right)^{n}|{n}\rangle\langle{n}|,$
(S40)

is a particular example of a state which can be considered to have negligible
representation for Fock numbers above $N$. It has

$\mathrm{Tr}\,\\{\hat{P}_{>N}\hat{\rho}_{T}\\}=\left(\frac{n_{\mathrm{th}}}{n_{\mathrm{th}}+1}\right)^{N+1}$
(S41)

which goes to $0$ in the limit $N\to\infty$. In practice, the choice of
$\sigma_{t}$ (i.e. $N$ and $M$) and $\alpha$ are restricted by experimental
limitations, and the nonzero $p_{e}$ leads to perturbations from the ideal
result (i.e. that described by $\hat{S}_{1,2}$).

The conditions derived so far are those under which our protocols become
equivalent to the operators $\hat{S}_{1,2}$. As a final remark, we note that
one can also study the conditions for the operators $\hat{S}_{1,2}$ to be
effectively unitary independently of how they are implemented. This can be
done by replacing $\hat{S}_{gg,i}$ with $\hat{S}_{i}$ in Eq. (S13). For a
general initial state $\hat{\rho}_{0}$, one then has

$\mathrm{Tr}\,\left\\{\hat{S}_{1}\hat{\rho}_{0}\hat{S}_{1}^{\dagger}\right\\}=1-\mathrm{Re}\,\left\\{\mathrm{e}^{-\mathrm{i}\phi}\chi_{0}(2\alpha)\right\\}=1,$
(S42)
$\mathrm{Tr}\,\left\\{\hat{S}_{2}\hat{\rho}_{0}\hat{S}_{2}^{\dagger}\right\\}=1-\frac{\pi}{2}\cos{\phi}\,W_{0}(\alpha)=1.$
(S43)

Here,
$W_{0}(\alpha)=2\pi^{-1}\mathrm{Tr}\,\\{\hat{\varPi}(\alpha)\hat{\rho}_{0}\\}$
and $\chi_{0}(\alpha)=\mathrm{Tr}\,\\{\hat{D}(\alpha)\hat{\rho}_{0}\\}$ are
the Wigner and characteristic functions of the initial state. These equations
can be satisfied either by choice of $\phi$ or $\alpha$. If we want this
equation to be satisfied for any $\phi$, this gives the necessary conditions
that $\chi_{0}(|\beta|>2|\alpha|)=0$ ($\hat{S}_{1}$) and
$W_{0}(|\beta|>|\alpha|)=0$ ($\hat{S}_{2}$). We also see that if $\phi$ is a
half-integer multiple of $\pi$, the operator $\hat{S}_{2}$ is unitary
independently of $\alpha$. For states that have a constant-phase
characteristic function (e.g. thermal states), there are also choices of
$\phi$ for which $\hat{S}_{1}$ is effectively unitary independently of
$\alpha$. To avoid confusion, we stress that $\phi$ does not enter any of the
conditions for the ECD and qcaMAP protocols to be equivalent to the operators
$\hat{S}_{1,2}$.

###  S2.2 Derivation of the Wigner functions $W_{1,2}(\beta)$ from the
operators $\hat{S}_{1,2}$

We give derivations of the Wigner functions $W_{1,2}(\beta)$ that result when
the operators $\hat{S}_{1,2}$ are applied to an initial state $\hat{\rho}_{0}$
and in particular the thermal state $\hat{\rho}_{T}$. Starting from

$W_{1,2}(\beta)=\frac{2}{\pi}\mathrm{Tr}\,\\{\hat{\varPi}(\beta)\hat{S}_{1,2}\hat{\rho}_{0}\hat{S}_{1,2}^{\dagger}\\}$
(S44)

where
$\hat{\varPi}(\beta)=\hat{D}(\beta)\hat{\varPi}\hat{D}^{\dagger}(\beta)$, one
uses the cyclic property of the trace to express $W_{1,2}(\beta)$ as the
expectation value of the operator
$\hat{S}_{1,2}^{\dagger}\hat{\varPi}(\beta)\hat{S}_{1,2}$ in the initial
state. This allows us to express the Wigner functions $W_{1,2}(\beta)$ in
terms of the Wigner function of the initial state $W_{0}(\beta)$, which is
given by

$W_{0}(\beta)=\frac{2}{\pi}\mathrm{Tr}\,\\{\hat{\varPi}(\beta)\hat{\rho}_{0}\\}.$
(S45)

Here, $\hat{\rho}_{0}$ is any state for which the operators $\hat{S}_{1,2}$
accurately describe the outcome of the ECD and qcMAP protocols (see Section
S2.1). One computes

$\hat{S}^{\dagger}_{1}\hat{\varPi}(\beta)\hat{S}_{1}=\frac{1}{2}\Bigl{\\{}\hat{\varPi}\left(\beta-\alpha\right)+\hat{\varPi}\left(\beta+\alpha\right)\\\
-2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi\right)\hat{\varPi}(\beta)\Bigr{\\}}$
(S46)

and consequently

$W_{1}(\beta)=\frac{1}{2}\Bigl{\\{}W_{0}\left(\beta-\alpha\right)+W_{0}\left(\beta+\alpha\right)-2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi\right)W_{0}(\beta)\Bigr{\\}}.$
(S47)

