title: Gaussian state entanglement witnessing through lossy compression
authors: Waldemar Klobus Pawel Cieslinski Lukas Knips Pawel Kurzynski Wieslaw Laskowski
text: 
  	
  	
  	
  
  Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
  
  
  Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
  
  
  Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
  Department für Physik, Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany
  Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
  
  
  Faculty of Physics, A. Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland
  
  	
  
  
  Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
  International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
  
  	
  
  We propose a method to witness entanglement between two continuous-variable systems in a Gaussian state. Its key ingredient is a local lossy state transfer from the original spatially separated systems onto two spatially separated qubits. The qubits are initially in a pure product state, therefore by detecting entanglement between the qubits we witness entanglement between the two original systems. This method greatly simplifies entanglement witnessing in complex systems. 
  
  	
  	Gaussian state entanglement witnessing through lossy compression
      Wiesław Laskowski
      
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  § INTRODUCTION
  
  
  Entanglement is considered one of the key resources in quantum information science <cit.>. It naturally emerges in the majority of many-body systems <cit.> and can be engineered on various experimental platforms <cit.>. However, despite the fact that entanglement seems to be all around us, its detection is challenging, especially in high-dimensional and continuous-variable systems. Detection of entanglement often requires a partial tomography of the system's state <cit.>, whose full description is determined by a number of measurements which grows exponentially with the dimension (and is infinite in the continuous case). 
  
  In this work we focus on the problem how to extract information about entanglement in the state of a complex bipartite system A. The main idea is to pair A with a simple system B <cit.>. A is assumed to be difficult to analyze, whereas B allows full analysis. In other words, we limit the interaction with A to the minimum, whereas we are allowed to perform full tomography on B. The goal is to learn whether A is entangled by studying solely B. At this point we stress that the two subsystems A and B can be defined as separate particles, or as different degrees of freedom of a single particle (e.g., path/polarization, time/polarization, etc., see <cit.>).
  
  In particular, we propose a method to detect entanglement between two continuous-variable systems in a Gaussian state by transferring their state onto a state of two qubits and then by analyzing the resulting two-qubit state. In order to develop some intuition, we first show how to design a protocol to detect entanglement between two qudits and then we generalize it to the continuous-variable case. The qubits are initially prepared in a separable state. Hence, any entanglement arising between them must stem from the initial entanglement between the more complex systems. 
  
  It is clear that such a state transfer cannot be perfect since the dimension of the system onto which the transfer is made is lower than the dimension of the original system. Therefore, the above process can be considered a lossy compression, which aims to preserve only the relevant information. In this case, we want to keep the information about entanglement and discard anything else. 
  
  
  
  
  § D-LEVEL SYSTEMS
  
  
  Before we analyze entanglement and continuous-variable systems, let us first discuss a single qudit (A) and a single qubit (B). We will introduce a unitary coupling operation which allows to transfer some properties of the system A to the system B. Later we will generalize the scheme to a pair: two qudits – two qubits.
  
  
  
  
   §.§ Single system
  
  
  
  As coupling operator we use a controlled rotation (CROT), i.e., a rotation of the qubit controlled by the state of the qudit. More precisely, CROT is defined for a bipartite system AB composed of a controlling d-level qudit state A, and a target qubit B, the state of which is rotated along the y-axis, by
  
      U_ CROT =  ∑_j=0^d-1 |j⟩⟨ j| ⊗exp(-i σ_y ξ_j) 
             =∑_j=0^d-1 |j⟩⟨ j| ⊗(cosξ_j  -  i sinξ_j σ_y ),
  
  where the rotation parameter ξ_j depends on the original state of the qudit, ξ_j = j π/2(d-1). After the coupling we ignore the subsystem A by tracing it out, and perform analysis on the qubit B. 
  
