# Number-Resolved Detection of Dark Ions in Coulomb Crystals

Fabian Schmid<EMAIL_ADDRESS>Max-Planck-Institut für Quantenoptik,
85748 Garching, Germany  Johannes Weitenberg  Max-Planck-Institut für
Quantenoptik, 85748 Garching, Germany  Jorge Moreno  Max-Planck-Institut für
Quantenoptik, 85748 Garching, Germany  Theodor W. Hänsch  Max-Planck-Institut
für Quantenoptik, 85748 Garching, Germany  Fakultät für Physik, Ludwig-
Maximilians-Universität München, 80799 München, Germany  Thomas Udem  Max-
Planck-Institut für Quantenoptik, 85748 Garching, Germany  Fakultät für
Physik, Ludwig-Maximilians-Universität München, 80799 München, Germany  Akira
Ozawa<EMAIL_ADDRESS>Max-Planck-Institut für Quantenoptik, 85748
Garching, Germany

###### Abstract

While it is straightforward to count laser-cooled trapped ions by fluorescence
imaging, detecting the number of dark ions embedded and sympathetically cooled
in a mixed ion crystal is more challenging. We demonstrate a method to track
the number of dark ions in real time with single-particle sensitivity. This is
achieved by observing discrete steps in the amount of fluorescence emitted
from the coolant ions while exciting secular motional resonances of dark ions.
By counting the number of fluorescence steps, we can identify the number of
dark ions without calibration and without relying on any physical model of the
motional excitation. We demonstrate the scheme by detecting H${}_{2}^{+}$ and
H${}_{3}^{+}$ ions embedded in a Be+ ion Coulomb crystal in a linear radio
frequency trap. Our method allows observing the generation and destruction of
individual ions simultaneously for different types of ions. Besides high-
resolution spectroscopy of dark ions, another application is the detection of
chemical reactions in real time with single-particle sensitivity. This is
demonstrated in this work.

Trapped and laser-cooled ions have been used to investigate fundamental light-
atom interactions [1, 2, 3, 4, 5], for observation of ion-neutral chemical
reactions [6, 7, 8, 9, 10], as well as for optical frequency standards [11,
12] and quantum computing [13, 14, 15].

Sympathetic cooling is a way for extending this technology to atomic or
molecular ions that do not possess suitable transitions for laser cooling. In
this scheme, the ions are trapped together with another ion species that can
be laser-cooled. Due to the mutual Coulomb interaction, the ions rapidly
thermalize, indirectly cooling all species. At sufficiently low temperatures,
the ions form regular Coulomb crystals in the trap [16]. The technique has
found application in precision spectroscopy of HD+ and H${}_{2}^{+}$ molecular
ions [17, 18, 19, 20, 21], optical atomic clocks based on quantum logic
spectroscopy of Al+ [22, 23], spectroscopy of highly-charged ions [24, 25],
and the study of chemical reactions with molecular ions [26, 7, 27, 28, 29].

While the number of laser-cooled ions can be easily measured by fluorescence
imaging, identification and counting of the non-fluorescing dark ions is more
difficult. One method is to eject the ions from the trap and to accelerate
them onto a detector in an electric field. The different ion species can be
distinguished by their arrival times [30, 31, 32, 33]. While this method
allows a quantitative measurement of the number of ions of each species, it
has the disadvantage that it is destructive and a new ion crystal has to be
prepared after each measurement.

In linear Paul traps, lighter ions are more tightly confined than heavier
ions. Lighter sympathetically cooled ions therefore form a dark region in the
center of the fluorescence image of an ion crystal consisting of a heavier
coolant species. The number of dark ions can then be obtained by comparing
experimental images with simulated ones [17, 34, 35]. This method is non-
destructive, and the ion images can be acquired quickly and post-processed
later. However, different dark ion species cannot be distinguished.

Instead, secular excitation has been used for non-destructive detection of
trapped ions. In this method the secular motion of the ions, i.e. the harmonic
motion in the time averaged trap potential, is excited resonantly by applying
an additional oscillating electric field. This transfers energy into the
motion of the surrounding coolant ions and thereby increases their
temperature. Due to the temperature dependence of the Doppler broadening this
leads to a change in the amount of fluorescence that can be observed from the
coolant ions. The secular motion of the ions in three-dimensional Coulomb
crystals has rich dynamics that can complicate the analysis of the secular
excitation spectra. For example, the frequencies of the secular resonances are
influenced by space charge effects and the mechanical coupling between the
ions [36, 37, 38]. The energy transfer to the coolant ions is expected to
increase with an increasing number of dark ions. Therefore, the fluorescence
change induced by motional excitation serves as a measure of the number of
dark ions. However, the relationship between the fluorescence change and the
number of excited dark ions is in general non-linear and is influenced by
various experimental parameters, such as the strength of the motional
excitation and the geometry of the mixed ion crystal. Therefore, evaluating
the number of dark ions quantitatively is challenging and often requires
intricate modeling and calibration of the signal using molecular dynamics
simulations [26, 39]. This problem has been limiting the usage of the secular
excitation method for highly precise spectroscopy so far.

In this work, we show that by properly choosing experimental parameters,
discrete steps in the secular excitation signal can be observed that are
identified with individual dark ions leaving the trap or being generated
within the trap. Hence such a signal is auto-calibrating and counting the
number of ions gives the ultimate accuracy. The signal does not have to be
calibrated using a physical model of the secular excitation. Spurious signals
at other frequencies that may arise from motional coupling have no influence
on the counting process and can be safely ignored.

We experimentally demonstrate this method by resonantly exciting the radial
motion of H${}_{2}^{+}$ and H${}_{3}^{+}$ ions embedded in a laser-cooled Be+
ion crystal. We observe concomitant changes in the amount of fluorescence from
the Be+ ions when the number of trapped H${}_{2}^{+}$ or H${}_{3}^{+}$ ions
changes due to chemical reactions with neutral rest gas molecules.

Spectroscopy of dark ions requires a scheme for detecting that the target
transition is being excited. This can for example be achieved by monitoring
that new ion species are created by state-dependent photoionization [40] or
resonance-enhanced multiphoton dissociation [39, 18, 19, 20]. The reliable
detection of single dark ions demonstrates that this detection scheme can be
single-event sensitive and that the spectroscopy will be ultimately limited
only by quantum projection noise. Nonlinearities in the signal intensity may
introduce a systematic frequency shift, especially when the spectrum consists
of multiple overlapping lines [39]. Accurate counting of the dark ions gives
rise to a spectroscopy signal with negligible nonlinearity.

Chemical reactions at ultracold temperatures can be investigated precisely for
a small number of atoms or ions after careful quantum-state preparation [41].
Our detection scheme can be employed to efficiently capture such events with
single-particle sensitivity.

Figure 1: Geometry of the ion trap setup. Radial (top) and axial (bottom)
view. The Coulomb crystal is not drawn to scale. Radial confinement is
generated by the RF voltage $U_{\mathrm{rf}}(t)$ that is applied to one
diagonal pair of blade electrodes. A static voltage $U_{\mathrm{ec}}$ is
applied to the endcap electrodes for axial confinement. The cooling beam
(purple) is directed along the trap axis ($z$). The fluorescence from the
laser-cooled Be+ ions is imaged onto two EMCCD cameras. The static
compensation voltage $U_{\mathrm{c}}$ and voltages applied to the two dc blade
electrodes are used to compensate stray fields in the trap caused by patch
potentials and electrode misalignment. A sinusoidal voltage
$U_{\mathrm{ex}}(t)$ is added to the compensation electrode in order to excite
the secular motion of the trapped ions.

We use a linear Paul trap to confine the ions. As shown in Fig. 1, it consists
of four blade electrodes that are spaced 0.45 mm from the trap axis and have
an axial length of 3.00 mm. Axial confinement is provided by two endcap
electrodes that are located 3.50 mm from the trap center.

The trap is driven with a radio frequency (RF) of 66.05 MHz with an amplitude
of around 120 V, and a static voltage of 400 V is applied to the endcaps. For
Be+ this results in a radial secular frequency of around 1.6 MHz,
corresponding to a Mathieu stability parameter $q\approx 0.07$ [5], and an
axial secular frequency of 645 kHz. The single-particle radial secular
frequency scales proportional to the ion’s charge-to-mass ratio. The
theoretical values are 4.8 MHz and 7.2 MHz for H${}_{3}^{+}$ and
H${}_{2}^{+}$, respectively.

The Be+ ions are laser-cooled by driving the 2s 2S1/2 ($F$=2) $\rightarrow$ 2p
2P3/2 ($F$=3) cycling transition with circularly polarized 313 nm light that
is generated by sum-frequency generation of two continuous wave fiber lasers
at 1051 nm and 1550 nm and subsequent frequency-doubling [42]. The cooling
transition has a natural linewidth of $\Gamma=2\pi\times 18$ MHz [43], and the
cooling beam contains two frequency components that are red-detuned from the
transition by 130 MHz and 460 MHz. We found that adding the far-detuned
component makes the system more robust against losing the trapped ions when
strongly driving secular excitations. Both frequency components can be
switched on or off separately and have similar intensities of around
$I_{\mathrm{sat}}$, where $I_{\mathrm{sat}}=765~{}\textup{W}/\textup{m}^{2}$
is the saturation intensity of the cooling transition [43]. The cooling beam
is aligned parallel to the trap axis and propagates through holes in the end-
cap electrodes. A weak magnetic field is applied parallel to the beam in order
to define the quantization axis. We use an electro-optic modulator to create
1.25 GHz sidebands on the cooling laser in order to re-pump the ions out of
the 2s 2S1/2 ($F$=1) dark state.

The trap is housed in an ultra-high vacuum chamber. The background pressure
measured with a cold-cathode ion gauge is around 4 $\times$ 10-11 mbar.

We load Be+ ions into the trap with the help of a beam of beryllium atoms from
an oven. The beam is sent through the trapping region where it is overlapped
with a 235 nm laser beam that resonantly ionizes the atoms [44]. During Be+
loading, initially only the far-detuned frequency component is applied to
efficiently cool the hot ions. Once fluorescence from trapped ions is
detected, the near-detuned frequency component is turned on. This strongly
increases the fluorescence signal, such that the ion crystal shape can be
monitored. The ionization laser is turned off when the desired ion crystal
size is reached.

We then turn on a home-built electron gun for 1 s with a current of about 95
µA. The electrons ionize some of the hydrogen molecules from the residual gas
in our vacuum chamber, and the resulting molecular ions become embedded in the
ion crystal.

As shown in Fig. 1, we use two imaging systems to observe the fluorescence
from the laser-cooled ions from the horizontal and vertical direction
perpendicular to the trap axis. In each system, the light is imaged onto an
electron-multiplying CCD (EMCCD) camera using a home-built objective with
magnification M $\approx 9$ and a numerical aperture of 0.23. The vertical
imaging system additionally contains a beam splitter that sends roughly half
of the light onto a photomultiplier tube (PMT).

Figure 2: Fluorescence images of an ion crystal consisting of around 2000 Be+
ions with a few embedded H${}_{2}^{+}$ and H${}_{3}^{+}$ ions, horizontal (a)
and vertical (b) view. (c) Coulomb crystal containing 2000 Be+ ions (red
spheres) and 5 H${}_{2}^{+}$ ions (light blue spheres) obtained by the
molecular dynamics simulation. (d) Simulated ion image for the crystal shown
in (c) at 10 mK.

Fig. 2(a) and (b) show typical fluorescence images of an ion crystal in our
setup. We believe that the asymmetric shape visible in Fig. 2(b) is due to
imperfect alignment of the trap electrodes.