If one instead of $\hat{S}_{1}$ uses $\hat{S}_{1}^{\prime}$ (Eq. (S5)), the
result is

$W_{1^{\prime}}(\beta)=\frac{1}{2}\Bigl{\\{}W_{0}\left[-\mathrm{i}(\beta-\alpha)\right]+W_{0}\left[-\mathrm{i}(\beta+\alpha)\right]-2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi+2|\alpha|^{2}\right)W_{0}(-\mathrm{i}\beta)\Bigr{\\}}.$
(S48)

Further, one computes

$\hat{S}^{\dagger}_{2}\hat{\varPi}(\beta)\hat{S}_{2}=\frac{1}{2}\Bigl{[}\hat{\varPi}(\beta-\alpha)+\hat{\varPi}(-\beta-\alpha)-\mathrm{e}^{\mathrm{i}(\phi+4\mathrm{Im}\,\\{\alpha^{*}\beta\\})}\hat{D}(2\beta)-\mathrm{e}^{-\mathrm{i}(\phi+4\mathrm{Im}\,\\{\alpha^{*}\beta\\})}\hat{D}(-2\beta)\Bigr{]}$
(S49)

and therefore

$W_{2}(\beta)=\frac{1}{2}\left[W_{0}(\beta-\alpha)+W_{0}(-\beta-\alpha)-\frac{4}{\pi}\mathrm{Re}\,\left\\{\mathrm{e}^{\mathrm{i}(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi)}\chi_{0}(2\beta)\right\\}\right].$
(S50)

Here we have identified the characteristic function [25] of the initial state

$\chi_{0}(\beta)\equiv\mathrm{Tr}\,\\{\hat{D}(\beta)\hat{\rho}_{0}\\}=\int\mathrm{d}^{2}\gamma\,\mathrm{e}^{2\mathrm{i}\mathrm{Im}\,\\{\beta\gamma^{*}\\}}W_{0}(\gamma).$
(S51)

Note that these Wigner functions are normalized because we have implicitly
assumed an $|\alpha|$ large enough such that there is no overlap between the
first two terms in each of Eqs. (S50) and (S47) (Section S2.1). For these
values of $|\alpha|$, the third term in each equation oscillates rapidly
enough to integrate to zero.

We now specialize to the thermal state, i.e. we set
$\hat{\rho}_{0}=\hat{\rho}_{T}$. The Wigner function of the thermal initial
state is

$W_{T}(\beta)=\frac{2\mathcal{P}}{\pi}\mathrm{e}^{-2\mathcal{P}|\beta|^{2}}$
(S52)

where $\mathcal{P}=(2n_{\mathrm{th}}+1)^{-1}$ is the purity of the thermal
state. The corresponding characteristic function is

$\chi_{T}(\beta)=\mathrm{e}^{-|\beta|^{2}/2\mathcal{P}}.$ (S53)

Replacing $W_{0}(\beta)$ and $\chi_{T}(\beta)$ in Eq. (S50) and Eq. (S47)
gives

$W_{1}(\beta)=\frac{\mathcal{P}}{\pi}\left[\mathrm{e}^{-2\mathcal{P}|\beta-\alpha|^{2}}+\mathrm{e}^{-2\mathcal{P}|\beta+\alpha|^{2}}-2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi\right)\mathrm{e}^{-2\mathcal{P}|\beta|^{2}}\right].$
(S54)
$W_{2}(\beta)=\frac{1}{\pi}\left[\mathcal{P}\left(\mathrm{e}^{-2\mathcal{P}|\beta-\alpha|^{2}}+\mathrm{e}^{-2\mathcal{P}|\beta+\alpha|^{2}}\right)-2\cos\left(4\mathrm{Im}\,\\{\alpha^{*}\beta\\}+\phi\right)\mathrm{e}^{-2|\beta|^{2}/\mathcal{P}}\right].$
(S55)

These are the Wigner functions given in the main text up to a redefinition of
the parameter $\phi$ ($\phi\to\phi+\pi$). Eq. (S54) is also obtained from
$W_{1}^{\prime}(\beta)$ if one inserts the thermal Wigner function and
redefines $\phi$ to absorb the geometric phase.

###  S2.3 Hot cat state coherence functions

In this section, we give the full two-dimensional hot cat coherence functions
as well as their derivations. As stated in the main text, the first-order
coherence function of a quantum state with density matrix $\hat{\rho}$ is
defined as

$g(x_{1},x_{2})\equiv\frac{|\langle{x_{1}}|\hat{\rho}|{x_{2}}\rangle|}{\sqrt{\langle{x_{1}}|\hat{\rho}|{x_{1}}\rangle\langle{x_{2}}|\hat{\rho}|{x_{2}}\rangle}}.$
(S56)

Here $x_{1,2}$ are real-valued dimensionless numbers, and $|{x_{1,2}}\rangle$
are eigenkets of the quadrature operator

$\hat{x}\equiv\frac{\hat{c}+\hat{c}^{\dagger}}{\sqrt{2}},$ (S57)

meaning that $\hat{x}|{x_{1}}\rangle=x_{1}|{x_{1}}\rangle$ and equivalently
for $x_{2}$. Due to the positive semidefiniteness of $\hat{\rho}$, the
coherence function is bounded: $1\geq g(x_{1},x_{2})\geq 0$.

