Entanglement in a qubit-qubit-tardigrade system

Quantum and biological systems are seldom discussed together as they seemingly demand opposing conditions. Life is complex, ‘hot and wet’ whereas quantum objects are small, cold and well controlled. Here, we overcome this barrier with a tardigrade—a microscopic multicellular organism known to tolerate extreme physicochemical conditions via a latent state of life known as cryptobiosis. We observe coupling between the animal in cryptobiosis and a superconducting quantum bit and prepare a highly entangled state between this combined system and another qubit. The tomographic data shows entanglement in the qubit-qubit-tardigrade system, with the tardigrade modelled as an ensemble of harmonic oscillators or collection of electric dipoles. The animal is then observed to return to its active form after 420 h at sub 10 mK temperatures and pressures below 6×10−6  mbar, setting a new record for the conditions that a complex form of life can survive.

. Sketch of the experiment. (a) A tardigrade in the tun state is positioned between the shunt capacitor plates, slightly offset from the Josephson junction, of the transmon qubit B. Qubit A is on the underside and is capacitively coupled to qubit B. The full chip is placed within a 3D copper cavity that is mounted inside a dilution refrigerator and connected to standard microwave electronics for probing. (b) Circuit diagram of the two qubits and the tardigrade tun. (c) Magnification of the revived tardigrade on transmon qubit. The tardigrade in the tun state is placed in the same position while still attached to a small piece of filter paper during the experiment.
In present experiments, we use specimens of a Danish population of Ramazzottius varieornatus Bertolani and Kinchin, 1993 (Eutardigrada, Ramazzottiidae). The species belongs to phylum Tardigrada comprising of microscopic invertebrate animals with an adult length of 50-1200 µm [17]. Importantly, many tardigrades show extraordinary survival capabilities [18] and selected species have previously been exposed to extremely low temperatures of 50 mK [19] and low Earth orbit pressures of 10 −19 mbar [20]. Their survival in these conditions is possible thanks to a latent state of life known as cryptobiosis [2,18]. Cryptobiosis can be induced in various extreme physicochemical ways, including freezing and desiccation. Specifically, during desiccation, tardigrades reduce volume and contract into an ametabolic state, known as a 'tun' . Revival is achieved by reintroducing the tardigrade into liquid water at atmospheric pressure. In the current experiments, we used desiccated R. varieornatus tuns with a length of 100-150 µm. Active adult specimens have a length of 200-450 µm. The revival process typically takes several minutes.
We place a tardigrade tun on a clean superconducting transmon qubit and observe coupling between the qubit and the tardigrade tun via a shift in the resonance frequency of the new qubit-tardigrade system. This joint qubit-tardigrade system is then entangled with a second superconducting qubit. We reconstruct the density matrix of this coupled system experimentally via quantum state tomography. Finally, the tardigrade is removed from the superconducting qubit and reintroduced to atmospheric pressure and room temperature. We observe the resumption of its active metabolic state in water. Notably, the tardigrade tun remained at a base temperature below 10 mK and pressures below 6 × 10 −6 mbar for 420 h (see Appendix B for more details). This is to-date the most extreme exposure to low temperatures and pressures that a tardigrade has been recorded to survive, clearly demonstrating that the state of cryptobiosis ultimately involves a suspension of all metabolic processes [21,22] given that all chemical reactions would be prohibited with all its constituent molecules cooled to their ground states. Figure 1 shows an illustration of the experiment. Two transmon qubits ('qubit A' and 'qubit B') are mounted back to back inside an oxygen-free high thermal-conductivity copper 3D microwave cavity with a TE 101 frequency of 4.521 GHz. The two transmons are positioned about 1 mm apart (twice the thickness of the silicon chips). The two qubits are directly coupled via the shunt capacitor plates on each qubit, with E A J = 5.23 GHz, E A C = 0.262 GHz for qubit A and E B J = 5.53 GHz, E B C = 0.286 GHz for qubit B. The coupling strength between the qubits is 66 MHz. We characterize the system using control and readout schemes typically used in superconducting qubits described in [23]. The single-qubit gate times were 50 to 120 ns, with the experimental pulse sequence consisting of a pulse train, synthesized using an arbitrary waveform generator (AWG5204 by Tektronix), followed by a 5 µs readout pulse and a 20 µs rest period for the qubit and cavity reset. The frequency of each qubit is tunable, but is always biased close to the maximum in our experiments. These correspond to f A = 3.048 GHz for qubit A and f B = 3.271 GHz for qubit B without tardigrade tun.