In order to measure the radial secular frequencies of the trapped ions, we
excite their motion by applying a sinusoidal voltage to the compensation
electrode that is located close to the trap electrodes (see Fig. 1). We sweep
the frequency range between 2 MHz and 10 MHz in 500 steps and for each
frequency collect photon counts with the PMT for 10 ms. This is slow enough
that the ion motion reaches steady state for each frequency point. Fig. 3
shows a typical resulting secular spectrum. Two peaks at 4.6 MHz and 7.1 MHz
can be observed which we attribute to H${}_{3}^{+}$ and H${}_{2}^{+}$ ions,
respectively. The electrode that we use for the secular excitation is low-pass
filtered with a 1 nF capacitor and is not impedance matched to the 50 $\Omega$
output impedance of the function generator that produces the excitation
signal. This makes the electric field amplitude at the position of the ions
frequency dependent. In addition, the efficiency of the motional excitation
might be different for H${}_{3}^{+}$ and H${}_{2}^{+}$ ions. We therefore use
two different excitation amplitudes for the frequency range from 2 MHz to 5.5
MHz and from 5.5 MHz to 10 MHz. The amplitudes are chosen to give roughly the
same signal strength per ion for the H${}_{2}^{+}$ and H${}_{3}^{+}$
resonances.

Figure 3: Secular spectrum showing the presence of H${}_{2}^{+}$ and
H${}_{3}^{+}$ ions embedded in the Be+ coolant ions.

We then repeatedly perform secular scans over a period of a few minutes. Fig.
4 shows how the height of the secular resonances changes over time in a
typical experiment. We normalize the peak heights of each scan to the off-
resonant scattering rate. This compensates for small drifts in the cooling
laser power and Be+ ion loss during the measurement. The height of the
H${}_{2}^{+}$ signal drops in steps until after around 360 s no peak is
visible anymore. The height of the H${}_{3}^{+}$ signal increases in steps
that coincide with drops of the H${}_{2}^{+}$ signal. We attribute this to
individual H${}_{2}^{+}$ ions reacting with molecular hydrogen from the
residual gas to form neutral hydrogen and H${}_{3}^{+}$ according to the
exothermic chemical reaction H${}_{2}^{+}$ \+ H${}_{2}\rightarrow$
H${}_{3}^{+}$ \+ H [45]. The clearly visible steps in the signals show that we
are observing individual ions being destroyed or created within the Coulomb
crystal. Sometimes the H${}_{2}^{+}$ signal drops without a corresponding
increase in H${}_{3}^{+}$ signal. One possible explanation is that the
H${}_{2}^{+}$ ion reacted with a different residual gas molecule such as N2,
O2, or H2O [46]. Another possibility is that the kinetic energy transferred to
the H${}_{3}^{+}$ molecule in the chemical reaction is large enough to
sometimes allow the product to escape from the trap. Rate constants for the
chemical reactions between H${}_{2}^{+}$ and typical residual gas molecules
are in the range of 2–7$\times 10^{-9}$ cm3 s-1 [45, 46]. An exponential fit
to the H${}_{2}^{+}$ ion number yields a total decay rate of 0.006 s-1 which
corresponds to a residual gas pressure in the range of $4\times 10^{-11}$ mbar
to $1\times 10^{-10}$ mbar, in rough agreement with the pressure indicated by
the vacuum gauge. The number of H${}_{3}^{+}$ ions subsequently drops in
similar steps over the timescale of a few minutes. We attribute this to proton
transfer reactions of the type H${}_{3}^{+}$ \+ X $\rightarrow$ HX+ \+ H2
which are possible with a number of residual gas molecules such as N2, CO,
CO2, and H2O [47].

Figure 4: Change of the fluorescence peak heights at the H${}_{2}^{+}$ (blue
circles) and H${}_{3}^{+}$ (orange squares) resonances over time. For each
measurement, the peak heights are normalized to the off-resonant scattering
rate. The stepwise changes of the peak heights are caused by individual ions
undergoing chemical reactions with residual gas molecules. Within each step,
the peak heights scatter by much less than the step-to-step difference. Each
secular scan therefore allows a reliable determination of the ion numbers.
Figure 5: Experimental secular excitation spectra for (a) 0–3 H${}_{3}^{+}$
ions and (b) 0–4 H${}_{2}^{+}$ ions. (c) and (d) are spectra obtained from
molecular dynamics simulations with 2000 Be+ ions and the same numbers of
H${}_{2}^{+}$ and H${}_{3}^{+}$ ions as in the experiment. The simulated
excitation amplitudes are 2.9 V/m for the H${}_{3}^{+}$ resonance and 2.5 V/m
for the H${}_{2}^{+}$ resonance.

One striking feature of Fig. 4 and Fig. 5 is that the fluorescence peak
heights are nearly linear in the number of dark ions. We use the LAMMPS
molecular dynamics code [48, 49] to further investigate this behavior. We
first simulate the stationary configuration of the mixed ion crystals without
applying the excitation as shown in Fig. 2(c). We generate simulated
fluorescence images, taking into account the motion of the ions and the spot
size and depth of field of the imaging objective. We then estimate the number
of trapped ions in the experiment and their temperature by comparing the
simulated images with the experimental ones [37, 34, 35]. A good agreement is
obtained between experimental and simulated images for around 2000 Be+ ions at
10 mK mixed with a few dark ions as shown in Fig. 2(d). In order to simulate
the secular excitation spectra, we add an oscillating electric field
perpendicular to the trap axis. The amount of fluorescence from the coolant
ions is calculated during the motional excitation. The simulation is repeated
for different excitation frequencies and numbers of dark ions. Fig. 5(a) and
(b) show experimental secular excitation spectra for different numbers of
H${}_{3}^{+}$ and H${}_{2}^{+}$ ions, and Fig. 5(c) and (d) show the
corresponding simulated spectra. The stepwise change of the fluorescence peak
height for different numbers of ions is reproduced in the simulations. Both
the simulation and the measurement suggest that the change in the amount of
fluorescence is roughly proportional to the number of dark ions that are being
excited at their secular frequency.

Figure 6: Average peak heights of the H${}_{2}^{+}$ (blue circles) and
H${}_{3}^{+}$ (orange squares) secular resonances. The error bars show the
standard error of the mean. The solid and dashed lines are fits of our model
curve $R(T_{\mathrm{s}})/R(T_{0})$ to the H${}_{2}^{+}$ and H${}_{3}^{+}$
data, respectively. Separate fits are performed for the H${}_{2}^{+}$ and
H${}_{3}^{+}$ resonances since the secular excitation strength is different
for the two species and two different fit parameters
$\alpha_{\mathrm{H}_{2}^{+}}$ and $\alpha_{\mathrm{H}_{3}^{+}}$ were
necessary.

We now derive a model for the dependence of the peak scattering rate on the
number of ions that are being excited. For a cycling cooling transition in an
atom at rest the photon scattering rate is given by

$w(\Delta)=\frac{s\Gamma/2}{1+s+(\frac{\Delta}{\Gamma/2})^{2}},$ (1)

where $\Delta$ is the laser detuning and $s=I/I_{\mathrm{sat}}$ is the
saturation parameter. During the secular excitation, the temperature of the
coolant ions increases to $T_{\mathrm{s}}$, and the scattering rate per Be+
ion is given by [39]

$R(T_{\mathrm{s}})=\int_{-\infty}^{\infty}w(\Delta-
kv)p(v,T_{\mathrm{s}})\mathrm{d}v,$ (2)

where $p(v,T)=\sqrt{m_{\mathrm{Be}}/\left(2\pi
k_{\mathrm{B}}T\right)}\exp{\left(-m_{\mathrm{Be}}v^{2}/(2k_{\mathrm{B}}T)\right)}$
is the Maxwell-Boltzmann distribution. $v$ is the velocity of the ion along
the cooling laser beam, $k$ is the cooling laser wave number, and
$m_{\mathrm{Be}}$ is the Be+ mass. We assume that the coolant ion temperature
$T_{\mathrm{s}}$ during the secular excitation is determined by the balance
between the heating due to the motional excitation and the average laser
cooling power. This condition is expressed as

$N_{\mathrm{Be}}p_{\mathrm{c}}(T_{\mathrm{s}})=N_{\mathrm{ex}}p_{\mathrm{h}},$
(3)

where $N_{\mathrm{Be}}$ is the number of Be+ ions and $N_{\mathrm{ex}}$ is the
number of dark ions that are being excited. $p_{\mathrm{h}}$ is the heating
power per excited dark ion. Note that the average cooling power per coolant
ion $p_{\mathrm{c}}(T_{\mathrm{s}})$ is temperature dependent because of the
Doppler broadening of the cooling transition

$p_{\mathrm{c}}(T_{\mathrm{s}})=-\hbar k\int_{-\infty}^{\infty}vw(\Delta-
kv)p(v,T_{\mathrm{s}})\mathrm{d}v.$ (4)

We confirmed the validity of our assumption of Eq. (3) using the molecular
dynamics simulations (see Supplemental Material [50]). The relative
fluorescence peak heights of the secular resonances shown in Fig. 4 correspond
to $R(T_{\mathrm{s}})/R(T_{0})$ where $T_{0}$ is the equilibrium temperature
of the Be+ ions in the absence of heating due to secular excitation. Under our
experimental conditions, $T_{0}$ is around 10 mK. For simplicity we use
$T_{0}=0$ K which introduces a marginal ($\sim 2\%$) difference in the
scattering rate. It is difficult to precisely estimate the heating rate
$p_{\mathrm{h}}$ from the experimental parameters. We therefore treat
$\alpha=p_{\mathrm{h}}/N_{\mathrm{Be}}$ as a free parameter in our model. We
evaluated $R(T_{\mathrm{s}})/R(T_{0})$ numerically and determined $\alpha$ to
best reproduce the experimental data. No other fit parameters were used. The
results are shown in Fig. 6 for different numbers of H${}_{2}^{+}$ and
H${}_{3}^{+}$ ions. For our parameters the model predicts a nearly linear
dependence of the scattering rate on the number of dark ions, which agrees
with the experimental observations.

With a larger number of dark ions, the model predicts that the fluorescence
peak height starts to saturate, and the fluorescence steps become smaller
accordingly. We have conducted experiments for a larger number of dark ions
and confirmed that up to eight dark ions can be reliably counted under our
experimental conditions. Additional molecular dynamics simulations with
different numbers of dark/coolant ions and different dark-ion species suggest
that our method is robust for a wide range of experimental parameters.

In conclusion, we have demonstrated that the secular excitation technique can
be used to detect dark ions embedded in laser-cooled ion Coulomb crystals with
single-particle resolution. Secular excitation only relies on the Coulomb
interaction between the different ion species. We therefore believe that this
method could be used for single-particle sensitive detection of a wide range
of atomic or molecular ions that do not possess suitable cycling transitions
for fluorescence detection.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 742247). T. W. Hänsch acknowledges support from the Max
Planck Foundation. The authors acknowledge the Max Planck Computing and Data
Facility (MPCDF) for providing computing time.