We compute the coherence function from $W_{1,2}(\beta)$ using the relation

$\langle{x_{1}}|\hat{\rho}|{x_{2}}\rangle=\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}p\,W\left(\frac{x_{1}+x_{2}}{2\sqrt{2}}+\frac{\mathrm{i}p}{\sqrt{2}}\right)\mathrm{e}^{\mathrm{i}p(x_{1}-x_{2})}$
(S58)

which can be derived e.g. from the expression for computing expectation values
from the Wigner function
$\langle\hat{A}\rangle=\mathrm{Tr}\,\\{\hat{A}\hat{\rho}\\}=\int\mathrm{d}^{2}\gamma\,2\mathrm{Tr}\,\\{\hat{\varPi}(\gamma)\hat{A}\\}W(\gamma)$
with $\hat{A}=|{x_{2}}\rangle\langle{x_{1}}|$. It is helpful to first remind
ourselves of the coherence function of a thermal state. Using Eq. (S52) in Eq.
(S58), we find the coherence function

$g_{T}(x_{1},x_{2})\equiv\exp\left\\{-\frac{(x_{1}-x_{2})^{2}}{4}\frac{(1-\mathcal{P}^{2})}{\mathcal{P}}\right\\}$
(S59)

which we denote $g_{T}$ since it is the coherence function of the thermal
state $\hat{\rho}_{T}$. The thermal state coherence function is independent of
the position on the diagonal $x_{1}+x_{2}$ and is a Gaussian in the distance
from the diagonal $x_{1}-x_{2}$. Its standard deviation

$\xi_{\mathrm{th}}\equiv\sqrt{\frac{2\mathcal{P}}{1-\mathcal{P}^{2}}}.$ (S60)

is termed the coherence length. When $\mathcal{P}^{2}\ll 1$,
$\xi_{\mathrm{th}}\approx\sqrt{2\mathcal{P}}$. In this case, the coherence
length is related to the quadrature standard deviation of the thermal state
$\sigma_{x}\equiv\sqrt{\mathrm{Tr}\,\\{\hat{x}^{2}\hat{\rho}_{0}\\}}=1/\sqrt{2\mathcal{P}}$
by the reciprocal relationship $\xi_{\mathrm{th}}=1/\sigma_{x}$. In the
opposite limit, $\mathcal{P}\to 1$, $\xi_{\mathrm{th}}\to\infty$ since in this
case $g(x_{1},x_{2})\to 1$.

We denote the coherence functions of the ECD and qcMAP states respectively as
$g_{1}(x_{1},x_{2})$ and $g_{2}(x_{1},x_{2})$. We also introduce the notation
$\bar{x}\equiv(x_{1}+x_{2})/2$ and $\Delta x\equiv x_{1}-x_{2}$ to make
expressions more concise. From Eq. (S56), Eq. (S58) and Eqs. (S55) and (S54),
we compute

$g_{1}(x_{1},x_{2})=\frac{\mathrm{e}^{-(\Delta
x)^{2}/2\xi_{\mathrm{th}}^{2}}\left|\cosh{(\sqrt{2}\alpha\mathcal{P}2\bar{x})}-\mathrm{e}^{-4\alpha^{2}/\xi_{\mathrm{th}}^{2}}\cosh{\left(\sqrt{2}\alpha\Delta
x/\mathcal{P}-\mathrm{i}\phi\right)}\right|}{\sqrt{\left[\cosh{(2\sqrt{2}\alpha\mathcal{P}x_{1})}-\mathrm{e}^{-4\alpha^{2}/\xi_{\mathrm{th}}^{2}}\cos(\phi)\right]\left[\cosh{(2\sqrt{2}\alpha\mathcal{P}x_{2})}-\mathrm{e}^{-4\alpha^{2}/\xi_{\mathrm{th}}^{2}}\cos(\phi)\right]}}.$
(S61)

and

$g_{2}(x_{1},x_{2})=\frac{\left|\mathrm{e}^{-(\Delta
x)^{2}/2\xi_{\mathrm{th}}^{2}}\cosh\left(\sqrt{2}\alpha\mathcal{P}2\bar{x}\right)-\mathrm{e}^{-(2\bar{x})^{2}/2\xi_{\mathrm{th}}^{2}}\cosh\left(\sqrt{2}\alpha\mathcal{P}\Delta
x-\mathrm{i}\phi\right)\right|}{\sqrt{\left[\cosh(2\sqrt{2}\alpha\mathcal{P}x_{1})-\mathrm{e}^{-2x_{1}^{2}/\xi_{\mathrm{th}}^{2}}\cos(\phi)\right]\left[\cosh(2\sqrt{2}\alpha\mathcal{P}x_{2})-\mathrm{e}^{-2x_{2}^{2}/\xi_{\mathrm{th}}^{2}}\cos(\phi)\right]}}$
(S62)

These expressions are exact, and we plot them as 2-dimensional functions of
$x_{1}$ and $x_{2}$ in Figure S5. In the remainder of this section, we will
explain the appearance of Figure S5 by making approximations.

