Generation of bipartite mechanical cat state by performing projective Bell state measurement
Roson Nongthombam Urmimala Dewan Amarendra K. Sarma


    
    
        n.roson@iitg.ac.in
        Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India
    
        d.urmimala@iitg.ac.in
        Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India
    
        aksarma@iitg.ac.in
        Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India

    

    
      Quantum state preparation and measurement of photonic and phononic Schrödinger cat states have gathered significant interest due to their implications for alternative encoding schemes in quantum computation. These scheme employ coherent state superpositions, leveraging the expanded Hilbert space provided by cavity or mechanical resonators in contrast to two-level systems. Moreover, such cat states also serve as a platform for testing fundamental quantum phenomena in macroscopic systems.
    In this study, we generate four bipartite phononic Bell cat states using an entanglement swapping scheme achieved through projective Bell state measurements on two superconducting qubits. Subsequently, we conduct a Bell inequality test on the bipartite cat state using the CHSH formulation. Given that the entangled cat states are generated through entanglement swapping, our approach could hold promising applications for the advancement of complex quantum network processors based on continuous variable systems.
    

    Generation of bipartite mechanical cat state by performing projective Bell state measurement.
    Amarendra K. Sarma
    April 29, 2024
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§ INTRODUCTION

        
        Observing the quantum behaviour of macroscopic mechanical structures continues to be a challenging endeavour and a significant step towards realizing quantum-acoustic state preparation and measurement. The intriguing aspect of these structures, being massive yet displaying quantum behaviour, makes them important platforms for implementing various quantum technological applications and for exploring fundamental physics questions. Recent experiments have unequivocally demonstrated the quantum properties of solid-state mechanical objects <cit.>. These include interfacing mechanical objects with the strong quantum nonlinearity of superconducting qubits, leading to the field of circuit quantum acoustodynamics (cQAD)  <cit.> which is analogous to the well-developed field of circuit quantum electrodynamics (cQED) <cit.>. The integration of superconducting qubits serves as a quantum-acoustic state preparation and measurement element for the mechanical system <cit.>. These state preparations and measurements are some of the basic building blocks for constructing acoustic quantum memories and processors.
        
        A very significant quantum-acoustic state that can be prepared from the superconducting qubit-mechanical integrated system is the phononic Schrödinger cat state, defined as quantum superpositions of quasi-classical coherent states.
        Such phononic cat state preparation has been studied in <cit.>. Similar photonic cat states have also been studied in other various quantum systems such as circuit quantum electrodynamics <cit.>, vibrational states of trapped ions <cit.>, propagating photon modes <cit.>. Preparing cat states has attracted wide research attention owing to its applications in implementing quantum metrology <cit.> and quantum information processing protocols based on continuous-variable cat states <cit.>, as well as testing fundamental quantum phenomena in macroscopic systems <cit.>. One of the most fundamental tests of quantum phenomena in a quantum system is the Bell inequality test <cit.>. Bell inequality tests, performed on pairs of spatially separated entangled quantum systems, demonstrate that quantum systems do not adhere to the principle of local causality. In this work, we use the Clauser–Horne–Shimony–Holt (CHSH) <cit.> formulation of the Bell test to conduct the Bell inequality test on a bipartite entangled phononic cat state generated through projective Bell state measurement on a pair of superconducting qubits <cit.>. The creation of bipartite cat states is demonstrated in photonic resonators <cit.>. In our scheme, 
        four bipartite phononic cat states, or four phononic Bell states, are generated on two phononic 
        crystal mechanical resonators, interacting piezoelectrically with a pair of superconducting qubits via capacitive coupling. The two qubits are connected through a microwave cavity resonator, and through virtual excitation of the cavity photon, they become entangled, creating qubit Bell state. Upon projective measurement of the Bell state of the qubits, each of the four bipartite phononic cat states can be distinguished. This measurement effectively swaps the entanglement from qubit-mechanical to mechanical-mechanical pairs. Such entanglement swapping schemes are pivotal in quantum repeaters, essential for realizing long-distance quantum communication and complex quantum networks <cit.>. Since the two qubits are connected via a microwave cavity bus, the phononic crystal resonator can be placed at far ends, enabling long-distance quantum state preparation. 
        By harnessing this capability and employing entanglement swapping schemes, the bipartite phononic cat states generated in this study hold promise for practical applications in implementing quantum network processors relying on continuous variable resonators. 