## References

  * Itano  _et al._ [2015] W. M. Itano, J. C. Bergquist, and D. J. Wineland, Early observations of macroscopic quantum jumps in single atoms, Int. J. Mass Spectrom. 377, 403 (2015).
  * Eichmann  _et al._ [1993] U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, D. J. Wineland, and M. G. Raizen, Young’s interference experiment with light scattered from two atoms, Phys. Rev. Lett. 70, 2359 (1993).
  * Stalgies  _et al._ [1996] Y. Stalgies, I. Siemers, B. Appasamy, T. Altevogt, and P. E. Toschek, The spectrum of single-atom resonance fluorescence, Europhys. Lett. 35, 259 (1996).
  * Eschner  _et al._ [2001] J. Eschner, Ch. Raab, F. Schmidt-Kaler, and R. Blatt, Light interference from single atoms and their mirror images, Nature 413, 495 (2001).
  * Leibfried  _et al._ [2003] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
  * Drewsen  _et al._ [2003] M. Drewsen, I. Jensen, J. Lindballe, N. Nissen, R. Martinussen, A. Mortensen, P. Staanum, and D. Voigt, Ion Coulomb crystals: a tool for studying ion processes, Int. J. Mass Spectrom. 229, 83 (2003).
  * Roth  _et al._ [2006] B. Roth, P. Blythe, H. Wenz, H. Daerr, and S. Schiller, Ion-neutral chemical reactions between ultracold localized ions and neutral molecules with single-particle resolution, Phys. Rev. A 73, 042712 (2006).
  * Willitsch  _et al._ [2008a] S. Willitsch, M. T. Bell, A. D. Gingell, S. R. Procter, and T. P. Softley, Cold reactive collisions between laser-cooled ions and velocity-selected neutral molecules, Phys. Rev. Lett. 100, 043203 (2008a).
  * Yang  _et al._ [2018] T. Yang, A. Li, G. K. Chen, C. Xie, A. G. Suits, W. C. Campbell, H. Guo, and E. R. Hudson, Optical control of reactions between water and laser-cooled Be+ ions, J. Phys. Chem. Lett. 9, 3555 (2018).
  * Tomza  _et al._ [2019] M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek, and P. S. Julienne, Cold hybrid ion-atom systems, Rev. Mod. Phys. 91, 035001 (2019).
  * Diddams  _et al._ [2001] S. A. Diddams, Th. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, An optical clock based on a single trapped 199Hg+ ion, Science 293, 825 (2001).
  * Huntemann  _et al._ [2016] N. Huntemann, C. Sanner, B. Lipphardt, Chr. Tamm, and E. Peik, Single-ion atomic clock with 3 $\times$ 10-18 systematic uncertainty, Phys. Rev. Lett. 116, 063001 (2016).
  * Blatt and Wineland [2008] R. Blatt and D. Wineland, Entangled states of trapped atomic ions, Nature 453, 1008 (2008).
  * Debnath  _et al._ [2016] S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, Demonstration of a small programmable quantum computer with atomic qubits, Nature 536, 63 (2016).
  * Pogorelov  _et al._ [2021] I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. Höfer, C. Wächter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz, Compact ion-trap quantum computing demonstrator, PRX Quantum 2, 020343 (2021).
  * Hornekær  _et al._ [2001] L. Hornekær, N. Kjærgaard, A. M. Thommesen, and M. Drewsen, Structural properties of two-component Coulomb crystals in linear Paul traps, Phys. Rev. Lett. 86, 1994 (2001).
  * Blythe  _et al._ [2005] P. Blythe, B. Roth, U. Fröhlich, H. Wenz, and S. Schiller, Production of ultracold trapped molecular hydrogen ions, Phys. Rev. Lett. 95, 183002 (2005).
  * Alighanbari  _et al._ [2020] S. Alighanbari, G. S. Giri, F. L. Constantin, V. I. Korobov, and S. Schiller, Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions, Nature 581, 152 (2020).
  * Patra  _et al._ [2020] S. Patra, M. Germann, J.-Ph. Karr, M. Haidar, L. Hilico, V. I. Korobov, F. M. J. Cozijn, K. S. E. Eikema, W. Ubachs, and J. C. J. Koelemeij, Proton-electron mass ratio from laser spectroscopy of HD+ at the part-per-trillion level, Science 369, 1238 (2020).
  * Schmidt  _et al._ [2020] J. Schmidt, T. Louvradoux, J. Heinrich, N. Sillitoe, M. Simpson, J.-Ph. Karr, and L. Hilico, Trapping, cooling, and photodissociation analysis of state-selected H${}_{2}^{+}$ ions produced by ($3+1$) multiphoton ionization, Phys. Rev. Appl. 14, 024053 (2020).
  * Kortunov  _et al._ [2021] I. V. Kortunov, S. Alighanbari, M. G. Hansen, G. S. Giri, V. I. Korobov, and S. Schiller, Proton-electron mass ratio by high-resolution optical spectroscopy of ion ensembles in the resolved-carrier regime, Nat. Phys. 17, 569 (2021).
  * Schmidt  _et al._ [2005] P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Spectroscopy using quantum logic, Science 309, 749 (2005).
  * Brewer  _et al._ [2019] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, 27Al+ quantum-logic clock with a systematic uncertainty below 10-18, Phys. Rev. Lett. 123, 033201 (2019).
  * Kozlov  _et al._ [2018] M. G. Kozlov, M. S. Safronova, J. R. Crespo López-Urrutia, and P. O. Schmidt, Highly charged ions: Optical clocks and applications in fundamental physics, Rev. Mod. Phys. 90, 045005 (2018).
  * Micke  _et al._ [2020] P. Micke, T. Leopold, S. A. King, E. Benkler, L. J. Spieß, L. Schmöger, M. Schwarz, J. R. Crespo López-Urrutia, and P. O. Schmidt, Coherent laser spectroscopy of highly charged ions using quantum logic, Nature 578, 60 (2020).
  * Baba and Waki [2002] T. Baba and I. Waki, Chemical reaction of sympathetically laser-cooled molecular ions, J. Chem. Phys. 116, 1858 (2002).
  * Willitsch  _et al._ [2008b] S. Willitsch, M. T. Bell, A. D. Gingell, and T. P. Softley, Chemical applications of laser- and sympathetically-cooled ions in ion traps, Phys. Chem. Chem. Phys. 10, 7200 (2008b).
  * Hall and Willitsch [2012] F. H. J. Hall and S. Willitsch, Millikelvin reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ion-atom hybrid trap, Phys. Rev. Lett. 109, 233202 (2012).
  * Okada  _et al._ [2013] K. Okada, T. Suganuma, T. Furukawa, T. Takayanagi, M. Wada, and H. A. Schuessler, Cold ion–polar-molecule reactions studied with a combined Stark-velocity-filter–ion-trap apparatus, Phys. Rev. A 87, 043427 (2013).
  * Schneider  _et al._ [2014] C. Schneider, S. J. Schowalter, K. Chen, S. T. Sullivan, and E. R. Hudson, Laser-cooling-assisted mass spectrometry, Phys. Rev. Appl 2, 034013 (2014).
  * Meyer  _et al._ [2015] K. A. E. Meyer, L. L. Pollum, L. S. Petralia, A. Tauschinsky, C. J. Rennick, T. P. Softley, and B. R. Heazlewood, Ejection of Coulomb crystals from a linear Paul ion trap for ion-molecule reaction studies, J. Phys. Chem. A 119, 12449 (2015).
  * Rösch  _et al._ [2016] D. Rösch, H. Gao, A. Kilaj, and S. Willitsch, Design and characterization of a linear quadrupole ion trap for high-resolution Coulomb-crystal time-of-flight mass spectrometry, EPJ Tech. Instrum. 3, 1 (2016).
  * Schmid  _et al._ [2017] P. C. Schmid, J. Greenberg, M. I. Miller, K. Loeffler, and H. J. Lewandowski, An ion trap time-of-flight mass spectrometer with high mass resolution for cold trapped ion experiments, Rev. Sci. Instrum. 88, 123107 (2017).
  * Germann  _et al._ [2014] M. Germann, X. Tong, and S. Willitsch, Observation of electric-dipole-forbidden infrared transitions in cold molecular ions, Nat. Phys. 10, 820 (2014).
  * Okada  _et al._ [2015] K. Okada, M. Ichikawa, M. Wada, and H. A. Schuessler, Quasiequilibrium characterization of mixed-ion Coulomb crystals, Phys. Rev. Appl. 4, 054009 (2015).
  * Roth  _et al._ [2007] B. Roth, P. Blythe, and S. Schiller, Motional resonance coupling in cold multispecies Coulomb crystals, Phys. Rev. A 75, 023402 (2007).
  * Zhang  _et al._ [2007] C. B. Zhang, D. Offenberg, B. Roth, M. A. Wilson, and S. Schiller, Molecular-dynamics simulations of cold single-species and multispecies ion ensembles in a linear Paul trap, Phys. Rev. A 76, 012719 (2007).
  * Ann  _et al._ [2019] B.-m. Ann, F. Schmid, J. Krause, T. W. Hänsch, T. Udem, and A. Ozawa, Motional resonances of three-dimensional dual-species Coulomb crystals, Journal of Physics B: Atomic, Molecular and Optical Physics 52, 035002 (2019).
  * Biesheuvel  _et al._ [2016] J. Biesheuvel, J.-Ph. Karr, L. Hilico, K. S. E. Eikema, W. Ubachs, and J. C. J. Koelemeij, High-precision spectroscopy of the HD+ molecule at the 1-p.p.b. level, Appl. Phys. B 123, 23 (2016).
  * Herrmann  _et al._ [2009] M. Herrmann, M. Haas, U. D. Jentschura, F. Kottmann, D. Leibfried, G. Saathoff, C. Gohle, A. Ozawa, V. Batteiger, S. Knünz, N. Kolachevsky, H. A. Schüssler, T. W. Hänsch, and Th. Udem, Feasibility of coherent xuv spectroscopy on the 1S–2S transition in singly ionized helium, Phys. Rev. A 79, 052505 (2009).
  * Ratschbacher  _et al._ [2012] L. Ratschbacher, C. Zipkes, C. Sias, and M. Köhl, Controlling chemical reactions of a single particle, Nat. Phys. 8, 649 (2012).
  * Wilson  _et al._ [2011] A. C. Wilson, C. Ospelkaus, A. P. VanDevender, J. A. Mlynek, K. R. Brown, D. Leibfried, and D. J. Wineland, A 750-mW, continuous-wave, solid-state laser source at 313 nm for cooling and manipulating trapped 9Be+ ions, Appl. Phys. B 105, 741 (2011).
  * Fuhr and Wiese [2010] J. R. Fuhr and W. L. Wiese, Tables of atomic transition probabilities for beryllium and boron, J. Phys. Chem. Ref. Data 39, 013101 (2010).
  * Lo  _et al._ [2013] H.-Y. Lo, J. Alonso, D. Kienzler, B. C. Keitch, L. E. de Clercq, V. Negnevitsky, and J. P. Home, All-solid-state continuous-wave laser systems for ionization, cooling and quantum state manipulation of beryllium ions, Appl. Phys. B 114, 17 (2013).
  * Bowers  _et al._ [1969] M. T. Bowers, D. D. Elleman, and J. King, Analysis of the ion-molecule reactions in gaseous H2, D2, and HD by ion cyclotron resonance techniques, J. Chem. Phys. 50, 4787 (1969).
  * Kim and Huntress [1975] J. K. Kim and W. T. Huntress, Ion cyclotron resonance studies on the reaction of H${}_{2}^{+}$ and D${}_{2}^{+}$ ions with various simple molecules and hydrocarbons, J. Chem. Phys. 62, 2820 (1975).
  * Burt  _et al._ [1970] J. A. Burt, J. L. Dunn, M. J. McEwan, M. M. Sutton, A. E. Roche, and H. I. Schiff, Some ion-molecule reactions of H${}_{3}^{+}$ and the proton affinity of H2, J. Chem. Phys. 52, 6062 (1970).
  * Thompson  _et al._ [2022] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  * Bentine  _et al._ [2020] E. Bentine, C. J. Foot, and D. Trypogeorgos, (py)LIon: A package for simulating trapped ion trajectories, Comput. Phys. Commun. 253, 107187 (2020).
  * [50] See Supplemental Material at [URL will be inserted by publisher] for additional details on the molecular dynamics simulations.