For the subsequent cooldown, a single cryptobiotic tardigrade on a 150 µm square piece of cellulose filter paper is placed on the substrate of qubit B, in between the shunt capacitor plates of the transmon qubit, with the tardigrade side closer to the substrate. The filter paper is sufficiently displaced from the transmon plane and interacts very weakly with the transmon electric fields, verified with numerical simulations, see figure 2(a). The maximum frequency of qubit B is shifted down by 8 MHz to f ′ B = 3.263 GHz, as shown in figure 2(b), while the maximum frequency of qubit A did not change. The shift can be attributed to the aging of the Josephson junctions and the addition of the tardigrade itself. The addition of the tardigrade results in a change of the effective dielectric medium and increases the transmon shunt capacitance of qubit B, consequently shifting its frequency. We simulate the electric fields and capacitance shifts using ANSYS Maxwell (classical electrodynamics software) where the tardigrade is modelled as a cube of length 100 µm positioned as in the experiment and with dielectric constant varied between 1.5 to 6, with the typical values for proteins thought to be around 4 to 30 [24] (the filter paper is also taken into account and verified to have small contribution). Figure 2(b) shows that the measured frequency shift corresponds to tardigrade dielectric constant of ε r ≈ 4.4 ± 1.5 (error due to tardigrade placement and size estimation). Note that this is in the lower range of the dielectric constants for proteins indicating that the contribution from the aging effect is small, in agreement with measurements on qubit A.
This macroscopic behaviour can also be understood with a microscopic model where the electric field from the shunt capacitors of the transmon qubit induces electric dipoles within small volumes in the tardigrade. The (dipole) coupling mechanism is represented by treating the electric dipoles in the tardigrade as effective harmonic oscillators interacting with the electric field of the qubit, which changes depending on whether the qubit is in the ground or excited state. This is a modification of the two-level systems used to model defects in dielectric structures [25]. We exclude all other coupling mechanisms with the qubit, justified by recalling that all biochemical processes are frozen at the realised temperatures. Due to the coupling, the combined system has different energy gap as compared to just the bare qubit. We model the charges inside the tardigrade as a collection of N harmonic oscillators (with possibly different frequencies) coupled to the qubit. Due to the small observed energy shift it is reasonable to assume that the coupling is weak and could be treated as a perturbation. We therefore split the Hamiltonian and write the following base term where σ z is the Pauli-z operator for the qubit with lower (higher) energy state denoted as |0 ⟩ (|1 ⟩) and a j (a † j ) is the lowering (raising) operator for the jth oscillator. Note that vanishing energy of the qubit has been set in the middle between its energy levels and the ground state of each oscillator is assigned energy zero. The perturbation term is given by the coupling, where again due to weak interaction we ignore so-called counter-rotating terms capable of simultaneous excitation of the qubit and an oscillator. Therefore, where for simplicity the same coupling g is used for each oscillator and we write σ + = |1 ⟩⟨0| and σ − = |0 ⟩⟨1|. One verifies that the ground state of this system is given by |g⟩ = |0 . . . 0 ⟩ for any coupling g, i.e. the qubit and all oscillators are in their respective ground states. The first excited state and its energy can be obtained via the second-order perturbation theory and show the following frequency gap to the ground state: Accordingly, if the sum is negative, the frequency difference is lowered in the presence of interaction. This happens when charges in the tardigrade oscillate faster than the qubit frequency. We shall also need the explicit form of the first excited state which is found to read: where the first subsystem denotes the qubit and the remaining subsystems describe the oscillators. The state |ψ 1 ⟩ is the superposition of a single excitation in each oscillator, i.e.