        We begin in Section II with a brief introduction of the hybrid electromechanical system under study and then discuss the generation of resonator-qubit Bell cat state.
        In Section III, we examine the projective Bell state measurement on the two qubits, which leads to entanglement swapping from qubit-qubit to resonator-resonator. We show how this Bell state measurement is realized and how four bipartite phononic Bell cat states are created as a result of the projective measurement. Moving on to Section IV, we conduct the Bell inequality test on the bipartite cat state using the CHSH formulation.


    
    
    

    

§ QUBIT-MECHANICAL RESONATOR ENTANGLEMENT.

        

        We consider two hybrid electromechanical systems, each comprising a mechanical resonator dispersively coupled to a superconducting qubit as shown in Fig. <ref>. Assuming that the two qubit-mechanical pairs are uncoupled (appendix <ref>), the Hamiltonian of the two hybrid systems reads as
         
        
            
    Ĥ_bq1   =   ħ/2Ω_1 |e_1⟩⟨ e_1| + ħω_1/2b̂_1^†b̂_1 + ħλ_1 b̂_1^†b̂_1 |e_1⟩⟨ e_1| ,  
    Ĥ_bq2   =   ħ/2Ω_2 |e_2⟩⟨ e_2|  + ħω_2/2b̂_2^†b̂_2 + ħλ_2 b̂_2^†b̂_2 |e_2⟩⟨ e_2| .

        
        Here, Ω_1 (Ω_2) and ω_1 (ω_2) are the qubit and mechanical frequency of the first (second) hybrid system. b̂_1 and b̂_2 are the operators of the two mechanical resonators. In the dispersive coupling, the detuning Δ_1 = ω_1-Ω_1 (Δ_2 = ω_2-Ω_2) is much larger than the coupling strength λ_1 (λ_2) between the qubit and the mechanical resonator. We prepare two entangled Bell-cat states by evolving the Hamiltonian Ĥ_bq1 and Ĥ_bq2. Initiating the qubits in the superposition state and the mechanical resonators in the coherent state, the state of the first hybrid system, after some time t, in the interaction frame, becomes
        
    |ψ⟩_1 = (|β_1 e^i λ_1t⟩|e_1⟩ + |β_1 ⟩|g_1⟩)/√(2),

        where |β_1⟩ is the coherent state of the first mechanical resonator, while |e_1⟩ and |g_1⟩ refer to the excited and the ground states of the first qubit, respectively.
        We get a similar state |ψ⟩_2 for the second hybrid system.
        At time interval t=(2n-1)π/λ_1, where n=1,2,3..., we get the Bell-cat state of the qubit-mechanical system.
        
        
        The fidelity and entanglement of the bipartite state |ψ⟩_1 in the presence of a noisy environment is shown in Fig. <ref>. The noisy environment is included by solving the Lindblad master equation.
        
    ρ̇̂̇_1    =    -i/ħi [ Ĥ_int,ρ̂_1 ] + γ_1(n_1+1) ℒ [ b̂_1 ] + γ_1n_1ℒ [ b̂_1^† ]
           + Γ_1 ℒ[ σ̂_- ] + Γ_1 ℒ[ σ̂_z ],

        where ℒ [ ô ] = ( 2 ôρ̂ô^† - ô^†ôρ̂ - ρ̂ô^†ô ) / 2 with ô∈{b̂_1, σ̂_-,σ̂_z }.
        γ_1 and Γ_1 are the decay rates of the mechanical resonator and the qubit, respectively.
        The entanglement of the qubit-mechanical bipartite system is computed using the relation E_N(ρ̂_1)=log_2||ρ̂^T_A_1||, where ρ̂^T_A_1 is the trace norm of the partial transpose of the bipartite mixed state ρ̂_1 <cit.>. As shown in the figure, the fidelity (F) reaches near one at the interval t=0.78μ s, 2.35μ s, and so on for coupling constant λ_1= 4MHz. Therefore, the qubit-mechanical bipartite system evolves into a Bell-cat state at the interval of π. 
        