# Geometric Visualizations of Single and Entangled Qubits

Li-Heng Henry Chang, Shea Roccaforte, Rose Xu, and Paul Cadden-Zimansky
<EMAIL_ADDRESS>Physics Program, Bard College
Annandale-on-Hudson, NY 12504

###### Abstract

The Bloch Sphere visualization of the possible states of a single qubit has
proved a useful pedagogical and conceptual tool as a one-to-one map between
qubit states and points in a 3-D space. However, understanding many important
concepts of quantum mechanics, such as entanglement, requires developing
intuitions about states with a minimum of two qubits, which map one-to-one to
unvisualizable spaces of 6 dimensions and higher. In this paper we circumvent
this visualization issue by creating maps of subspaces of 1- and 2-qubit
systems that quantitatively and qualitatively encode properties of these
states in their geometries. In particular, a toroidal map of 2-qubit states
illuminates the non-trivial topology of the state space while allowing one to
simultaneously read off, in distances and angles, the level of entanglement in
the 2-qubit state and the mixed state properties of its constituent qubits. By
encoding states and their evolutions with little to no need of mathematical
formalism, these maps may prove particularly useful for understanding
fundamental concepts of quantum mechanics and quantum information at the
introductory level. Interactive versions of the visualizations introduced in
this paper are available at quantum.bard.edu.

##  I Introduction

The conventional visualization model for the states of a single qubit is a
Bloch Sphere, which provides a one-to-one correspondence between points in a
visualizable geometric space and the quantum mechanical state space. However,
many of the interesting and unintuitive properties of quantum mechanics occur
only in multi-qubit systems. One of the shortcomings of the conventional Bloch
Sphere is that it characterizes only individual qubits rather than
correlations between them that dominate the state space of multi-qubit
systems. The increasing dimensionality of parameters needed to describe an
$n$-qubit system grows exponentially with $n$, making visualizations of even
entangled 2-qubit states an intrinsically hard problem. Nevertheless, the
conceptual and pedagogical utility of visualizable maps and the growing
interest in the study and teaching of quantum information concepts has
informed the development of several approaches to visualize $n\geq{2}$ qubit
states, Hilbert spaces, and entanglement [1, 2, 3, 4, 5]. Strategies to
visualize 2-qubit systems include a tetrahedron nested in an octahedron to map
equivalence classes of different states [6], two 1-qubit Bloch spheres that
track the 2-qubit correlations using color-map ellipsoids [7], embedded
steering ellipsoids [8], or embedded axes [9], and separate vectors on three-
sphere maps that characterize the two 1-qubit states and their entanglement
[11, 10]. Past works have established visual models that are useful for
capturing the fullest extent of the 2-qubit state space, albeit using
complicated parameterizations that require intimate knowledge of the quantum
formalism.

Our approach to visualizing single and entangled qubit states takes its cue
from 2- and 3-D Minkowski diagrams, reduced-dimensional cross-sections of the
full 4-D space-time map that preserve many of its properties. Mathematically,
our cross-sections of Hilbert spaces correspond to states with representations
specified by only real numbers, which has the added advantage that one can
choose to introduce the corresponding basic linear algebra formalism for the
mapped states without the need for complex numbers. Geometrically, the real
cross-section of a 1-qubit state is mapped to a circular cross-section of the
familiar Bloch sphere, a “Bloch Circle”, while the real cross-section of all
pure 2-qubit states is mapped to a solid annular toroid (a donut with a donut-
shaped hole [12]). The toroidal geometry is chosen to simultaneously allow the
encoding of the degree of entanglement of the 2-qubit state in a single length
and the states of the underlying 1-qubit mixed states in lengths and angles.
Moreover, the geometry demonstrates the difference in abundance of separable
and entangled states, illuminates the non-trivial topology of the full 2-qubit
state-space, and builds intuition for why maximally entangled 2-qubit states
are so frequently employed in basic quantum information algorithms. Thus, our
toroid model can serve as an intuitive tool for teaching and learning qubit
concepts at an introductory level.

To construct our 2-qubit toroid visualization, we first review some
underappreciated geometrical properties of the 1-qubit Bloch Sphere model by
analyzing a Bloch Circle cross-section of it. We then concatenate two Bloch
Circles to construct a torus model that maps separable 2-qubit pure states.
Finally, the separable torus is extended inward and outward by a finite amount
to create a toroidal volume that maps 2-qubit entangled states. The toroid
model straightforwardly displays both the 2-qubit state of a system and the
corresponding mixed states of each of the constituent qubits, while clearly
delineating the 2-qubit states corresponding to separable, entangled, and
maximally entangled qubits. This model thus helps learners build intuitions
about 2-qubit separable and entangled states, unitary evolutions between them,
the corresponding evolutions of the underlying 1-qubit states, pure and mixed
states, and how the relative “size” of a state space grows with the number of
qubits.

##  II Single Qubits and Bloch Circle Geometry

The Poincaré or Bloch Sphere provides a canonical map with 1-to-1
correspondence between 1-qubit states (up to a global phase uncertainty) and
points within a 2-sphere ball, a “statepoint.” Pure states composed of a
superposition of an orthogonal basis $\ket{0}$ and $\ket{1}$ are represented
by column vectors $\ket{\psi}=a\ket{0}+b\ket{1}\rightarrow\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}\cos\left(\frac{\theta}{2}\right)\\\
\sin\left(\frac{\theta}{2}\right)e^{i\phi}\end{pmatrix}$, with the
parametrization terms generally introduced as a mix of algebraic,
probabilistic, and geometric significance: the $a$ and $b$ probability
amplitudes are coeficients of the linear expansion corresponding to the
respective probabilities $|a|^{2}$ and $|b|^{2}$ of obtaining states $\ket{0}$
and $\ket{1}$ on measurement, while $\theta$ and $\phi$ are the polar and
azimuthal coordinates of the statepoint on the surface of the sphere.

Figure 1:  (a) A Bloch Circle obtained by taking a cross-section of the
conventional Bloch Sphere (upper left corner). The statepoint $S$ on the
perimeter of a circle corresponds to a pure state
$\ket{\psi}=a\ket{0}+b\ket{1}$ represented by $\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right)=\left(\begin{smallmatrix}\cos(\frac{\theta}{2})\\\
\sin(\frac{\theta}{2})\end{smallmatrix}\right)$. By choosing the radius of the
Bloch Circle to be $R=\frac{1}{2}$, the vector components $a$ and $b$
correspond to the lengths of the lines drawn from the statepoint to the basis
state points $\ket{0}\rightarrow\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$ and
$\ket{1}\rightarrow\left(\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right)$,
which form the legs of a right triangle with angle $\frac{\theta}{2}$. A
perpendicular dropped from the state point to the diameter joining the basis
points cuts this diameter into two segments with lengths equal to the
probabilities $a^{2}$ and $b^{2}$ of finding $\ket{0}$ and $\ket{1}$ on
measurement, respectively. The density matrix for the pure state in this basis
is $\rho=\left(\begin{smallmatrix}a^{2}&ab\\\
ab&b^{2}\end{smallmatrix}\right)=\left(\begin{smallmatrix}a^{2}&c\\\
c&b^{2}\end{smallmatrix}\right)$, where each component directly corresponds to
the labeled line segments. (b) The density matrix representation for a mixed
state is $\rho^{\prime}=\left(\begin{smallmatrix}a^{\prime 2}&c^{\prime}\\\
c^{\prime}&b^{\prime 2}\end{smallmatrix}\right)$ with its corresponding
statepoint $S^{\prime}$ inside the Bloch Circle; again the matrix entries are
equal to the labeled line segments. The degree of increasing mixedness of this
state compared to a pure state can be parametrized either by the shortening
length of the green line $r$ (radius) or the increasing length of the line
segment $|s|$ drawn perpendicularly to $r$ from the state point to the Bloch
Circle circumference, with $|s|=0$ a pure state and $|s|=\frac{1}{2}$ a
maximally mixed state. (c) A change of basis representation for any state can
be read off geometrically using the same constructions with a different choice
of basis/measurement axis. In particular, the same state in (b) is here shown
with the corresponding lines drawn with respect to a different basis axis,
creating line segments equal to the entries of the density matrix
representation for this state in the new basis:
$\rho^{\prime\prime}=\left(\begin{smallmatrix}a^{\prime\prime
2}&c^{\prime\prime}\\\ c^{\prime\prime}&b^{\prime\prime
2}\end{smallmatrix}\right)$.

Our initial task is to give all these parameters geometrical significance by
reviewing the properties of the Bloch Sphere (Fig. 1). This task is
accomplished first by taking the radius of the Bloch Sphere to be
$R=\tfrac{1}{2}$ as opposed to the more common choice of an $R=1$ unit
sphere[13]. Second, we focus only on the subspace of states corresponding to
$\phi=0$, i.e. a cross-section of the sphere represented by real values of $a$
and $b$ that we term a Bloch Circle. This dimensional reduction makes the
plane geometric relations between magnitudes and angles easiest to see, and
allows for pure 1-qubit quantum states to be specified by a single angle
$\theta$ ranging from $-\pi$ to $\pi$. With these choices the parametrization
angle $\frac{\theta}{2}$ and lengths corresponding to the magnitudes
$a=\cos(\frac{\theta}{2})$, $b=\sin(\frac{\theta}{2})$, $|a|^{2}$, and
$|b|^{2}$ are easily related to a statepoint located at the perimeter of the
circle and the choice of “measurement” axis corresponding to the
representation (here through the antipodal $\ket{0}$ and $\ket{1}$ states) as
shown in Fig. 1. We note one virtue of this geometrical diagram is that it is
basis independent, so that the corresponding column vector representations and
measurement probabilities for a different choice of basis can be intuited
geometrically rather than algebraically. An interactive visualization of this
static diagram allowing for different choices of state and measurement axis is
found at quantum.bard.edu; proofs of the geometrical properties are found in
Appendix A.

Another useful representation of the pure 1-qubit state is the density matrix
$\ket{\psi}\bra{\psi}\rightarrow\begin{pmatrix}a\
b\end{pmatrix}\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}|a|^{2}&ab^{*}\\\
a^{*}b&|b|^{2}\end{pmatrix}=\begin{pmatrix}|a|^{2}&c^{*}\\\
c&|b|^{2}\end{pmatrix}=\rho$. Again, by looking at the subspace of states on
the Bloch Circle when all entries are real,

$\rho=\begin{pmatrix}{\color[rgb]{0,0,1}a^{2}}&{\color[rgb]{0.5,0,1}{c}}\\\
{\color[rgb]{0.5,0,1}{c}}&{\color[rgb]{1,0,0}b^{2}}\end{pmatrix},$

with each entry corresponding to lengths of line segments shown in Fig. 1(a)
and the off-diagonal matrix element $c$ positive for statepoints on the right-
hand side of the Bloch Circle and negative on the left-hand side.