Note that this model reduces to the well-known Tavis-Cummings model in the case of identical oscillators [26]. Accordingly, the state |e⟩ is entangled for all non-zero coupling strengths g, although the weak coupling means that this entanglement is weak. We have verified experimentally that the lifetime of the first excited state is T 1 = 3 µs, half of the lifetime of the excited state of the bare qubit. This indicates that state |e⟩ is almost as isolated from the environment, i.e. not further correlated with other systems, as the bare qubit. Most importantly, this relaxation time is still much longer than the quantum gate and measurement times.
Since it is technologically demanding to address the tardigrade independently of the qubit, entanglement present in state |e⟩ cannot be verified directly. This opens up possibility of alternative explanations of observed frequency shift. One such alternative is a semi-classical model similar to the simulations of figure 2. Within this model one asserts that the qubit in state |0 ⟩ produces no electric field and hence induces no electric dipoles within the tardigrade treated as a dielectric. The combined state of qubit B and tardigrade is therefore represented as |g⟩ = |0 ⟩ |/ d ⟩, where |/ d ⟩ describes lack of dipoles (note that the use of quantum notation for the tardigrade degree of freedom is necessitated by unambiguously quantum nature of the qubit). Similarly, the qubit in state |1 ⟩ gives rise to electric field which induces dipoles in the tardigrade tun, i.e. |e⟩ = |1 ⟩ |d⟩. These dipoles modify the capacitance and hence the frequency. Note that within this model neither |g⟩ nor |e⟩ is entangled. This motivates the second part of our experiment.
In the second part, we couple the qubit B-tardigrade system with qubit A using a pulse sequence shown in figure 3. The controlled-NOT gate uses a pulse shape given by speeding up waveforms by inducing phases to harmful transitions protocol [27,28]. This pulse sequence ideally produces the entangled state , where we have used subscripts to demarcate the subsystems. To verify the entangled state, we performed quantum state tomography in the four-dimensional subspace of the whole combined tripartite system. We applied 16 different combinations of one-qubit gates on qubit A and dressed states of the joint qubit B-tardigrade system. We then jointly readout the state of both qubits using the cavity [29]. Maximum likelihood estimation [30] was then employed to prevent the resulting density matrix from having nonphysical properties. Comparing the experimental density matrix with the expected density matrix, we find a state fidelity of F = 82 % [31].
Let us now describe entanglement with the tardigrade tun according to the microscopic model and a family of semi-classical models. We consider the microscopic model more fundamental and thus first expand the dressed states |g BT ⟩ and |e BT ⟩ back into the larger qubit-tardigrade subspace using equation (4). Since the states of the oscillators involved in equation (4) are orthogonal, the tardigrade is effectively modelled by a qubit. The density matrix of this three-qubit system (qubit A-qubit B-tardigrade) can then be reconstructed from the tomographic data, see Appendix C. Using tangles based on negativity as entanglement quantifiers [32][33][34], we observe that entanglement in various bipartitions (N X:Y ) of the tripartite system as well as genuine tripartite entanglement (π-tangle) are monotonically increasing functions of θ, the coupling strength between qubit B and tardigrade tun, see figure 3(c). In particular, for any finite interaction strength the tardigrade is entangled with both qubits. The same conclusion is reached within the semi-classical model presented above and in fact in any model where the coupling to the superconducting qubit is through its electric field, see Appendix F. Note that this is achieved without directly measuring the tardigrade.
The results reported here have been obtained in a single experimental sequence, i.e. using one tardigrade in its tun state. Two more experiments have been conducted and we wish to point out that it is very important for the revival of the animal to change the external temperature and pressure gently. See Appendix B for their profiles that led to the successful revival.