        

    
    
    
    
    

§ GENERATION OF BIPARTITE CAT STATE.

        
         After the interaction time of t_1=π/λ_1 (t_2=π/λ_2), the state of the qubit-mechanical bipartite state (Eq.<ref>) become |ψ⟩_1 = (|-β_1⟩|e_1⟩ + |β_1 ⟩|g_1⟩)/√(2) (|ψ⟩_2 = (|-β_2⟩|e_2⟩ + |β_2 ⟩|g_2⟩)/√(2)). The state  of the combined system |Ψ⟩ is then given by the tensor product of the states of the two hybrid systems, i.e.,|Ψ⟩=|ψ⟩_1|ψ⟩_2
        
    |Ψ⟩   =   1/2[e_1e_2-β_1-β_2⟩+|e_1g_2-β_1+β_2⟩
           + |g_1 e_2 β_1-β_2⟩+|g_1 g_2 β_1 β_2⟩],

        where |β_2⟩ is the coherent amplitude of the second resonator. In terms of Bell's basis, |ϕ^±⟩=(1/√(2))(|e_1e_2⟩± i|g_1g_2⟩ and |ψ^±⟩=(1/√(2))(|e_1g_2⟩± i|g_1e_2⟩, the wave function of the combined system (Eq. <ref>) can be rearranged to 
        
    |Ψ⟩   =   1/2√(2)[|ϕ^+⟩(|-β_1-β_2⟩-i|β_1 β_2⟩) 
           + |ϕ^-⟩(|-β_1-β_2⟩+i|β_1 β_2⟩)  
           + |ψ^+⟩( |-β_1 β_2⟩ -i|β_1-β_2⟩ ) 
           + |ψ^-⟩( |-β_1 β_2⟩ +i|β_1-β_2⟩ ) ].

        Therefore, by measurement the Bell's states |ϕ^±⟩ and |ψ^±⟩, the two mechanical resonators are projected into the bipartite cat states |C'_∓⟩=𝒩'_∓(|-β_1-β_2⟩∓ i|β_1 β_2⟩) and |C_∓⟩=𝒩_∓( |-β_1 β_2⟩∓ i|β_1-β_2⟩ ), respectively. The Bell state measurement of the two qubits is performed by first switching on the interaction between them. This interaction, which will entangle the two qubits, is achieved by red detuning the second qubit from the cavity and blue detuning the same qubit from the second resonator (appendix <ref>). Such detuning arrangement allows the exchange of virtual cavity photons with the qubits. However, this virtual interaction between the two qubits only produces two of the four Bell states, |ψ^±⟩. To generate all four Bell states, we continuously drive the two qubits, resulting in two dressed states. When the difference in the frequencies of the two qubits is much larger than their interaction strength and the phase of the two drives, which is inverted in the middle of the two-qubit interaction, is the same, all four Bell states are generated. 
        They are mapped onto the computational basis as |ϕ^+⟩→(0_10_2), |ϕ^-⟩→(1_11_2), |ψ^+⟩→(0_11_2), and |ψ^-⟩→(1_10_2) <cit.> (appendix <ref>). Therefore, based on measuring outcomes of the two qubits in the computational basis, all the four bipartite cat states of the resonators are generated.
        

        The above protocol is realized in the noisy environment by first independently evolving the two qubit-mechanical hybrid systems under the Lindblad master equation and then performing the projective measurement on the density matrix of the combined system, ρ̂=ρ̂_1ρ̂_2. The reduced density matrices of the four bipartite Bell-cat states after the projective Bell state measurement on the two qubits read as,
        
    ρ̂_C'_∓   =   ⟨ϕ^±|ρ̂|ϕ^±⟩/tr(⟨ϕ^±|ρ̂|ϕ^±⟩),
          ρ̂_C_∓ = ⟨ψ^±|ρ̂|ψ^±⟩/tr(⟨ψ^±|ρ̂|ψ^±⟩).