We are mainly interested in understanding $g_{1,2}$ for $\mathcal{P}\ll 1$ and
for $(x_{1},x_{2})$ away from the origin $(0,0)$ (since our nonzero entries in
$\langle{x_{1}}|\hat{\rho}|{x_{2}}\rangle$ are centered around the four points
$(\pm\sqrt{2}\alpha,\pm\sqrt{2}\alpha)$,
$(\pm\sqrt{2}\alpha,\mp\sqrt{2}\alpha)$, and we must have
$\alpha>\sigma_{x}=1/\sqrt{2\mathcal{P}}$). When the purity is not close to 1,
the terms $\propto\cos(\phi)$ in the denominators of $g_{1,2}$ are generally
negligible ($g_{1}$) or nonzero only close to the coordinate axes $(g_{2})$.
We therefore drop these terms, which makes the denominators of $g_{1,2}$
equal. Using the hyperbolic trig relations, we rewrite the denominators to be
$[\cosh{(2\sqrt{2}\alpha\mathcal{P}2\bar{x})}+\cosh{(2\sqrt{2}\alpha\mathcal{P}\Delta
x)}]/2$. Finally, we approximate $\cosh{(x)}\approx\exp{(|x|)/2}$ for both
$\bar{x}$ and $\Delta x$. This gives

$g_{1}(x_{1},x_{2})\approx\frac{\exp\left\\{-(\Delta
x)^{2}/2\xi_{\mathrm{th}}^{2}\right\\}}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(|\Delta
x|-2|\bar{x}|)\right\\}}}+\frac{\exp\left\\{-(|\Delta
x|-2\sqrt{2}\alpha)^{2}/2\xi_{\mathrm{th}}^{2}\right\\}}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(2|\bar{x}|-|\Delta
x|)\right\\}}}.$ (S63) $g_{2}(x_{1},x_{2})\approx\frac{\exp\left\\{-(\Delta
x)^{2}/2\xi_{\mathrm{th}}^{2}\right\\}}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(|\Delta
x|-2|\bar{x}|)\right\\}}}+\frac{\exp\left\\{-(2\bar{x})^{2}/2\xi_{\mathrm{th}}^{2}\right\\}}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(2|\bar{x}|-|\Delta
x|)\right\\}}},$ (S64)

These expressions differ only in the nominator of the last term. The
denominators are approximately indicator functions for the $x_{1}x_{2}>0$
quadrants (first term denominator) and $x_{1}x_{2}<0$ quadrants (second term
denominator) of the $(x_{1},x_{2})$ plane, i.e.

$\frac{1}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(|\Delta
x|-2|\bar{x}|)\right\\}}}\approx[x_{1}x_{2}>0]=\left\\{\begin{array}[]{cc}1&x_{1}x_{2}>0\\\
0&x_{1}x_{2}<0\end{array}\right.,$ (S65)
$\frac{1}{\sqrt{1+\exp\left\\{2\sqrt{2}\alpha\mathcal{P}(2|\bar{x}|-|\Delta
x|)\right\\}}}\approx[x_{1}x_{2}<0]=\left\\{\begin{array}[]{cc}0&x_{1}x_{2}>0\\\
1&x_{1}x_{2}<0\end{array}\right..$ (S66)

(The notation $[\ \ ]$ for the conditional expressions is called the Iverson
bracket). Using this observation, we arrive at our final expressions for
$g_{1,2}$:

$g_{1}(x_{1},x_{2})\approx
g_{T}(x_{1},x_{2})[x_{1}x_{2}>0]+g_{T}(|x_{1}-x_{2}|,2\sqrt{2}\alpha)[x_{1}x_{2}<0].$
(S67) $g_{2}(x_{1},x_{2})\approx g_{T}(|x_{1}|,|x_{2}|).$ (S68)

When $\mathcal{P}\ll 1$, the approximation Eq. (S65) loses accuracy in Eq.
(S63), and in this limit we should instead replace $[x_{1}x_{2}>0]$ by $1$ in
Eq. (S67) to get an accurate approximation for $g_{1}(x_{1},x_{2})$. Eqs.
(S67) and (S68) agree well with the exact functions plotted in Figure S5.

Along the line $l(s)\equiv(x_{1}(s),x_{2}(s))=\sqrt{2}\alpha(2s-1,-1)$,
$s\in[0,\infty]$ in the $(x_{1},x_{2})$ plane, $g_{1}$ and $g_{2}$ are equal
within our approximations leading up to Eqs. (S67) and (S68). This is the line
along which we plot the hot cat state coherence functions in Figure 1C in the
main text.

Fig. S5. Hot Schrödinger Cat State Coherence Functions. Shown is the thermal
state coherence function $g_{T}(x_{1},x_{2})$ Eq. (S59) (left column), and the
two cat coherence functions $g_{1}(x_{1},x_{2})$ Eq. (S61) (center column) and
$g_{2}(x_{1},x_{2})$ Eq. (S62) (right column) for a cold (top row) and hot
(bottom row) $n_{\mathrm{th}}$. The plot uses $\alpha=3$ and $\phi=0$.