These geometric properties of statepoints plotted on a Bloch Circle can be
extended to mixed qubit states, such as those derived from partial traces of a
multi-qubit state or as a linear combination of pure qubit density matrices
weighted by epistemic probabilities. The resulting matrices have the form
$\rho^{\prime}=\begin{pmatrix}|a^{\prime}|^{2}&c^{\prime*}\\\
c^{\prime}&|b^{\prime}|^{2}\end{pmatrix}$ with
$|c^{\prime}|\leq|a^{\prime}b^{\prime}|$. There is a one-to-one correspondence
between these states and statepoints in the interior of the Bloch Sphere, with
the $\phi$ argument of $c^{\prime}=|c^{\prime}|e^{i\phi}$ determining the
azimuthal angle of the statepoint. Again we restrict our visualization to the
real subspace ($\phi=0$) of points within the Bloch Sphere corresponding to
density matrices
$\rho=\begin{pmatrix}{\color[rgb]{0,0,1}a^{2}}&{\color[rgb]{0.5,0,1}{c}}\\\
{\color[rgb]{0.5,0,1}{c}}&{\color[rgb]{1,0,0}b^{2}}\end{pmatrix}$, which have
analogous correspondence between line segment lengths, matrix parameters, and
physical significance as the pure-state case (Fig. 1(b)). The degree to which
a state is mixed can be visualized geometrically either by the length of the
radius from the center of the Bloch Circle to the statepoint (which
monotonically increases with increasing purity from 0 to $\frac{1}{2}$) or the
line segment of length $|s|=\sqrt{\left(\frac{1}{2}\right)^{2}-r^{2}}$ drawn
from the statepoint to the Bloch Circle perimeter perpendicular to the radius
(which monotonically increases with increasing mixedness from 0 to
$\frac{1}{2}$). Below we show that $s$ is also a useful parameter in
visualizing 2-qubit states (see also Appendix C). As with pure-state
representations, the geometrical lengths can be used to read off the density
matrix representation of a state for any choice of basis, see Fig. 1(c), and
can be used in a basis-independent fashion to calculate mixed states formed
through linear combinations of other mixed states (Appendix B). With this
review of the Bloch Sphere and Circle geometry, we can now move to a
visualization of 2-qubit states that incorporates features that track the
individual qubit states and their degree of entanglement.

##  III Toroid Model for a 2-qubit system

Figure 2: (a) Two 1-qubit pure states, each plotted on Bloch Circles at angles
$\theta_{1}$ and $\theta_{2}$, are encoded in a torus map of a separable
2-qubit state where the location of a statepoint (pink) uniquely determines
the two angles with respect to the green and black rings. The four white dots
on the torus map the basis states
$\ket{00},\ket{01},\ket{10}\text{and}\ket{11}$. (b) The separable torus
surface in (a) is expanded outward and inward to form a toroidal volume with a
cross-section in the shape of an annulus. The separable torus is embedded in
the center of this toroid with the line from the surface of the toroid to the
statepoint of length $r=\frac{1}{2}$, corresponding to the radial coordinate
of the individual 1-qubit states. Points in the toroid volume not on this
embedded separable torus surface correspond to entangled qubit states (see
Fig. 3).

The simplest 2-qubit systems are the separable states, those with no
entanglement that are composed of the tensor product of two pure 1-qubit
states:

$\ket{\chi}=\ket{\psi_{1}}\otimes\ket{\psi_{2}}=\ket{\psi_{1}\psi_{2}}\rightarrow\begin{pmatrix}a_{1}\\\
b_{1}\end{pmatrix}\otimes\begin{pmatrix}a_{2}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}a_{1}a_{2}\\\ a_{1}b_{2}\\\ b_{1}a_{2}\\\
b_{1}b_{2}\end{pmatrix}=\begin{pmatrix}\alpha\\\ \beta\\\ \gamma\\\
\delta\end{pmatrix}$

As the general 1-qubit states are each independently parametrized by a polar
and azimuthal angle corresponding to their Bloch Sphere statepoint location, a
full accounting of only the separable 2-qubit states would require a 4-D space
in which to plot a 2-qubit statepoint. We again circumvent the challenge of
this visualization by focusing on a subspace of these states formed from
tensor products of 1-qubit states located on the Bloch Circle. This
restriction results in column vector representations with real entries, the
first of which, $\alpha$, is always positive or zero. Since each of these
1-qubit states is parametrized by angles $\theta_{1}$ and $\theta_{2}$ ranging
from $-\pi$ to $\pi$ on their respective Bloch Circles, a natural
visualization of the corresponding 2-qubit state is a statepoint on a surface
that can encode each of these angles, namely a torus. Fig. 2(a) shows such a
statepoint where the angles of the corresponding 1-qubit statepoints can be
read off using the location of a separable 2-qubit statepoint on this
“separable torus.” The statepoint locations of the standard 2-qubit
orthonormal basis set used to expand the state
$\ket{\chi}=\alpha\ket{00}+\beta\ket{10}+\gamma\ket{01}+\delta\ket{11}$ are
shown in the same figure for reference.

To extend the visualization of statepoints to entangled (non-separable) pure
2-qubit states, we first note that the full space of these states can be
parametrized by the four complex coefficients $\alpha,\beta,\gamma,\delta$
constrained by a bounding normalization condition
$|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1$ and subjected to a
choice of global phase that allows $\alpha$ to be taken to be real and non-
negative. The resulting space is a 6-D projective plane, which involves both
an unvisualizably high dimension as well as a non-trivial topology that is
challenging to visualize even in lower dimensions. To circumvent these
challenges, we follow our established approach of looking at only the real-
coefficient subspace of the full state space, which, with the $\alpha\geq 0$
phase choice and normalization condition, is a bounded 3-D region. The
visualization of this subset of states and their corresponding Bloch Circle
1-qubit states is aided by the fact – proved in Appendix C – that though the
angles of the 1-qubit statepoints are independent, their degree of mixedness,
as measured by the length of their radius $r$ or the length of the
corresponding $s$ parameter, are the same for each qubit. Also shown in
Appendix C is that this degree of mixedness correlates with the degree of
entanglement of the 2-qubit state and can be calculated directly from the
2-qubit expansion coefficients $s={\alpha\delta-\beta\gamma}$, which has a
magnitude $|s|=\sqrt{\left(\frac{1}{2}\right)^{2}-r^{2}}$ that varies from 0
for separable states up to $\frac{1}{2}$ for maximally entangled states as
each of the 1-qubit states vary from pure ($r=\frac{1}{2}$) to maximally mixed
($r=0$).

Figure 3: Entangled states. (a) An entangled 2-qubit statepoint plotted on the
2-qubit toroid next to the corresponding 1-qubit states plotted on Bloch
Circles. The pink statepoint, being off the embedded central separable torus,
indicates it is an entangled state with the degree of entanglement
parametrized by the green-line distance between the statepoint and the toroid
surface. This length correlates to the radial length of each of the 1-qubit
states and monotonically decreases with increasing entanglement. As in Fig.
2(a), the angular coordinates of the 1-qubit state can again be read off from
the angular location of the statepoint in the toroid. (b) For each of the
infinitely many maximally entangled 2-qubit states, the state is plotted not
as a point but as a closed-curve torus knot on either the inner or outer
surfaces of the toroid. The corresponding 1-qubit states are always maximally
mixed for these states with a 50/50 probability outcome for all possible
measurements. One useful property of these states that can be visualized is
their role as part of a continuous evolution between 1-qubit states with
discretely differing angles: a torus knot continuously connects states to
multiple regions of the toroid as the 1-qubit statepoints evolve continuously
through the center of their Bloch Circles.

To plot statepoints of these 2-qubit states in a bounded 3-D space, our three
real, bounded parameters are the 1-qubit angles $\theta_{1}$, $\theta_{2}$,
and the $s$ entanglement parameter. Starting from just the separable 2-qubit
states, which have zero entanglement ($s=0$), and the pair of angles
parametrized on the separable torus surface, we can extend this torus shell
inward (for $s<0$) and outward (for $s>0$) by a $\frac{1}{2}$ magnitude to
create a volume to map all the real 2-qubit states over the full range of $s$.
The separable torus can be taken to have a cross-sectional radius of any size
greater than $\frac{1}{2}$, so that the inward extension leaves an empty
torroidal space inside of the torroidal volume rather than resulting in a
filled-in “donut.” The resulting state space is thus an annular toroidal
volume bounded by two torus surfaces (see Fig. 2(b) and Fig. 6 in Appendix E).
Any state point not on the central separable shell where $s=0$ is an entangled
state (see Fig. 2(b)), with the outer ($s=\frac{1}{2}$) and inner
($s=-\frac{1}{2}$) torus surfaces of this volume corresponding to maximally
entangled qubits. As with the separable states, the angles $\theta_{1}$ and
$\theta_{2}$ of the 1-qubit states can be read off directly from the location
of the 2-qubit statepoint in this volume, while the shared radius $r$ of the
1-qubit states is directly visualizable in the distance from the statepoint to
the surface of the toroid. Interactive visualizations of 2-qubit statepoints
plotted in this volume and the corresponding 1-qubit states plotted on Bloch
Circles, from which Fig. 2 and 3 are taken, are found at quantum.bard.edu.

Continuous paths through the 2-qubit toroid correspond to unitary evolutions
of 2-qubit states within the real-coefficient subspace. As detailed in
Appendix 6, each state in this subspace is in a one-to-one correspondence with
a point in the toroid with one notable exception: each $r=0$ maximally
entangled state, located at the bounding surfaces of the toroid, is mapped not
to a point but to a unique torus knot on these surfaces that winds once around
the surface with respect to each angle (Fig. 3(b)). The mapping of these
states to knots rather than points is a manifestation of the non-trivial
projective plane topology of the full state space. Unitary evolutions of
2-qubit systems that pass through these maximally entangled states, such as
Bell states, are common in quantum information algorithms. One of their
virtues that can be seen in the toroid visualization is that, by providing a
path through the $r=0$ states, they allow for the continuous evolution of the
1-qubit states from one set of angles to another without passing through any
of the intervening angles; states evolving in one section of the torus can use
a torus knot “highway” to reach a different section. For these states the
$\theta_{1}$ and $\theta_{2}$ angles cease to have any physical significance
for the 1-qubit states, which are always maximally mixed, though their
constant difference/sum uniquely determines which torus knot they correspond
to on the outer/inner surface of the 2-qubit toroid (see Appendix E).

One intuition about the correlations between 2-qubit pure states and their
corresponding 1-qubit states that can be reinforced by these visualizations is
that there is not a simple one-to-one mapping between them. While the
separable 2-qubit states map to unique 1-qubit states in this subspace
(reflective of a one-to-one mapping in the full space up to a relative phase
between the individual qubits), there is a two-to-one mapping for the general
entangled 2-qubit states (one for each possible sign of the same-magnitude $s$
parameter at the same $\theta_{1}$ and $\theta_{2}$ angles) and infinitely
many maximally entangled states corresponding to the 1-qubit pair where each
qubit is in its maximally mixed state (reflective of the continuum of torus
knots on the outer and inner surfaces of the toroid).

##  IV Conclusion

The increasing need to introduce basic quantum information concepts at
multiple education levels, including teaching students with formal backgrounds
and future prospects not limited to conventional physics training,
necessitates the development of new tools for conveying these concepts that do
not rely on first mastering the linear algebra quantum formalism. A virtue of
the subspace visualization approach presented here is that it can be deployed
as a pedagogical tool in different ways for different audiences. For students
with minimal mathematical training, the points and lines of the Bloch Circle
develop intuitions about qubit states, pure and mixed; choice of measurement
and bases; and probabilities and probability amplitudes; while the 2-qubit
toroid develops intuitions about the difference between separable and
entangled states; the relation of pure 2-qubit states to the 1-qubit states of
their entangled qubits; and the multiple paths of unitary evolutions of these
states. For students with some background in linear algebra, these
visualizations can aid in understanding the actions of algorithms in the
quantum formalism that can be restricted to real-number calculations. Indeed,
in addition to the measurement algorithm, 1-qubit unitary quantum logic gates
such as X, Z, and Hadamard, and 2-qubit gates such as CNOT, Controlled Z, and
SWAP are closed transformations within the real-state subspace whose actions
can be visualized. At the more advanced undergraduate level, the proofs in the
Appendices relating calculations in the quantum formalism to lengths and
angles in the diagrams, can form the basis for multiple exercises where visual
intuition and mastery of the formalism can be developed in tandem.

##  Appendix A Geometric Interpretation of the Bloch Circle

In this appendix we prove the equivalence between geometrical lengths drawn on
a Bloch Circle and the magnitudes of matrix elements and measurement
probabilities for 1-qubit state representations.