        Constructing the reduced density matrix (Eq. <ref>) on the number basis will require a huge subspace, and performing joint state tomography on the resonators will be experimentally challenging. So, we construct the density matrix into a two-level system subspace by measuring the displaced phonon number parity observable P̂_β_1=D̂_β_1P̂D̂^†_β_1, where D̂_β_1 is the displacement operator and P̂ the phonon number parity operator <cit.>. By applying the projector Ô=|β_1⟩⟨β_1| + |-β_1⟩⟨-β_1| to the displaced parity operator P̂_β_1, the Pauli operators for the resonator mode become
        
    X̂_β_1   =   ÔP̂_0Ô^†, Î_β_1 = Ô[P̂_β_1+P̂_-β_1]Ô^†, 
          Ŷ_β_1 = ÔP̂_-iπ/8β^*_1Ô^†, Ẑ_β_1 = Ô[P̂_β_1 - P̂_-β_1] Ô^†.

        We have assumed a large orthonormal cat state, i.e., ⟨β_1|-β_1⟩<<1.
        The basis for these operators is (|β_1⟩, |-β_1⟩) which is analogous to the qubit basis (|e⟩, |g⟩). Similar Pauli operators and basis (|β_2⟩, |-β_2⟩)  can be generated for the second resonator using the same approach. 
        
        In Fig. <ref>, we plot the joint density matrix of the two resonators in the joint basis states (|β_1 β_2⟩, |-β_1 β_2⟩, |β_1 -β_2⟩, and |-β_1 -β_2⟩). We observe four bipartite cat states having fidelities F_C'_-=0.946, F_C'_+=0.946, F_C_-=0.95, and F_C_+=0.95 and entanglements E_C'_-=0.875, E_C'_+=0.875, E_C_-=0.884, and E_C_+=0.884. The dip in the density matrix element is attributed to the decoherence effect of the two qubits and the relaxation of the two resonators. In addition to the decays resulting from direct contact with the noisy environment, the other factors that lead to the infidelities of the prepared state include read-out errors and errors while performing the Bell measurement.
        The four bipartite cat states resemble the traditional qubit Bell states generated using a Hadamard and CNOT gate in a quantum circuit. Going by this similarity, the scheme proposed here could be used to implement an entanglement gate for a continuous variable resonator qubit, and may find applications in the bosonic based quantum processors.
        

    
    
    

    

§ BELL'S TEST OF THE RESONATOR BIPARTITE CAT STATE

        
        The bipartite entangled cat states of the two resonators generated on a conditional Bell state measurement of the two qubits can be used as a platform to test the Bell's inequality in a macroscopic quantum system. Here, we perform this test by calculating the expectation values of all the correlations of the measurement outcomes measured locally at the two resonators and then determine the Clauser–Horne–Shimony–Holt (CHSH) value S=⟨ X_1X_2⟩ + ⟨ X_1Y_2⟩ - ⟨ Y_1X_2⟩+⟨ Y_1Y_2⟩, where, X_1 (X_2) and Y_1 (Y_2) are the observables of the first (second) resonator. The observables are measured in the resonator qubit subspace discussed above. As per CHSH inequality, a system is said to be classically correlated if |S|≤2 and quantumly if 2<|S|≤2√(2). The correlation of the observables is measured by first choosing two arbitrary values of β'_1 and β'_2 corresponding to  X_1, Y_1 and X_2, Y_2, respectively. We then rotate the resonator detector basis by coherently displacing the observables X_1 and Y_1 to X_α=D_-iαX_1D_iα and 
        Y_α=D_-iαY_1D_iα or
        
    X_α   =    X_1  cos2(αβ'^*_1+α^*β'_1) +Y_1sin2(αβ'^*_1+α^*β'_1),
          Y_α = Y_1  cos2(αβ'^*_1+α^*β'_1)-X_1sin2(αβ'^*_1+α^*β'_1).