We conclude by revisiting Bohr's assertion on the incompatibility of conducting quantum experiments with living organisms. Our present investigation is perhaps the closest realisation combining biological matter and quantum matter available with present-day technology. While one might expect similar physical results from inanimate object with similar composition to the tardigrade, we emphasise that entanglement is observed with charges within a volume of organism that retains its biological functionality post experiment. At the same time, the tardigrade survived the most extreme and prolonged conditions it has ever been exposed to, demonstrating that cryptobiosis (latent life) is truly ametabolic. We hope this will stimulate further experiments with the states of the animal being more and more macroscopically distinguishable. Our work provides a first step in the exciting direction of creating hybrid systems consisting of living matter and quantum bits.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.

Acknowledgment
This work was supported by the National Research Foundation and the Ministry of Education of Singapore, and the Polish National Agency for Academic Exchange NAWA Project No. PPN/PPO/2018/1/00007/U/00001.
The authors thank Dr Farshad Foroughi (CNRS Grenoble) for the assistance in the fabrication of the transmon qubits used in the experiment.

Ethical Statements
No ethical approval was required for this study, as Nanyang Technological University's Institutional Animal Care and Use Committee's guidelines (https://www.ntu.edu.sg/research/research-integrity-office/ institutional-animal-care-and-use-committee/guidelines) state that projects involving invertebrates do not require approval.

Appendix A. Tardigrades in the tun state
In the tun state, the tardigrade legs are withdrawn and the body is longitudinally contracted to appear more cuboid in structure, see figure 4. In our experiments, adult specimens of Ramazzottius varieornatus Bertolani and Kinchin, 1993 (Eutardigrada, Ramazzottiidae) were collected in February 2018 from a roof gutter in Nivå, Denmark (N 55 • 36.53", E 12 • 30'00.90"). The roof gutter sample, containing the tardigrades, was frozen under wet conditions and stored at −20 • C until October 2020, when the sample was thawed, diluted in ultrapure water (Millipore Milli-Q® Reference, Merck, Darmstadt, Germany) and examined for active adult tardigrades with the help of a stereomicroscope. Single adult tardigrades were removed from the sample with the aid of hand pulled glass Pasteur pipettes and desiccated on filter paper substrates for use in the experiments. Subsequent attempts to mechanically remove the tardigrades from the filter paper irreversibly damage their cuticle, and hamper their ability to revive in water.

Appendix B. Temperature and pressure profiles
We have attempted three experimental sequences, each one with a different tardigrade. A successful revival was only achieved in the third experimental trial. In this trial a slower venting process was undertaken to implement gentler pressure gradients on the warmup (room pressure was reintroduced in about 1 h, while typically it takes only about 15 min). There are no pressure sensors inside the mixing chamber where the qubit and tardigrade are located. However, the sensors in the outer chambers set an upper limit on the pressures of the mixing chamber. The pressure and temperature profiles in the refrigerator for the experimental run that yielded a successful revival, from 22 January 2021 to 25 January 2021 (cooldown) and 11 February 2021 to 14 February 2021 (warmup), are shown in figure 5. The cooldown took about 2 d to reach sub 10 mK temperatures while the pressure drop to around 6 × 10 −3 mbar took about half an hour. The final pressure is likely to be much lower than 10 −6 mbar [35]. The warmup took about 3 d for temperature and an hour for pressure to return to room values.
After the cooldown, the pressure and temperature stabilises to around 6 × 10 −6 mbar and 10 mK, with some fluctuations. A typical 12 h period is shown in figure 6.

Appendix C. Reconstruction of the tripartite density matrix within the microscopic model
The experimentally determined 4 × 4 matrix of the tripartite system composed of qubit A, qubit B and tardigrade tun is not necessarily physical, i.e. positive semi-definite with unit trace, due to experimental errors and noise. Maximum likelihood method was used to reconstruct a physical density matrix ρ closest to the experimental data, by following the procedure described in [30]. This density matrix is presented in the main text. Note that it is written in the basis {|0g⟩ , |0e⟩ , |1g⟩ , |1e⟩}, where the combined states |g⟩ and |e⟩ of the qubit B-tardigrade system are used.