Figure 4: (a) A pure 1-qubit state $\ket{\psi}$ that lies on the periphery of
the Bloch Circle with line segments formed as in Fig. 1(a). (b) A statepoint
plotted along the vertical basis axis corresponding to the mixed state
$\rho=\begin{pmatrix}p_{0}&0\\\ 0&p_{1}\end{pmatrix}$. (c) The statepoint for
the mixed state in (b) when the 1-qubit state has been rotated by an angle
$\theta$. Geometrical lengths formed by drawing a perpendicular to the basis
axis correspond to elements of the state’s density matrix
$\rho=\begin{pmatrix}a^{2}&c\\\ c&b^{2}\end{pmatrix}$.

In Fig. 4(a) a point $S$ is plotted on a Bloch Circle with radius
$R=\frac{1}{2}$ to represent a pure state $\ket{\psi}$. A basis of
representation/measurement is chosen by drawing any diameter of the circle
$\overline{AB}$ and designating the point $A$ to be the state
$\ket{0}\rightarrow\begin{pmatrix}1\\\ 0\end{pmatrix}$ and point $B$ to be the
state $\ket{1}\rightarrow\begin{pmatrix}0\\\ 1\end{pmatrix}$. Lines are drawn
between these three points and the center of the Bloch Circle $C$, and a line
from $S$ perpendicular to $\overline{AB}$ is drawn that intersects this
diameter at point $D$. The location of the pure statepoint $S$ relative to the
chosen basis can be parametrized either by the angle $\theta=\angle{ACS}$ or
$\theta/2=\angle{ABS}$. As the latter is one angle of the right triangle
$\triangle ABS$ with unit hypotenuse, we can alternately parametrize the state
$S$ in this basis by specifying the lengths of the legs of this triangle,
$\ket{\psi}\rightarrow\begin{pmatrix}a\\\
b\end{pmatrix}\equiv\begin{pmatrix}\overline{BS}\\\
\overline{AS}\end{pmatrix}=\begin{pmatrix}\cos(\tfrac{\theta}{2})\\\
\sin(\tfrac{\theta}{2})\end{pmatrix}$, which is the conventional column-vector
representation of a pure qubit state in the real-state subspace corresponding
to a cross-section of the Bloch Sphere taken at azimuthal angle $\phi=0$. Note
the convention that $\overline{AS}=b$ is positive for statepoints on the right
half of the circle, negative on the left half, and that it would be multiplied
by $e^{i\phi}$ if other azimuthal angle cross-sections were chosen.

To visualize the density matrix representation of this same state in this
basis, we note that from the similar right triangles $\triangle
ASB\sim\triangle ADS\sim\triangle SDB$ we have the equal ratios
$\frac{\overline{BD}}{\overline{BS}}=\frac{\overline{BS}}{\overline{AB}}\Rightarrow\overline{BD}=a^{2}$,
$\frac{\overline{AD}}{\overline{AS}}=\frac{\overline{AS}}{\overline{AB}}\Rightarrow\overline{AD}=b^{2}$,
and
$\frac{\overline{SD}}{\overline{AS}}=\frac{\overline{BS}}{\overline{AB}}\Rightarrow\overline{SD}=ab$.
These three line segments are thus equal in magnitude to the density matrix
elements: $\ket{\psi}\bra{\psi}\rightarrow\rho=\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}=\begin{pmatrix}\overline{BD}&\overline{SD}\\\
\overline{SD}&\overline{AD} \end{pmatrix}$. Note again the convention that the
off-diagonal element $\overline{SD}=ab$ can be taken to be positive for
statepoints on the right half of the circle and modified by $e^{\pm{i\phi}}$
for other subspace cross-sections. The diagonal matrix elements are always
zero or positive, with the two line segments $\overline{BD}$ and
$\overline{AD}$ of the chosen measurement diameter exactly equal to the
probabilities that the state $S$ will become the state at $A$ or $B$ on
measurement.

The cutting of the basis diameter into pieces equal to probabilities of
measurement outcomes can be extended to plotting mixed statepoints. Consider a
state that results from a qubit having the epistemic probabilities $p_{0}$ and
$p_{1}$ of being in the states $\ket{0}$ or $\ket{1}$ respectively. This state
is represented by the density matrix

$p_{0}\begin{pmatrix}1\\\ 0\end{pmatrix}\begin{pmatrix}1\
0\end{pmatrix}+p_{1}\begin{pmatrix}0\\\ 1\end{pmatrix}\begin{pmatrix}0\
1\end{pmatrix}=\begin{pmatrix}p_{0}&0\\\ 0&p_{1}\end{pmatrix},$

which can be plotted as a point along the measurement diameter that cuts it
into segments equal to these probabilities as shown in Fig. 4(b).

To create a general mixed-state point within the circle, we note that the
probabilities $p_{0}$ and $p_{1}$ determine the radial coordinate of the
statepoint, $r=(p_{0}-p_{1})/2$, and that changing the angular coordinate of
the statepoint can correspond to a rotational transformation on the density
matrix by
$R(\theta)\equiv{\begin{pmatrix}\cos(\tfrac{\theta}{2})&-\sin(\tfrac{\theta}{2})\\\
\sin(\tfrac{\theta}{2})&\cos(\tfrac{\theta}{2})\end{pmatrix}}$. Thus the state

$\rho=R(\theta)\begin{pmatrix}p_{0}&0\\\
0&p_{1}\end{pmatrix}R^{\dagger}(\theta)=\begin{pmatrix}p_{0}\cos^{2}(\tfrac{\theta}{2})+p_{1}\sin^{2}(\tfrac{\theta}{2})&(p_{0}-p_{1})\sin(\tfrac{\theta}{2})\cos(\tfrac{\theta}{2})\\\
(p_{0}-p_{1})\sin(\tfrac{\theta}{2})\cos(\tfrac{\theta}{2})&p_{1}\cos^{2}(\tfrac{\theta}{2})+p_{0}\sin^{2}(\tfrac{\theta}{2})\end{pmatrix}$

corresponds to the statepoint shown in 4(c). Again drawing a perpendicular
from this point to cut the measurement axis $\overline{AB}$ at point $D$, we
have the following geometric relations:

$\displaystyle\overline{DS}$
$\displaystyle=r\sin(\tfrac{\theta}{2}+\tfrac{\theta}{2})=\tfrac{1}{2}(p_{0}-p_{1})\times
2\cos{(\tfrac{\theta}{2})}\sin{(\tfrac{\theta}{2})}=(p_{0}-p_{1})\sin(\tfrac{\theta}{2})\cos(\tfrac{\theta}{2}),$
$\displaystyle\overline{AD}$
$\displaystyle=R-\overline{DC}=\tfrac{1}{2}-r\cos(\tfrac{\theta}{2}+\tfrac{\theta}{2})$
$\displaystyle=\tfrac{1}{2}\left[(p_{0}+p_{1})\left(\cos^{2}{(\tfrac{\theta}{2})}+\sin^{2}{(\tfrac{\theta}{2}})\right)-(p_{0}-p_{1})\left(\cos^{2}{(\tfrac{\theta}{2})}-\sin^{2}{(\tfrac{\theta}{2})}\right)\right]$
$\displaystyle=p_{1}\cos^{2}(\tfrac{\theta}{2})+p_{0}\sin^{2}(\tfrac{\theta}{2}),$
$\displaystyle\overline{BD}$ $\displaystyle=1-\overline{AD}$
$\displaystyle=\left[\left(p_{0}+p_{1}\right)\left(\cos^{2}{(\tfrac{\theta}{2})}+\sin^{2}{(\tfrac{\theta}{2})}\right)-\left(p_{1}\cos^{2}(\tfrac{\theta}{2})+p_{0}\sin^{2}(\tfrac{\theta}{2})\right)\right]$
$\displaystyle=p_{0}\cos^{2}{(\tfrac{\theta}{2})}+p_{1}\sin^{2}{(\tfrac{\theta}{2})}.$

By matching matrix elements with these line segments we have the geometric
interpretation of a Bloch Circle for a general mixed state where the diagonal
line segments again correspond to measurement probabilities for the chosen
basis diameter and the magnitude of the off-diagonal element is equal to the
distance from the statepoint to this diameter:

$\rho=\begin{pmatrix}\overline{BD}&\overline{SD}\\\
\overline{SD}&\overline{AD}\end{pmatrix}.$

##  Appendix B Linear Combinations of Mixed States on a Bloch Circle

Figure 5: Geometric picture of taking linear combination of states using the
epistemic probabilities $p_{1}$ and $p_{2}$. With the states $\rho_{1}$ and
$\rho_{2}$ plotted by the statepoints $S_{1}$ and $S_{2}$, the state
$\rho=p_{1}\rho_{1}+p_{2}\rho_{2}$ is plotted by the statepoint $S$ that cuts
the line segment joining $S_{1}$ and $S_{2}$ in the ratio of the epistemic
probabilities.

An especially useful construction on the Bloch Circle, which also holds for
the Bloch Sphere, is visualizing the resulting state point when one or more
states are combined in a mixed fashion with epistemic probabilities. To begin
with, consider any two 1-qubit states $S_{1}$ and $S_{2}$ with density
matrices $\rho_{1}$ and $\rho_{2}$ plotted as statepoints on the Bloch Circle.
We wish to find the statepoint corresponding to the linear combination
$\rho=p_{1}\rho_{1}+p_{2}\rho_{2}$, where the epistemic probabilities of each
state $p_{1}$ and $p_{2}$ are normalized: $p_{1}+p_{2}=1$. Without loss of
generality we choose to work in the basis corresponding to the Bloch Circle
diameter that is perpendicular to the line $\overline{S_{1}S_{2}}$ drawn
between $S_{1}$ and $S_{2}$ with $S_{2}$ to the right of this diameter and
$S_{1}$ (see Fig. 5 for the two possible cases). From the correspondence
between statepoints and the matrix representations established in Appendix A,
the representations of $\rho_{1}=\begin{pmatrix}a_{1}^{2}&c_{1}\\\
c_{1}&b_{1}^{2}\end{pmatrix}$ and $\rho_{2}=\begin{pmatrix}a_{2}^{2}&c_{2}\\\
c_{2}&b_{2}^{2}\end{pmatrix}$ have the same diagonal elements,
$a_{1}^{2}=a_{2}^{2}=a^{2}$ and $b_{1}^{2}=b_{2}^{2}=b^{2}$, corresponding to
how the diameter is cut by the perpendicular, and different off-diagonal
elements corresponding to the distances from the statepoints to this diameter.
From our choice of basis and the sign conventions of these off-diagonal
elements, the length of $\overline{S_{1}S_{2}}=c_{2}-c_{1}$. Using the
normalization we have

$\rho=p_{1}\rho_{1}+p_{2}\rho_{2}=p_{1}\begin{pmatrix}a^{2}&c_{1}\\\
c_{1}&b^{2}\end{pmatrix}+p_{2}\begin{pmatrix}a^{2}&c_{2}\\\
c_{2}&b^{2}\end{pmatrix}=\begin{pmatrix}a^{2}&p_{1}c_{1}+p_{2}c_{2}\\\
p_{1}c_{1}+p_{2}c_{2}&b^{2}\end{pmatrix}.$

Since this resulting matrix also shares the same diagonal elements, and has
real off-diagonal element, its statepoint $S$ must lie along the
$\overline{S_{1}S_{2}}$ line as well, with its distance from this line
determined by its off-diagonal element $c=p_{1}c_{1}+p_{2}c_{2}$.
Geometrically the distance from $S_{1}$ to this statepoint is

$\displaystyle\overline{S_{1}S}$ $\displaystyle=p_{1}c_{1}+p_{2}c_{2}-c_{1}$
$\displaystyle=(1-p_{2})c_{1}+p_{2}c_{2}-c_{1}$
$\displaystyle=p_{2}(c_{2}-c_{1})$ $\displaystyle=p_{2}\overline{S_{1}S_{2}}.$

Similarly, $\overline{SS_{2}}=p_{1}\overline{S_{1}S_{2}}$, so that
$\frac{\overline{S_{1}S}}{\overline{SS_{2}}}=\frac{p_{2}}{p_{1}}$. Thus the
statepoint $S$ has cut the line between the two constituent points in
proportion to the epistemic probability weights, a generalization of locating
the mixed point formed from the two pure states $\ket{0}$ and $\ket{1}$
discussed in Appendix A. Algorithmically, this composition mimics the location
of the center-of-mass of two masses, with the mass weighting of coordinates
replaced by the epistemic probability weighting. Just as with the mass
algorithm, the coordinate location of a mixed state can be extended to any
number of statepoints in the composition: for a state
$\rho=\sum_{i}p_{i}\rho_{i}$ composed of states located at points
$\vec{r_{i}}$ on a Bloch Circle or Sphere with normalized epistemic
probabilities $p_{i}$, the statepoint of $\rho$ is located at
$\vec{r}=\sum_{i}p_{i}\vec{r_{i}}$.