        Here, α is the coherent displacement amplitude of the resonator. By changing the amplitude α, we are able to rotate the measurement basis direction and perform measurements at all possible orientations of the detectors. The |S| value for the state |C_+⟩ is shown in Fig. <ref>. We observe maximum |S| value when β'_1,2=β_1,2 and at the displacement amplitude α_1=-0.14 and α_2=0.41. Furthermore, the α_1, α_2 and β'_1 amplitudes corresponding to the maximum |S| value satisfy the relations 4α_2β'_1=3π/4 and 4α_1β'_1=-π/4. 
        The corresponding observables (Eq. <ref>) at these amplitudes become X_α_1=(X_1-Y_1)/√(2) and Y_α_1=(X_1+Y_1)/√(2). Therefore, the |S| value observed in the figure exceeds the classical bound limit and attains a maximum of 2.44, which is less than the ideal quantum bound limit 2√(2). As we increase the decay rates of the resonators, the maximum attainable |S| value decreases, as shown in Fig <ref>(b). The expectation values of all the measurement correlations are also shown in Fig. <ref>(a). The behaviour of these correlations as we change the displacement amplitude α resembles the one observed in two-qubit Bell test experiments <cit.>. By integrating two phononic crystal resonators into the experimental arrangement typically employed for conducting a loophole-free Bell test with two superconducting qubits <cit.>, our proposed approach could be employed to examine the Bell inequality of a phononic cat state. Additionally, the bipartite phononic cat state generated through entanglement swapping in this work could hold significant practical implications for the advancement of complex quantum network processors based on continuous variable resonators. 
        

    
    
    
    
    

§ CONCLUSION

        
        In conclusion, we propose a scheme to generate four bipartite phononic cat states by performing a projective Bell state measurement on two superconducting qubits. Initially uncoupled phononic crystal resonators, each coupled to a different superconducting qubit, become entangled through entanglement swapping from qubit-mechanical to mechanical-mechanical interactions. Displaced phonon parity measurement is done for generating joint density matrix of the two resonators in two-level subspace. These joint density matrices resemble those of traditional qubit Bell states generated using a Hadamard and CNOT gate in a quantum circuit. Subsequently, we investigate the Bell inequality test using the CHSH formulation. The expectation values of the measurement correlations and the S values of the CHSH inequality test obtained here are akin to those observed in <cit.> for two superconducting qubits. The bipartite phononic
        cat state generated through entanglement swapping in
        this work may find useful in implementing quantum network processors
        based on continuous variable resonators. Furthermore, the scheme presented in this work also serves
        as a platform for studying Bell inequality test in continuous variable system.
        

    
    
    

    

§ ACKNOWLEDGEMENT

        R.N. gratefully acknowledges support of a research fellowship from CSIR, Govt. of India. U.D. gratefully acknowledges a research fellowship from MoE, Government of India. A.K.S. acknowledges the STARS scheme, MoE, government of India (Proposal ID
        2023-0161).

    
    
    

    

    

§ DISPERSIVE HAMILTONIAN

        
        The interaction part of the complete Hamiltonian, including the cavity resonator connecting the two qubits, is given by the Jaynes-Cummings interaction
        
    Ĥ_int   =    g_1(b̂^†_1σ̂_-^1+b̂_1σ̂_+^1)+g_2(b̂^†_2σ̂_-^2+b̂_2σ̂_+^2)
          +G_1(â^†σ̂_-^1+âσ̂_+^1)+ G_2(â^†σ̂_-^2+âσ̂_+^2).

        Here, g_1 (g_2) and G_1 (G_2) are the coupling strength of mechanical-qubit and qubit-cavity interactions, respectively. â and â^† are the creation and annihilation operators of the cavity resonator. We blue detune the first resonator from the first qubit (δ_1=ω_1-Ω_1), second qubit from the cavity resonator (Δ_2=ω-Ω_2), and red detune the cavity from the first qubit (Δ_1=Ω_1-ω), second resonator from the second qubit (δ_2=Ω_2-ω_2). Here, ω refers to the cavity resonance frequency. Because of the detuning, we can transform the Jaynes-Cummings interaction (Eq. A1) into a dispersive one by performing a unitary operation U=exp{ g_1/δ_1(b̂^†_1σ̂_-^1-b̂_1σ̂_+^1) + g_2/δ_2(b̂_2σ̂_+^2-b̂^†_2σ̂_-^2) + G_1/Δ_1(âσ̂_+^1-â^†σ̂_-^1) + G_2/Δ_2(â^†σ̂_-^2-âσ̂_+^2)} on Ĥ_int.
        