##  S3 Numerical model

Based on our characterization of the experimental setup, we introduce the
numerical _ab initio_ model of our experiment and compare its predictions to
the experimentally measured data.

###  S3.1 Method

We perform all numerical work using QuTiP version 4.7 [45]. We simulate the
ECD and qcMAP protocols as follows: Displacement operators and thermal initial
states are implemented using QuTiP’s built-in functions. The time evolution
operations $\hat{T}(t)$ are implemented using QuTiP’s built-in functions to
simulate the cavity-qubit dynamics during time evolution, as described in the
next paragraph. The qubit and cavity-conditional qubit operations are
implemented by simulating the qubit-cavity dynamics under a driving
Hamiltonian, as described in more detail below. This lets us compute the
density matrix resulting from the state preparation protocols. We obtain the
result of the Wigner function measurement from this density matrix by
computing the expectation value of the observable

$\hat{M}(\beta)\equiv\frac{2}{\pi}\hat{\varPi}(\beta)\left(|{g}\rangle\langle{g}|-|{e}\rangle\langle{e}|\right).$
(S69)

This observable gives the outcome of the Wigner function measurement when
decoherence and nonlinearities during the measurement sequence are neglected
[15].

We simulate the time-evolution operators $\hat{T}(t)$ by using QuTiP’s built-
in mesolve function to solve the following Lindblad equation

$\frac{\partial}{\partial
t}\hat{\rho}(t)=\mathcal{L}\hat{\rho}(t)\equiv-\frac{\mathrm{i}}{\hbar}\left[\hat{H},\,\hat{\rho}(t)\right]+\left(\gamma_{1}\mathcal{D}\left[|{g}\rangle\langle{e}|\right]+\frac{\gamma_{2}}{2}\mathcal{D}\left[\hat{\sigma}_{z}\right]+\Gamma\mathcal{D}\left[\hat{c}\right]\right)\hat{\rho}(t)$
(S70)

from $t=t_{0}$, where $\hat{\rho}(t_{0})$ is the total cavity-qubit state
before the $\hat{T}$ operation is to be applied, until the final time $t$. The
state $\hat{\rho}(t)$ is then used as input for the next step of the protocol.
Here $\gamma_{1}=1/T_{1}$, $\gamma_{2}=1/T_{2}^{*}$ are the dissipation and
dephasing rates of the qubit and $\Gamma=1/T_{1,\mathrm{c}}$ is the
dissipation rate of the cavity. $\mathcal{D}$ is the dissipator superoperator,
defined as

$\mathcal{D}[\hat{A}]\hat{\rho}(t)\equiv\hat{A}\hat{\rho}(t)\hat{A}^{\dagger}-\frac{1}{2}\left(\hat{A}^{\dagger}\hat{A}\hat{\rho}(t)+\hat{\rho}(t)\hat{A}^{\dagger}\hat{A}\right),$
(S71)

for an arbitrary operator $\hat{A}$, and $\hat{H}$ is the Hamiltonian in the
interaction picture of the cavity and qubit including the dominant higher-
order perturbations

$\frac{1}{\hbar}\hat{H}\equiv-\chi_{\mathrm{qc}}\hat{c}^{\dagger}\hat{c}|{e}\rangle\langle{e}|-\left(\frac{K_{\mathrm{c}}}{2}+\frac{\chi_{\mathrm{qc}}^{\prime}}{2}|{e}\rangle\langle{e}|\right)\hat{c}^{\dagger}\hat{c}^{\dagger}\hat{c}\hat{c}.$
(S72)

We choose the parameters
$K_{\mathrm{c}},\chi_{\mathrm{qc}}^{\prime},\gamma_{1},\gamma_{2},\Gamma$ to
be the values measured in the experiment (Table S1).

For the qubit operations, we use QuTiP’s built-in mesolve function to solve
the Lindblad equation

$\frac{\partial}{\partial
t}\hat{\rho}(t)=\mathcal{L}\hat{\rho}(t)-\frac{\mathrm{i}}{\hbar}\left[\hat{H}_{\mathrm{drive}},\,\hat{\rho}(t)\right]$
(S73)

from time $t_{0}$, where $\hat{\rho}(t_{0})$ is the total cavity-qubit state
before the qubit operation is applied, to time $t$, when the next operation is
applied, and we use $\hat{\rho}(t)$ as initial state for the following
operation. The driving Hamiltonian is

$\frac{1}{\hbar}\hat{H}_{\mathrm{drive}}\equiv\frac{\Omega(t)}{2}\left(\mathrm{e}^{\mathrm{i}\phi}|{e}\rangle\langle{g}|+\mathrm{e}^{-\mathrm{i}\phi}|{g}\rangle\langle{e}|\right)$
(S74)

with

$\Omega(t)\equiv\frac{\theta\mathrm{e}^{-(t-t_{0}-T/2)^{2}/2\sigma_{t}^{2}}}{\sqrt{2\pi\sigma_{t}^{2}}\operatorname{erf}\left(T/2^{3/2}\sigma_{t}\right)}.$
(S75)