##  Appendix C Shared Radius of Constituent Qubits of 2-qubit States

A pure 2-qubit state $\ket{\chi}$ in the real-state subspace is represented by
a column vector with real entries:

$\ket{\chi}\rightarrow\begin{pmatrix}\alpha\\\ \beta\\\ \gamma\\\
\delta\end{pmatrix}.$

The normalization condition $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}=1$
bounds the magnitude of the entries, and, with the choice of a global phase,
we always take $\alpha$ to be positive when non-zero, so that $0\leq\alpha\leq
1$, and $-1\leq\beta,\gamma,\delta\leq 1$. A representation of the density
matrix $\rho$ of this state is obtained by taking the outer product of the
column vector representation:

$\ket{\chi}\bra{\chi}\rightarrow\begin{pmatrix}\alpha\\\ \beta\\\ \gamma\\\
\delta\end{pmatrix}\begin{pmatrix}\alpha&\beta&\gamma&\delta\end{pmatrix}=\rho=\begin{pmatrix}\alpha^{2}&\alpha\beta&\alpha\gamma&\alpha\delta\\\
\alpha\beta&\beta^{2}&\beta\gamma&\beta\delta\\\
\alpha\gamma&\gamma\beta&\gamma^{2}&\gamma\delta\\\
\alpha\delta&\beta\delta&\gamma\delta&\delta^{2}\\\ \end{pmatrix}.$

Density matrices for the constituent single qubits are found by taking two
partial traces of $\rho$:

$\displaystyle\rho_{1}=Tr(\rho,1)=\begin{pmatrix}\alpha^{2}+\gamma^{2}&&\alpha\beta+\gamma\delta\\\
\alpha\beta+\gamma\delta&&\beta^{2}+\delta^{2}\end{pmatrix}\equiv\begin{pmatrix}x&&y\\\
y&&z\end{pmatrix}$
$\displaystyle\rho_{2}=Tr(\rho,2)=\begin{pmatrix}\alpha^{2}+\beta^{2}&&\alpha\gamma+\beta\delta\\\
\alpha\gamma+\beta\delta&&\gamma^{2}+\delta^{2}\end{pmatrix}\equiv\begin{pmatrix}w&&u\\\
u&&v\end{pmatrix}.$

Each of these states can be plotted as statepoints on a Bloch Circle, located
at distances $r_{1}$ and $r_{2}$ from the center respectively. From the
geometric properties derived in Appendix A these distances are the hypotenuses
of right triangles satisfying

$\displaystyle r_{1}^{2}$
$\displaystyle=(x-\tfrac{1}{2})^{2}+y^{2}=(\alpha^{2}+\gamma^{2}-\tfrac{1}{2})^{2}+(\alpha\beta+\gamma\delta)^{2},$
$\displaystyle r_{2}^{2}$
$\displaystyle=(w-\tfrac{1}{2})^{2}+u^{2}=(\alpha^{2}+\beta^{2}-\tfrac{1}{2})^{2}+(\alpha\gamma+\beta\delta)^{2}.$

Subtracting these two equations, regarding the difference of the first terms
in the expressions as the difference of squares, and using the normalization
condition, we find

$\displaystyle r_{1}^{2}-r_{2}^{2}$
$\displaystyle=\left[\left(x-\tfrac{1}{2}\right)+\left(w-\tfrac{1}{2}\right)\right]\left[\left(x-\tfrac{1}{2}\right)-\left(w-\tfrac{1}{2}\right)\right]+\left(y-u\right)^{2}$
$\displaystyle=(x+w-1)(x-w)+(y-u)^{2}$
$\displaystyle=\left[\alpha^{2}+(\alpha^{2}+\gamma^{2}+\beta^{2}-1)\right](\gamma^{2}-\beta^{2})+\left[(\alpha\beta)^{2}+(\gamma\delta)^{2}-(\alpha\gamma)^{2}-(\beta\delta)^{2}\right]$
$\displaystyle=(\alpha^{2}-\delta^{2})(\gamma^{2}-\beta^{2})+(\alpha^{2}-\delta^{2})(\beta^{2}-\gamma^{2})=0.$

Since $r_{1}$ and $r_{2}$ are non-negative, it follows that
$r_{1}=r_{2}\equiv{r}$. Thus the statepoints of the constituent single qubits
of a pure 2-qubit state have the same radius $r$ on the Bloch Circle, a result
which also holds for states in the full, pure 2-qubit state space and
corresponding 1-qubit states plotted on a Bloch Sphere.

The shared $r$ is closely related to an alternate shared parameter that is
useful for visualizing 2-qubit states: $s\equiv\alpha\delta-\beta\gamma$. The
parameter $s$ is proportional to the quantum concurrence of the qubit pair and
is a direct measure of the degree of entanglement, varying from $s=0$ for
separable states to $|s|=\frac{1}{2}$ for maximally entangled states [14, 15].
This property may be seen in its relation to $r$, which can be calculated from
the above equations and the normalization condition:

$\displaystyle s^{2}$
$\displaystyle=\left(\alpha\delta-\beta\gamma\right)^{2}$
$\displaystyle=\tfrac{1}{4}-\tfrac{1}{4}+\left(\alpha^{2}\gamma^{2}+\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}+\beta^{2}\delta^{2}\right)-\left(\alpha\gamma+\beta\delta\right)^{2}$
$\displaystyle=\tfrac{1}{4}-\left[\tfrac{1}{4}\left(\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)^{2}-4\left(\alpha^{2}\gamma^{2}+\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}+\beta^{2}\delta^{2}\right)\right)+\left(\alpha\gamma+\beta\delta\right)^{2}\right]$
$\displaystyle=\left(\tfrac{1}{2}\right)^{2}-\left[\left(\tfrac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right)\right)^{2}+\left(\alpha\gamma+\beta\delta\right)^{2}\right]$
$\displaystyle=\left(\tfrac{1}{2}\right)^{2}-\left[\left(\alpha^{2}+\beta^{2}-\tfrac{1}{2}\right)^{2}+\left(\alpha\gamma+\beta\delta\right)^{2}\right]$
$\displaystyle=\left(\tfrac{1}{2}\right)^{2}-r^{2},$ $\displaystyle\Rightarrow
s=\pm\sqrt{\left(\tfrac{1}{2}\right)^{2}-r^{2}}.$

Visually, this result shows that $|s|$ and $r$ form the legs of a right
triangle with $\frac{1}{2}$ being the hypotenuse. From the geometry of the
Bloch Circle (see Appendix A), this means that the magnitude of $s$ is the
length of the line segment drawn perpendicular to $r$ from the statepoint to
the perimeter of the Bloch Circle (see Fig. 1(c)). The sign associated with
$s$ allows the encoding of an additional bit of information useful in mapping
the 2-qubit states.

##  Appendix D Maximally Entangled States

The results of Appendix C allow us to write explicit column-vector
representations of all maximally entangled 2-qubit states. The
$|s|=\frac{1}{2}$ or $r=0$ condition entails that the 1-qubit density matrices
have the form $\rho_{1}=\rho_{2}=\begin{pmatrix}\frac{1}{2}&&0\\\
0&&\frac{1}{2}\end{pmatrix}$ and are plotted at the center of the Bloch
Circles. From the diagonal elements and the partial traces of Appendix C, the
column vector entries of the maximally entangled 2-qubit states must satisfy
$\alpha^{2}+\gamma^{2}=\beta^{2}+\delta^{2}=\alpha^{2}+\beta^{2}=\gamma^{2}+\delta^{2}=\frac{1}{2}$,
so that each coefficient has a maximum magnitude of $\frac{1}{\sqrt{2}}$.
Again using our choice of global phase to always take $\alpha$ to be positive
when non-zero, we can parametrize all its possible values for a maximally
entangled state using a single angle $\xi$ ranging from $-\pi$ to $\pi$,
$\alpha=\frac{1}{\sqrt{2}}\cos{\frac{\xi}{2}}$, and use the diagonal element
equalities to write the other possible coefficient values as
$\beta=\pm\frac{1}{\sqrt{2}}\sin{\frac{\xi}{2}}$,
$\gamma=\pm\frac{1}{\sqrt{2}}\sin{\frac{\xi}{2}}$, and
$\delta=\pm\frac{1}{\sqrt{2}}\cos{\frac{\xi}{2}}$. The relative signs of these
coefficients are fixed by the off-diagonal equalities of the 1-qubit density
matrices, $\alpha\beta+\gamma\delta=\alpha\gamma+\beta\delta=0$, which
dictates the two possible sets of maximally entangled states

$\ket{\chi_{+}}\rightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\frac{\xi}{2})\\\
-\sin(\frac{\xi}{2})\\\ \sin(\frac{\xi}{2})\\\
\cos(\frac{\xi}{2})\end{pmatrix}\;\textrm{and}\;\ket{\chi_{-}}\rightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\frac{\xi}{2})\\\
\sin(\frac{\xi}{2})\\\ \sin(\frac{\xi}{2})\\\
-\cos(\frac{\xi}{2})\end{pmatrix}.$

The subscripts labeling these sets are chosen to reflect the fact that
$s=+\frac{1}{2}$ for the first set and $s=-\frac{1}{2}$ for the second. All
possible maximally entangled 2-qubit states in the real-state subspace are
thus specified up to a global phase by this choice of sign and a single angle
over a $2\pi$ range.

##  Appendix E Mapping 2-Qubit States

In this section we establish formulas for the mappings between the 1-qubit
geometric parameters $s,\theta_{1},\theta_{2}$; the 2-qubit pure-state column
vector entries $\alpha,\beta,\gamma,\delta$; and the statepoints on the two
1-qubit Bloch circles and the 2-qubit annular torriod.

Figure 6: A gradient shows the possible $s$ values of pure 2-qubit quantum
states as they range from $-\frac{1}{2}$ to $\frac{1}{2}$. (a) As the shared
radius of either 1-qubit statepoint $r=|s|$ evolves from $\frac{1}{2}$ to $0$
and back over this range, a statepoint corresponding to states with $\theta=0$
traces out a vertical radius twice over this range as it moves from the center
of the Bloch Circle to its edge and back again. Statepoints corresponding to
other angles can be graphically found by rotating this radius by the angle
$\theta$ for each qubit. (b) To create a one-to-one correspondence between the
2-qubit geometric parameters $s,\theta_{1},\theta_{2}$ and statepoints, the
$s$ values are mapped one-to-one to a vertical line. With fixed $\theta_{1}$,
all possible $\theta_{2}$ statepoints are created by rotating this line about
a collinear center point below it by $\theta_{2}$ to form an annulus. All
possible $\theta_{1}$ statepoints are created by rotating this annulus about a
vertical axis outside of it by $\theta_{1}$ to form a toroid.