    Ĥ_dis   =   ħ(η_1σ̂_z^1+η_2σ̂_z^2)â^†â + ħλ_1σ̂_z^1b̂_1^†b̂_1 + ħλ_2σ̂_z^2b̂_2^†b̂_2 
           
                + ħ J(σ̂_1^-σ̂_2^+ + σ̂_1^+σ̂_2^-) + ħχ_1(âb̂_1^†+b̂_1â^†)σ̂_1^z 
          
                +ħχ_2(âb̂_2^†+b̂_2â^†)σ̂_2^z,

        where η_1=G_1^2/Δ_1, η_2=G_2^2/Δ_2, λ_1=g_1^2/δ_1, λ_2=g_2^2/δ_2,
        J=G_1G_2(1/Δ_1-1/Δ_2), χ_1=g_1G_1(1/δ_1-1/Δ_1) and χ_2=g_2G_2(1/Δ_2-1/δ_2). The first term in the Hamiltonian A2 can be neglected since the cavity remains in the ground state. By choosing the detuning as Δ_1=Δ_2, δ_1=Δ_1 and δ_2=Δ_2, we can also neglect the fourth, fifth, and sixth terms <cit.>. The remaining two interacting terms are the ones used in Hamiltonian <ref> and <ref>. Thus, we see that by changing the detuning, we can independently evolve the two qubit-resonator pairs. After the two pairs evolve up to a certain time period, we bring back the qubit-qubit interaction (fourth term) to perform Bell state measurement <cit.>. This can be done in two ways. First, by choosing Δ_1 ≠Δ_2, or second, by red detuning the second qubit from the cavity and blue detuning the second qubit from the second resonator. In the second approach the coupling constant J becomes J'=G_1G_2(1/Δ_1+1/Δ'_2), where Δ'_2=Ω_2-ω. The rearrangement of the detuning in the second approach ensures that the last term in the Hamiltonian <ref> remains zero. This interaction allows us to perform the necessary Bell test on the two qubits. 

        

§ BELL STATE OF THE TWO QUBITS

        
        In order to distinguish all the Bell states of the two qubit, we resonantly drive the qubits individually. 
        In the interaction frame, the Hamiltonian of the two qubit inteaction is 
        
    Ĥ = ħ (J'σ̂_1^+σ̂_2^- +
                 ∑_j=1,2A_je^-iΦ_jσ_j^+)+H.c.

        Here, A_j and Φ_j are the Rabi frequency and phase, respectively, of the drive applied to the qubits. We have assumed that J'>>λ_1,λ_2. The drive produces two dressed states |±⟩_j = (1/√(()2))(|g_j⟩± e^iΦ_j|e⟩_j). The qubit cannot go into transitions between different dressed states under the conditions Φ_1=Φ_2=Φ and |A_1-A_2|>>|J'|. Then the Hamiltonian <ref> in the dressed state basis reduces to <cit.>.
        
    Ĥ_eff = 1/2ħ J'S_z1S_z2 + ħ∑_j=1,2A_jS_zj.

         Here, S_zj=|+⟩_j⟨+|_j - |-⟩_j⟨-|_j. If the phases of both the driving fields are reversed right in the middle of the
         two-qubit interaction time, then the dressed state |+⟩_1 |+⟩_2 evolves to |++⟩_t = exp(iJ't/2)|+⟩_1 |+⟩_2. At time τ=π/2J', the dressed state become |++⟩_τ=(1/2)(i|ϕ^-⟩ + |ϕ^+⟩ + i|ψ^-⟩ + |ψ^+⟩). In the computational basis (|0⟩, |1⟩), the state |++⟩_τ can be obtained by applying the unitary operator
         
    U=[ 1 0 0 i; 0 1 i 0; 0 i 1 0; i 0 0 1 ]

         Therefore, in the computational basis the Bell state |ϕ^+⟩ is mapped to |0_10_2⟩, i.e., U|0_10_2⟩=|ϕ^+⟩. Similarly, |ϕ^-⟩, |ψ^+⟩, and |ψ^-⟩ are mapped onto |1_11_2⟩, |0_11_2⟩, |1_10_2⟩. 

                