Here $T$ is the pulse duration so that $t=T+t_{0}$, $\operatorname{erf}$ is
the error function, and $\theta$ is the pulse area over the interval $T$:

$\int_{t_{0}}^{t_{0}+T}\mathrm{d}\tau\,\Omega(\tau)=\theta.$ (S76)

We use $T=4\sigma_{t}$ as this is the value used in the experiment. We take
$\theta$, $\sigma_{t}$ and $\phi$ as the parameters of the pulse and choose
them according to the desired operation to be modelled. In particular,
$\sigma_{t}=6$ ns for the global qubit operations, and $\sigma_{t}=20$ ns for
the cavity-selective disentanglement pulse. We compensate the free evolution
times for the finite length of the qubit pulses, so that the maximum of the
disentanglement pulse occurs at $t=\pi/\chi_{\mathrm{qc}}$ in both protocols,
and in the ECD protocol, the maximum of the qubit echo pulse occurs at
$t=\pi/2\chi_{\mathrm{qc}}$.

###  S3.2 Comparison of Simulation and Data

We run our simulations with all parameters of the Lindbladian taking the
values that we measure experimentally (Table S1), and the pulse parameters
being the experimentally used values. We present a comparison of the
simulation results to the experimental Wigner map data in Figure S7. Here we
simulated with $n_{\mathrm{th}}=3.48$, $\alpha=3.06$ for the ECD protocol and
$\alpha=3.47$ for the ECD protocol, and $\phi=\pi$. To facilitate a comparison
of the simulations to the data, we displace the simulated Wigner functions so
that their fringe pattern aligns with that of the data. We do this by
identifying the coherence fringe in the simulated Wigner functions
corresponding most closely to the coherence fringe centered at $\beta=0$ in
the data. We then displace the simulated Wigner functions so that the
identified fringe is also centered at $\beta=0$. Specifically, the simulated
ECD Wigner function was displaced by $0.654\mathrm{i}$ and the simulated qcMAP
Wigner function was displaced by $0.111-0.353\mathrm{i}$. We additionally
rotate the qcMAP state clockwise by $0.163$ radians so that the centers of the
displaced thermal states lie along the $\mathrm{Re}\,\\{\beta\\}$ axis. The
ECD state was not rotated.

Using the same simulation parameters as for the Wigner maps (including the
final displacement and rotations), we also simulate the Wigner function
linecuts along the $\mathrm{Re}\,\\{\beta\\}$ and $\mathrm{Im}\,\\{\beta\\}$
axes for the values of $n_{\mathrm{th}}$ that were reported in Figure 2D-G. We
present the computed linecuts in Figure S8.

Fig. S7. Comparison of simulated Wigner maps to measured data. (A)
Experimental data obtained for the ECD protocol (also displayed in Figure 1B).
(B) Experimental data obtained for the qcMAP protocol (also displayed in
Figure 1C). (C) Result of the numerical simulation of the ECD protocol. (D)
Result of the numerical simulation of the qcMAP protocol.

Fig. S8. Wigner function linecuts obtained from simulation. The line colors
are chosen in analogy to the line colors in Figure 1D-G. (A) Linecut along
$\mathrm{Re}\,\\{\beta\\}$ through the ECD Wigner function. (B) Linecut along
$\mathrm{Im}\,\\{\beta\\}$ through the ECD Wigner function. (C) Linecut along
$\mathrm{Re}\,\\{\beta\\}$ through the qcMAP Wigner function. (D) Linecut
along $\mathrm{Im}\,\\{\beta\\}$ through the qcMAP Wigner function.

##  S4 Analysis of Imperfections

The experimentally achieved protocols differ from the ideal ECD and qcMAP
protocols analyzed theoretically in Section S2. In this section, we study the
identified differences in isolation using a combination of analytical and
numerical methods. By comparing the results to the ideal protocols and the
data, we can attribute features in the data which are not seen in the
theoretical Wigner functions to particular experimental imperfections.

###  S4.1 Residual cavity-qubit entanglement

The experimental imperfections lead to a finite probability $p_{e}$ of the
qubit being in the excited state at the end of the protocol. From Eq. (S69)
and Eq. (S34), one computes that the expected result of the measurement is

$W_{\mathrm{meas.}}(\beta)\equiv\mathrm{Tr}\,\left\\{\hat{M}(\beta)\hat{U}\hat{\rho}_{0}|{g}\rangle\langle{g}|\hat{U}^{\dagger}\right\\}=p_{g}W_{g}(\beta)-p_{e}W_{e}(\beta).$
(S77)

Here, we have introduced the qubit-conditional Wigner functions

$W_{g,e}(\beta)\equiv\frac{2}{\pi}\mathrm{Tr}\,\\{\hat{\varPi}(\beta)\hat{\rho}_{g,e}\\}.$
(S78)

Imperfect disentanglement between the cavity and qubit will lead to $p_{e}>0$,
$p_{g}<1$. In this case, the output of the Wigner function measurement is not
the final state Wigner function, but rather the weighted difference of the
Wigner functions of the qubit-conditional states $\hat{\rho}_{g}$ and
$\hat{\rho}_{e}$, with the weights being $p_{g}$ and $p_{e}$. As long as
$p_{g}\gg p_{e}$ and/or the Wigner functions $W_{g}(\beta)$ and $W_{e}(\beta)$
do not overlap significantly, the residual qubit-cavity entanglement is a
perturbative imperfection to measurement of the final cavity state Wigner
function.