  The above operation resembles the von Neumann measurement apparatus <cit.>, with the exception that the pointer is not a continuous-variable system, but a single qubit. As a result, the measurement of the observable (in this case A=∑_j j|j⟩⟨ j|) cannot be perfect due to the fact that one can encode at most a single bit of classical information on a single qubit. Nevertheless, we are going to show that after the CROT operation some important information about the qudit's state can be decoded from the qubit's state.
  
  As an example let us consider a d-level system being in the state
  
      |ψ (p) ⟩_A = ∑_k=0^d-1√(d-1  k p^k (1-p)^d-1-k) |k⟩
   
  parametrized by a single unknown parameter p.	The probability amplitudes are given by the Bernoulli distribution.
  Applying the coupling operation (<ref>) on the qudit–qubit pair |ψ(p)⟩_A ⊗ | 0 ⟩_B, we get 
  
      |Ψ⟩_AB   =    U_ CROT(|ψ(p)⟩_A ⊗ | 0 ⟩_B) 
         =   ∑_j=0^d-1√(d-1  j p^j (1-p)^d-1-j) |j⟩_A 
         ⊗   (cosξ_j |0⟩_B + sinξ_j |1⟩_B ).
   
  The reduced density matrix of the system B is given by	
  
      ρ_B =  Tr_A |Ψ⟩_AB⟨Ψ|	 = ∑_j=0^d-1d-1  j p^j (1-p)^d-1-j×
            (
      [    cos^2ξ_j 1/2sin 2ξ_j; 1/2sin 2ξ_j   sin^2 ξ_j ]),
  
  allowing to extract information about the parameter p by, for example, a measurement along σ_z, Tr (ρ_B σ_z).
  In the limit of infinite dimensions d, 
  
      ρ_B d →∞=(
      [ cos ^2(π  p/2)  1/2sin (π  p);  1/2sin (π  p) sin ^2(π  p/2) ]),
  
  ρ_B becomes a pure state |Ψ⟩_B = cos(π  p/2) |0⟩ + sin(π  p/2) |1⟩.  Hence, p = (1/π) arccos Tr  (ρ_B σ_z).
  
  
  
  
  
   §.§ Entangled systems
  
  
  
  Let us now suppose the system A is composed of a pair of d-dimensional qudits in the state |ψ⟩_A= ∑_j,l=0^d-1 a_jl |jl⟩, which we want to couple with a pair of qubits B. In order to do this, we use the coupling operator U_ CROT^⊗ 2 for each pair of subsystems, such that the CROT operator couples the first (second) qudit to its respective qubit. 
  
  If both qubits are initially in the state |0 ⟩, then application of U_ CROT^⊗ 2 to the total system |ψ⟩_A |00⟩_B results in
  
      | Ψ⟩_AB   =    U_ CROT^⊗ 2(|ψ⟩_A ⊗ |00⟩_B) 
         =   ∑_j,l=0^d-1 a_jl   |j  l ⟩_A  ⊗[	cosξ_j cosξ_l |00⟩_B 
             +cosξ_j sinξ_l |01⟩_B  +sinξ_j cosξ_l |10⟩_B 
             +sinξ_j sinξ_l |11⟩_B  ].
  
  In general, the state | Ψ⟩_AB can be highly four-partite entangled, which results in separable subsystems. Therefore, if we want to transfer entanglement from the system A to B, we are obligated to do a conditional (projective) measurement on the system A. One of the good candidates is the local projection onto the state |++⟩_A=|+⟩ |+⟩ with |+⟩  =1/√(d)∑_k=0^d-1 |k⟩. 
  After successful projection, the resulting state reads
  
      𝒩 |++⟩_A ∑_j,l=0^d-1  a_jl [ cosξ_j cosξ_l |00⟩_B 
             +cosξ_j sinξ_l |01⟩_B  +sinξ_j cosξ_l |10⟩_B 
             +sinξ_j sinξ_l |11⟩_B  ],
  
  where (1/𝒩)^2 is the probability of projecting the system A of two qudits onto |+⟩ |+⟩. 
  