Starting with the mapping between the geometric parameters and statepoints,
consider the plotting of states for all possible $s$ values for the case
$\theta_{1}=\theta_{2}=0$. As shown in Appendix C and Fig. 6(a), while $s$
ranges from $-\frac{1}{2}$ to $\frac{1}{2}$ either of the 1-qubit statepoints
will move from the center of the Bloch Circle to the edge and back again along
the vertical radius corresponding to $\theta=0$. Each statepoint along this
radius thus corresponds to two values of $s$ with the exception of the pure
state $s=0$ statepoint at the edge. 1-qubit statepoints for other angles from
$-\pi$ to $\pi$ for each qubit are found by rotating this vertical radius to
sweep out the full circle. The general two-to-one mapping of
$s,\theta_{1},\theta_{2}$ parameters and statepoints holds with the exception
of the pure-state perimeter, where the mapping is one-to-one, and the
maximally mixed point at the center, where there infinitely many choices of
$\theta_{1},\theta_{2}$ that map to this statepoint for the single qubits.

To make the geometric mapping everywhere one-to-one, we first map the $s$
values when $\theta_{1}=\theta_{2}=0$ to statepoints on a vertical line in a
one-to-one fashion as shown in Fig. 6(b). Other values of $\theta_{2}$ are
plotted by rotating this line about a central point below it to form an
annulus, and other values of $\theta_{1}$ are plotted by rotating this annulus
azimuthally about a vertical axis outside of it into the 3rd dimension to form
the annular torroid shown in Fig. 2.

To correlate the 2-qubit column-vector entries to this mapping, consider the
following states $\ket{\chi_{0}}$ that map one-to-one to all possible values
of $s$:

$\ket{\chi_{0}}\rightarrow\begin{pmatrix}\sqrt{\frac{1}{2}+\sqrt{\left(\frac{1}{2}\right)^{2}-s^{2}}}\\\
0\\\ 0\\\
\text{sgn}(s)\sqrt{\frac{1}{2}-\sqrt{\left(\frac{1}{2}\right)^{2}-s^{2}}}\end{pmatrix}=\begin{pmatrix}\sqrt{\frac{1}{2}+r}\\\
0\\\ 0\\\ \\\
\text{sgn}(s)\sqrt{\frac{1}{2}-r}\end{pmatrix}\equiv\begin{pmatrix}\alpha_{0}\\\
0\\\ 0\\\ \delta_{0}\end{pmatrix}.$

The normalized coefficients $\alpha_{0}\geq\delta_{0}$ vary continuously over
the full range of $s$ producing states in the chosen real-state subspace. From
the partial traces of Appendix C, the density matrices of the single qubits
corresponding to $\ket{\chi_{0}}$ are

$\rho_{1}=\rho_{2}=\begin{pmatrix}{\alpha_{0}}^{2}&&0\\\
0&&{\delta_{0}}^{2}\end{pmatrix}.$

With off-diagonal elements equal to $0$ and
${\alpha_{0}}^{2}\geq{\delta_{0}}^{2}$, these states correspond to the
continuum of statepoints lying along the $\theta_{1}=\theta_{2}=0$ vertical
lines of Fig. 6 for all possible $s$ values. To find 2-qubit column vectors
for all other angles, we again sweep out this line by rotating each of the
1-qubit states. Formally, this is accomplished in the 2-qubit space using the
tensor product of two of the 1-qubit rotation matrices $R(\theta)$ of Appendix
A to create a unitary rotation operator
$R_{1,2}\equiv{R(\theta_{1})\otimes{R(\theta_{2})}}$ in the 2-qubit space and
use this on $\ket{\chi_{0}}$:

$\ket{\chi}\rightarrow R_{1,2}\begin{pmatrix}\alpha_{0}\\\ 0\\\ 0\\\
\delta_{0}\end{pmatrix}=\begin{pmatrix}\alpha_{0}\cos(\theta_{1}/2)\cos(\theta_{2}/2)+\delta_{0}\sin(\theta_{1}/2)\sin(\theta_{2}/2)\\\
\alpha_{0}\cos(\theta_{1}/2)\sin(\theta_{2}/2)-\delta_{0}\sin(\theta_{1}/2)\cos(\theta_{2}/2)\\\
\alpha_{0}\sin(\theta_{1}/2)\cos(\theta_{2}/2)-\delta_{0}\cos(\theta_{1}/2)\sin(\theta_{2}/2)\\\
\alpha_{0}\sin(\theta_{1}/2)\sin(\theta_{2}/2)+\delta_{0}\cos(\theta_{1}/2)\cos(\theta_{2}/2)\end{pmatrix}.\\\
$

This parametrization of $\ket{\chi}$ serves as a functional map from any
choice of $s,\theta_{1},\theta_{2}$ or their corresponding statepoints to the
chosen real subspace of 2-qubit states. However, this map is not bijective, as
can be seen by examining the maximally mixed $s=\pm\frac{1}{2}$ states that
map to the outer and inner surfaces of the torroid. For the outer surface,
$s=\frac{1}{2},\alpha_{0}=\delta_{0}=\frac{1}{\sqrt{2}}$, and, with aid of
trigonometric identities, the above mapping simplifies to

$\displaystyle\ket{\chi_{+}}\rightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\frac{\theta_{1}-\theta_{2}}{2})\\\
-\sin(\frac{\theta_{1}-\theta_{2}}{2})\\\
\sin(\frac{\theta_{1}-\theta_{2}}{2})\\\
\cos(\frac{\theta_{1}-\theta_{2}}{2})\end{pmatrix}.$

For the inner surface
$s=-\frac{1}{2},\alpha_{0}=-\delta_{0}=\frac{1}{\sqrt{2}}$, and

$\ket{\chi_{-}}\rightarrow\frac{1}{\sqrt{2}}\begin{pmatrix}\cos(\frac{\theta_{1}+\theta_{2}}{2})\\\
\sin(\frac{\theta_{1}+\theta_{2}}{2})\\\
\sin(\frac{\theta_{1}+\theta_{2}}{2})\\\
-\cos(\frac{\theta_{1}+\theta_{2}}{2})\end{pmatrix}.$

Comparing these states with the full set of maximally mixed states derived in
Appendix D, we see that the sign and angle needed to uniquely determine these
states are given by $\text{\text{sgn} }(s=\pm{\frac{1}{2}})$ and
$\xi=\theta_{1}\mp\theta_{2}$. With the angles restricted to a $2\pi$ range
there will be infinitely many combination of $\theta_{1}$ and $\theta_{2}$
that map to the same maximally mixed state. Geometrically the locus of all
statepoints on either the outer or inner surface of the torroid
($\ket{\chi_{+}}$ or $\ket{\chi_{-}}$ states) corresponding at a single state
with fixed $\xi=\theta_{1}\mp\theta_{2}$ value is a torus knot – a closed loop
as shown in Fig. 3(b) that winds once around each torus angle. The knots on
the outer and inner surfaces wind with opposite helicity dependent on whether
$\xi$ is determined by the difference or sum of $\theta_{1}$ and $\theta_{2}$.

Though this analysis shows there is a many-to-one mapping of the
$s,\theta_{1},\theta_{2}$ geometric parameters to unique maximally mixed
states, there is a one-to-one mapping from these parameters to all other
2-qubit states in our real subspace as can be shown by establishing a function
that maps $\alpha,\beta,\gamma,\delta$ to the geometric parameters.

Applying the geometric properties of the Bloch sphere derived in Appendix A to
the density matrices of the constituent qubits, we can find the geometric
parameters $s,\theta_{1},\theta_{2}$ in terms of the $u,w,x,y$ 1-qubit density
matrix elements, which are in turn functions of the $\ket{\chi}$ vector
elements (see Appendix C):

$\displaystyle s$ $\displaystyle=\alpha\delta-\beta\gamma,$ $\displaystyle r$
$\displaystyle=\sqrt{\left(\tfrac{1}{2}\right)^{2}-s^{2}}=\sqrt{\left(x-\tfrac{1}{2}\right)^{2}+y^{2}}=\sqrt{\left(w-\tfrac{1}{2}\right)^{2}+u^{2}},$
$\displaystyle\theta_{1}$
$\displaystyle=2\tan^{-1}{\frac{y}{|\frac{1}{2}-x|}}=2\cos^{-1}{\frac{y}{r}},$
$\displaystyle\theta_{2}$
$\displaystyle=2\tan^{-1}{\frac{u}{|\frac{1}{2}-w|}}=2\cos^{-1}{\frac{u}{r}}.$

With the exception of the singular, maximally mixed, $r=0$ cases, these
functions complete the one-to-one map between column-vector entries, geometric
parameters, and toroid statepoints. For the maximally mixed cases, as shown
above, there is a one-to-one map between column-vector entries and torus
knots. The annular toroid thus has a unique point or knot for each unique pure
2-qubit state in the real subspace.

###### Acknowledgements.

The authors wish to thank Hal Haggard and Ian Durham for their helpful
comments on the manuscript. We gratefully acknowledge the Bard Summer Research
Institute and the Bard Physics Program for supporting the research work.

## References

  * [1] Dagmar Bruß, “Characterizing entanglement”, J. Math. Phys. 43, 4237 (2002).
  * [2] R. A. Bertlmann H. Narnhofer, and W. Thirring, “Geometric picture of entanglement and Bell inequalities,” Phys. Rev. A 66, 032319 (2002).
  * [3] Frank B. Verstraete, Frank and Jeroen Dehaene, and Bart De Moor, “On the geometry of entangled states,” J. Mod. Opt. 49, 1277 (2010).
  * [4] Ingemar Bengtsson and Karol Zyczkowski Geometry of Quantum States: An Introduction to Quantum Entanglement, (Cambridge University Press, 2008).
  * [5] Jon Magne Leinaas, Jan Myrheim, and Eirik Ovrum, “Geometrical aspects of entanglement,” Phys. Rev. A 74, 012313 (2006).
  * [6] J. E. Avron, G. Bisker, and O. Kenneth , “Visualizing two qubits,” J. Math. Phys. 48, 102107 (2007).
  * [7] Joseph B. Altepeter, Evan R. Jeffrey, Milja Medic, and Prem Kumar, “Multiple-Qubit Quantum State Visualization,” in The Conference on Lasers and Electro‐Optics (CLEO)/The International Quantum Electronics Conference (IQEC) (Optical Society of America, Washington, DC, 2009), 10842880.
  * [8] Sania Jevtic, Matthew Pusey, David Jennings, and Terry Rudolph, “Quantum Steering Ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
  * [9] Omar Gamel, “Entangled Bloch spheres: Bloch matrix and two-qubit state space,” Phys. Rev. A, 93 (6), 062320 (2016).
  * [10] Chu-Ryang Wie, “Two-Qubit Bloch Sphere,” Physics, 2 (3), 383-396 (2020).
  * [11] R. Mosseri, “Two-Qubit and Three-Qubit Geometry and Hopf Fibrations,” in Topology in Condensed Matter, edited by M. I. Monastyrsky (Springer, Berlin, Heidelberg, 2006), p. 187.
  * [12] Rian Johnson, “Knives Out” (T-Street Productions, 2019).
  * [13] Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition (Cambridge University Press, New York, NY, 2011).
  * [14] Sam A. Hill and William Wootters, “Entanglement of a Pair of Quantum Bits,” Phys. Rev. Lett 78 5022 (1997).
  * [15] Roland Hildebrand, “Concurrence revisited,” J. Math. Phys. 48 102108 (2007).