###  S4.2 Comparison of simulations with different parameters

To understand the effect of each difference between our experiment and the
ideal ECD and qcMAP protocols, we run our simulations with different sets of
parameters which are chosen to isolate each difference. We present the results
of these simulations in Figures S9 (ECD) and S10 (qcMAP). We go through the
parameters used for each panel of these figures in the next paragraphs.

‘Experimental parameters’ (panel A) refers to the simulations as described in
Section S3, where we chose the parameters to match the experiment as closely
as possible (however, in Figures S9 and S10, we do not rotate or displace the
Wigner functions before plotting).

‘Reference parameters’ (panel B) refers to a set of parameters chosen to match
the ideal ECD and qcMAP protocols considered in Section S2.1. These
simulations have no Kerr nonlinearities, negligible width of the non-selective
qubit pulses, infinite coherence times, and a disentanglement pulse width
which was optimized by sweeping $\sigma_{t}$ for these parameters and choosing
the $\sigma_{t}$ that minimized $p_{e}$ in this scenario. Specifically, the
reference parameters are: $K_{\mathrm{c}}=\chi^{\prime}_{\mathrm{qc}}=0$,
$\Gamma=\gamma_{1}=\gamma_{2}=0$, disentanglement pulse $\sigma_{t}=6$ ns,
other qubit pulses $\sigma_{t}=10^{-13}$ s.

Panels C-F show simulations using the reference parameters but with a subset
of the parameters set to their experimental values, as explained in the figure
captions.

Fig. S9. Comparison of simulations with different parameters for ECD. We
simulate the hot cat state preparation protocols with different sets of
parameters, which are chosen to isolate the differences between our experiment
and the ideal ECD protocol. All simulations shown used $n_{\mathrm{th}}=3.48$
and $\alpha=3.06$. (A) Simulation with all parameters taking the values
measured or used in the experiment, as in Fig. S7C. (B) Simulation with
parameters chosen to match the ideal ECD protocol (see Section S4.2). (C)
Simulation with the reference parameters but the disentanglement pulse
standard deviation $\sigma_{t}$ set to the experiment value of 20 ns. (D)
Simulation with the reference parameters but the Kerr nonlinearity parameters
$K_{\mathrm{c}}$ and $\chi_{\mathrm{qc}}^{\prime}$ taking their experimentally
measured values. (E) Simulation with the reference parameters but the
$\sigma_{t}$ of all qubit pulses set to 6 ns, the minimum pulse $\sigma_{t}$
that our experimental instrumentation can achieve. (F) Simulation with the
reference parameters but with the coherence times $T_{1}$, $T_{2}^{*}$ and
$T_{1,\mathrm{c}}$ taking their experimental values.

Fig. S10. Comparison of simulations with different parameters for qcMAP. We
simulate the hot cat state preparation protocols with different sets of
parameters, which are chosen to isolate the differences between our experiment
and the ideal qcMAP protocol. All simulations shown used
$n_{\mathrm{th}}=3.48$ and $\alpha=3.47$. (A) Simulation with all parameters
taking the values measured or used in the experiment, as in Fig. S7D. (B)
Simulation with parameters chosen to match the ideal qcMAP protocol (see
Section S4.2). (C) Simulation with the reference parameters but the
disentanglement pulse standard deviation $\sigma_{t}$ set to the experiment
value of 20 ns. (D) Simulation with the reference parameters but the Kerr
nonlinearity parameters $K_{\mathrm{c}}$ and $\chi_{\mathrm{qc}}^{\prime}$
taking their experimentally measured values. (E) Simulation with the reference
parameters but the $\sigma_{t}$ of all qubit pulses set to 6 ns, the minimum
pulse $\sigma_{t}$ that our experimental instrumentation can achieve. (F)
Simulation with the reference parameters but with the coherence times $T_{1}$,
$T_{2}^{*}$ and $T_{1,\mathrm{c}}$ taking their experimental values.

###  S4.3 Free evolution timing errors

In the experiment, all cavity and qubit operation pulses take a finite time.
Cavity displacement pulses take 16 ns, and qubit operations take $4\sigma_{t}$
to complete (24 ns for non-selective qubit pulses, 80 ns for the
disentanglement pulse). The design of the experimental protocol takes this
into account by optimizing the pulse timings, with a step size of 4 ns.
Nevertheless, here we theoretically investigate the effects of perturbing the
time argument of the free evolution operator $\hat{T}(t)$ in the qcMAP
protocol. We show that such perturbations, if present, lead to bending
distortions of the hot cat state fringes which are not seen for cold cat
states.