In order to isolate the degrees of freedom related to the tardigrade tun we rewrite the density matrix ρ by expanding states |g⟩ and |e⟩. To this end we use equation (4) of the main text, which we here repeat for convenience: where we introduced qubit notation for the tardigrade degrees of freedom using the fact that states |0 . . . 0 ⟩ and |ψ 1 ⟩ are orthogonal. For example, the operator |0g⟩ ⟨0e| corresponding to the matrix element of ρ is mapped to the operator in tensor product of three Hilbert spaces: and so on. The expression on the right-hand side is written in the following order H A ⊗ H B ⊗ H T , where the Hilbert spaces describe qubit A, qubit B and effective qubit for the tardigrade tun, respectively. The reconstructed density matrix written in this space explicitly depends on the coupling strength θ and reads: where ρ ij are the estimated matrix elements of ρ. Note that ρ ABT admits 28 zero values reflecting the assumption that the system is not energetic enough for simultaneous excitation of qubit B and the tardigrade, i.e. terms with |011⟩ , |111⟩ , ⟨011| , ⟨111| have vanishing corresponding matrix elements. The density matrix ρ ABT was used to compute entanglement presented in the main text.

Appendix D. Entanglement quantifiers
To quantify various types of entanglement in the tripartite system, we utilise so-called tangles previously defined in [34]. Let us begin with (doubled) negativity [36] being well-known quantifier of bipartite entanglement: where ||M|| = Tr √ M † M is the trace norm of matrix M and T X stands for the partial transpose over subsystem X. These quantities for various bipartitions involving the tardigrade are plotted in figure 3(c) of the main text. We see that the bipartite entanglement between the tardigrade and the other subsystems, N T:AB , N A:T and N B:T , start off close to zero and increase with the coupling strength. The bipartite entanglement between qubit A and the qubit B-tardigrade subsystem, N A:BT , is close to 0.5 (negativity maximum) because the prepared state is close to the Bell-state |ψ + ⟩.
In order to quantify genuine tripartite entanglement, i.e. entanglement irreducible to pairwise entanglements, we compute so-called π-tangle as follows: where where where δ is the anharmonicity and ℏω = √ E C E J − E C , with E C = e 2 2C being the energy of the shunt capacitor with capacitance C, and E J is the Josephson energy associated with the Josephson junction.
By using the transmon as a qubit, we restrict ourselves only to the ground and first excited state, allowing us to neglect the anharmonicity and rewrite the Hamiltonian reduced to this subspace as [37]: The presence of the tardigrade tun increases the relative permittivity across the shunt capacitor, ε r . The relative permittivity was modelled using finite element method on ANSYS Maxwell. The altered capacitance, C ′ = ε r C, then lowers the charging energy and consequently the qubit frequency.
where H T is an arbitrary tardigrade Hamiltonian and h T is also arbitrary tardigrade interaction part.
We now show that this model cannot explain our data without entanglement. The idea is to use the condition for the lack of entanglement (17) to show that it implies an increase of frequency of the qubit B-tardigrade system, opposite to the experimental findings. To this end, consider the ground state eigen-equation for the combined qubit B-tardigrade system, H b g |t⟩ = E g b g |t⟩, and set to H T |t⟩ = 0. Using (18) in the eigen-equation shows that |t⟩ is an eigenstate of the interaction part, h T |t⟩ = |t⟩. Therefore b g and |b e ⟩ are eigenstates of operator − ℏωq 2 σ z + ℏg 2 σ x , which results in the frequency gap between the ground and excited states to be Hence, without entanglement the new frequency gap is larger than the bare qubit frequency, contrary to the experimental findings.