  
  At this point it is worth to consider an example. Let A be in the maximally entangled state corresponding to a_jl= δ_jl/√(d). Then, after the coupling operation, the overlap of the resulting state |Ψ⟩_B with the maximally entangled state 1/√(2)(|00⟩ +  |11⟩) decreases with d, 
  but asymptotically approaches π^2/(π^2 + 4) ≈ 0.712 as d →∞.
  
  
  Notice that in the special case of d=2, the operation swaps the state of the system A to the system B,
  
      |Ψ⟩_B d=2=𝒩^-1∑_j,l=0^1  a_jl [ cos(j π/2 ) cos(l π/2 ) |00⟩
            	+cos(j π/2 ) sin(l π/2 ) |01⟩   +sin(j π/2 ) cos(l π/2 ) |10⟩
            	+sin(j π/2 ) sin(l π/2 ) |11⟩] 
            =   a_00 |00⟩+ a_01 |01⟩ + a_10 |10⟩  + a_11|11⟩.
  
  
  For a general d the resulting state |Ψ⟩_B is separable if the input state |ψ⟩_A is separable. This is because a factorization of the amplitudes a_jl=a_j' a_l” allows to factorize the resulting state |Ψ⟩_B,
  
      | Ψ⟩_B |ψ_prod⟩_A=∑_j=0^d-1 a_j'(cosξ_j |0⟩ + sinξ_j|1⟩) 
            ⊗∑_l=0^d-1 a_l”(cosξ_l|0⟩ + sinξ_l|1⟩).
  
  	
  	
  	
  	
  
  	
  Let us now consider a general mixed state of two qudits, i.e., ρ_A = ∑_i,j,k,l=0^d-1ρ_ij,kl |i⟩⟨ j|⊗ |k ⟩⟨ l|. If we denote ρ_AB as the state of the total system after the coupling,
  
      ρ_AB = U_ CROT^⊗ 2 (ρ_A ⊗ |00⟩_B ⟨ 00| ) (U_ CROT^⊗ 2)^†,
  
  projecting the subsystem A onto |++⟩ results in subsystem B becoming
  
      ρ_B =  ∑_i,j,k,l=0^d-1∑_m,n,p,q=0^1 ρ_ij,kl  a^m_i a^n_k a^p_j a^q_l    |m⟩⟨ p|⊗ |n ⟩⟨ q|,
  
  where
  
      a^α_β=
      cosβπ/2(d-1),   for α=0, 
      sinβπ/2(d-1),   for α=1.
  
  Note that for d=2, we have ρ_B=ρ_A, the same as in the case of pure states.
  
  
  Additionally, we also show that a separable state of two qudits is mapped onto a separable state of two qubits. Consider the separable state of two qudits ρ_A^sep =∑_λp_λρ^λ_1 ⊗ρ^λ_2, where
  
      ρ^λ_1    =   ∑_i,j=0^d-1ρ^λ_1,ij |i ⟩⟨ j|, 
      ρ^λ_2    =   ∑_k,l=0^d-1ρ^λ_2,kl |k ⟩⟨ l|,
  
  and hence
  
      ρ_A^sep = ∑_λp_λ∑_i,j,k,l=0^d-1ρ^λ_1,ijρ^λ_2,kl |i⟩⟨ j|⊗ |k ⟩⟨ l|.
  
  Performing analogous calculations as in the general case for mixed states and taking into account the linearity of all operations, we get a separable state:
  
      ρ_B ρ_A^sep=∑_λp_λ
         ×   ∑_i,j,k,l=0^d-1∑_m,n,p,q=0^1 ρ^λ_1,ijρ^λ_2,kl  a^m_i a^n_k a^p_j a^q_l    |m⟩⟨ p|⊗ |n ⟩⟨ q| 
             = ∑_λp_λ( ∑_i,j=0^d-1∑_m,p=0^1 ρ^λ_1,ij  a^m_i a^p_j |m⟩⟨ p| ) 
            ⊗( ∑_k,l=0^d-1∑_n,q=0^1 ρ^λ_2,kl  a^n_k a^q_l |n ⟩⟨ q| ) .
  