In the ideal qcMAP protocol, the free evolution time is
$\pi/\chi_{\mathrm{qc}}$. Here we take the free evolution time to be
$\pi/\chi_{\mathrm{qc}}+\tau$, where $\tau$ is the timing error. Following the
analysis in Section S2.1, the effective operator of the protocol becomes

$\hat{S}_{1}=\frac{1}{\sqrt{2}}\left[1-\exp\left\\{\mathrm{i}\left(\chi_{\mathrm{qc}}\tau\hat{n}+\phi\right)\right\\}\hat{\varPi}\right]\hat{D}(\alpha).$
(S79)

The Wigner function resulting from the application of this operator on a
cavity state can be computed using the same methods as in Section S2.2. The
result is

$W_{1}(\beta)=\frac{1}{2}\biggl{[}W_{0}(\beta-\alpha)+W_{0}(-\alpha-\beta\mathrm{e}^{-\mathrm{i}\chi_{\mathrm{qc}}\tau})\\\
-\frac{4}{\pi}\mathrm{Re}\,\left\\{\mathrm{e}^{\mathrm{i}\varphi}\mathrm{Tr}\,\left\\{\hat{D}\left[2\beta+\alpha(\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}-1)\right]\exp\left\\{\mathrm{i}\chi_{\mathrm{qc}}\tau\hat{n}\right\\}\hat{\rho}_{0}\right\\}\right\\}\biggr{]}$
(S80)

with

$\varphi=\phi+2\mathrm{Im}\,\left\\{\alpha^{*}\beta\left[1+\exp\left(-\mathrm{i}\chi_{\mathrm{qc}}\tau\right)\right]\right\\}+|\alpha|^{2}\sin\left(\chi_{\mathrm{qc}}\tau\right).$
(S81)

For a thermal state $\hat{\rho}_{0}=\hat{\rho}_{T}$,

$\mathrm{Tr}\,\left\\{\hat{D}\left[2\beta+\alpha(\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}-1)\right]\exp\left\\{\mathrm{i}\chi_{\mathrm{qc}}\tau\hat{n}\right\\}\hat{\rho}_{0}\right\\}=\\\
=\frac{1}{1+n_{\mathrm{th}}\left(1-\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}\right)}\exp\left\\{-\left(\frac{1}{2}+\frac{n_{\mathrm{th}}\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}}{1+n_{\mathrm{th}}\left(1-\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}\right)}\right)\left|2\beta+\alpha\left(\mathrm{e}^{\mathrm{i}\chi_{\mathrm{qc}}\tau}-1\right)\right|^{2}\right\\}.$
(S82)

Taking $\tau=0$ recovers Eq. (S55) as expected. To understand the effect of a
small $\tau$, we linearize Eq. (S81) and Eq. (S82) in $\tau$. Taking $\alpha$
to be real, the coherence term in Eq. (S80) linearized in $\tau$ is

$\frac{4}{\pi}\cos\left\\{4\alpha\mathrm{Re}\,\\{k^{*}\beta\\}+\phi^{\prime}+4\chi_{\mathrm{qc}}\tau|\beta|^{2}n_{\mathrm{th}}(1+n_{\mathrm{th}})\right\\}\\\
\cdot\exp\left\\{-2(2n_{\mathrm{th}}+1)\left(|\beta|^{2}+\chi_{\mathrm{qc}}\tau\alpha\mathrm{Im}\,\\{\beta\\}\right)\right\\}.$
(S83)

Here $k\equiv-\chi_{\mathrm{qc}}\tau/2+\mathrm{i}$ and
$\phi^{\prime}\equiv\phi+\chi_{\mathrm{qc}}\tau(|\alpha|^{2}+n_{\mathrm{th}})$.
This expression contains three additional effects compared to the $\tau=0$
case: 1) The phase shift has changed from $\phi$ to $\phi^{\prime}$. 2) Since
the left Gaussian has moved in the $\mathrm{Im}\,\\{\beta\\}$ direction, the
center of the fringes has also moved in the $\mathrm{Im}\,\\{\beta\\}$
direction, and $k$ has obtained a real part. 3) The presence of $|\beta|^{2}$
in the cosine argument causes a bending distortion of the fringes. The
$|\beta|^{2}$ term vanishes when $n_{\mathrm{th}}\to 0$, so that noticeable
bending of the fringes occurs only for hot cats.

For illustration, we plot Eq. (S80) in Figure S11 for $\alpha=3.47$,
$n_{\mathrm{th}}=3.48$, $\phi=\pi$, and a timing error of $\tau=20$ ns. We
emphasize that this value of $\tau$ is much larger than the 4 ns steps with
which we optimize the pulse timings in our experiment. Figure S11 bears some
resemblance to Figures S7 and S8; however, our numerical simulations do not
contain any timing error, and the fringe bending observed in our experiment is
mainly due to the Kerr nonlinearity (see Section S4.2).

Fig. S11. Fringe bending due to timing errors. (A) Wigner function Eq. (S80)
for a timing error of 20 ns. The Wigner function has been rotated and
displaced before plotting to remove the extra rotation due to the timing error
and center the fringe pattern at $\beta=0$. (B) Linecut through the Wigner
function in panel A along $\mathrm{Re}\,{\beta}$. (C) Linecut through the
Wigner function in panel A along $\mathrm{Im}\,{\beta}$.