  
  Please note that the condition for ρ_A to be separable so that the resulting ρ_B is also separable is only sufficient, not necessary. There are instances of entangled states ρ_A which are not mapped into entangled ρ_B, hence the scheme effectively works as an entanglement witness.
  
  
  
  
  § CONTINOUS-VARIABLE SYSTEMS
  
  
  We will now generalize our scheme to the case in which the system A is being described by a continuous-variable state. In this regard we limit our considerations to the broad family of Gaussian states.
  
  
  
  
   §.§ Single system
  
  
  If the first subsystem has a continuous spectrum, the coupling operator reads
  
  
      U_ CROT = ∫_-∞^∞ x |x⟩⟨ x| ⊗ (cos x  - i sin x σ_y).
  
  Next, consider a Gaussian state
  
      |ψ(σ,m)  ⟩_A = ∫ x 1/(2πσ^2)^1/4 e^-(x-m)^2/4σ^2 |x⟩
   
  specified by two parameters (σ, m).  After applying the coupling operation to |ψ(σ,m)⟩_A ⊗ | 0 ⟩_B,	we get 
  
      U_ 	CROT( |ψ(σ,m)⟩_A ⊗ | 0 ⟩_B )
         =   ∫ x  1/(2πσ^2)^1/4 e^-(x-m)^2/4σ^2 |x⟩⊗ (cos x  |0⟩ + sin x  |1⟩ ).
   
  The reduced state of the qubit B is	
  
      ρ_B=  1/2( +   e^-2 σ^2(
      [   cos 2m   sin 2m;   sin 2m - cos 2m ]))
  
  and can be visualized by a Bloch vector b⃗ lying in the xz-plane. The vector b⃗ makes an angle 2m with the z-axis and its norm is e^-2 σ^2. 
  The parameters of the original Gaussian state can be recovered from a tomography on the qubit. In particular, σ^2 = -(1/4)ln ||b⃗||^2 and m=acot(b_z/b_x)/2.
  
  
  
   §.§ Entangled systems
  
  
  Now, we consider the lossy entanglement transfer from the bipartite Gaussian state onto the two-qubit state. In order to do this, we use the coupling operator of (<ref>) for each respective pair of subsystems, U_ CROT^⊗ 2, such that the first (second) complicated subsystem interacts with its respective qubit. 
  
  After the coupling operation, we project the system A of the two particles onto a product of Gaussian states
  
      |x_1^+ x_2^+(Γ)⟩ =∫ x_1 ∫ x_2 1/(2πΓ^2)^1/2 e^-(x_1^2+x_2^2)/4Γ^2 |x_1⟩|x_2⟩.
  
  This is an analogy to the projection onto a uniform superposition that we used in the two-qudit case. This time the projection is parametrized by a single parameter Γ, which corresponds to the standard deviation. Note that in the limit Γ→∞ the Gaussian function becomes a uniform superposition over the whole space, akin to what was considered in the qudit case. Such a projection can be interpreted as a projection onto a ground state of a harmonic oscillator, for which the parameter Γ can be manipulated by the oscillator's frequency. 
  
  As an example we consider two particles in an entangled Gaussian state
  
      |ψ (σ,Σ) ⟩_A = 
            ∫_-∞^∞ x_1 ∫_-∞^∞ x_21/(2 πσΣ)^1/2 e^-(x_1+x_2)^2/8σ^2 e^-(x_1-x_2)^2/8Σ^2 |x_1⟩ |x_2⟩.
  
  This state is entangled whenever σ≠Σ. Since it is a pure state, its entanglement can be measured by the purity of a subsystem, which in this case is given by <cit.>
  
      P=2σΣ/σ^2 + Σ^2.
   
  
  
  After the coupling operation, the state |ψ (σ,Σ) ⟩_A⊗ |00⟩_B becomes
  
  
  
  
      U_	CROT^⊗ 2 (	|ψ (σ,Σ) ⟩⊗ |00⟩) 
         =   ∫ x_1 ∫ x_2 1/(2 πσΣ)^1/2 e^-(x_1+x_2)^2/8σ^2 e^-(x_1-x_2)^2/8Σ^2 |x_1⟩ |x_2⟩
         ⊗   [
      cos x_1 cos x_2 |00⟩  +cos x_1 sin x_2 |01⟩
          +   sin x_1 cos x_2|10⟩  	+sin x_1 sin x_2 |11⟩].
  
  After projecting the system A onto |x_1^+ x_2^+(Γ)⟩, the state of system B becomes
  
      𝒩 (a_+ |00⟩ + a_- |11⟩),
  
  where
  
      a_±=± e^-2 σ ^2 Γ^2/σ ^2+Γ^2+e^-2 Σ ^2 Γ^2/Σ ^2+Γ^2/√((σ ^2+Γ^2) (Σ ^2+Γ^2)/σΣΓ^2).
  
  The purity of the qubit subsystem B is   
  
      P_q = 1/2(sech^2(2 Γ^4 (σ -Σ ) (σ +Σ )/(σ ^2+Γ^2) (Σ ^2+Γ^2))+1),
  
  which becomes in the limit of Γ→∞
  
  
      lim_Γ→∞ P_q = 1/2(sech^2(2 (σ -Σ ) (σ +Σ ))+1).
  
  Comparing (<ref>) and (<ref>), we see that in both cases the states are entangled for Σ≠σ. Also, in both cases the situation σ-Σ→∞ corresponds to the maximally entangled states (see Fig. <ref>).
  
  
  
  In Fig. <ref> we present how the purity of a subsystem depends on Γ. We analyze the extreme case (σ - Σ) →∞. In this case the purity is 1/2 ( sech^2 (2 Γ^2)+1) and decreases with Γ. Already for Γ above 1, the purity is close to 1/2.
  
  
  
  
  
  
  
  
  
  
   §.§ Possible realization
  
  
  
  Here, we give examples for possible implementations of the above scheme. The first concept is in a sense an inverted Stern-Gerlach scenario. We focus on a single system, since the entangled case is a straight-forward generalization.
  
  Consider a spin-1/2 particle, say a silver atom, propagating along the z-axis.
  The continuous-variable state of interest is encoded in the transversal degree of freedom, say, a spatial state along the x-axis |ψ(x)⟩. The spin of the particle is initially pointing up along the z-axis. In addition, consider a region in which there is nonzero magnetic field B⃗ = B(x) ŷ pointing along the y-axis with a gradient along the x-axis. We assume that in the region in which |ψ(x)⟩ is supported one can use the approximation B(x) ≈ B_0 x. This magnetic field region starts at z=z_0 and ends at z=z_1 (0<z_0<z_1). Outside of this region there is no magnetic field.
  The particle starts at z=0 and moves towards the magnetic field region with velocity v. It spends the time t=(z_1-z_0)/v within the magnetic field region. The magnetic field causes a position dependent rotation of spin about the y-axis 
  
      |↑_z⟩→cosα(x) |↑_z⟩ + sinα(x) |↓_z⟩,
  
  where α(x) ∝ B_0 x (z_1-z_0)/v. This conditional rotation can be associated with the CROT operation. This way the state |ψ(x)⟩ is lossy transferred onto the spin state.
  
  Another possible implementation is the interaction of different degrees of freedom of photons. 
  A natural choice for the degree of freedom of the d-level system is using path-encoding as it easily allows to manipulate, say, the polarization state of the photon depending on the path state using waveplates for building up the CROT operation as given in Eq. (<ref>).
  This general concept can be combined with a plethora of different degrees of freedom.
  Wavelength division multiplexers allow coupling frequency-bin encoded qudits to qubits using this scheme.
  Similarly, when using, e.g., orbital angular momentum (OAM) for the qudit-system, the OAM-encoding can first be translated to path-encoding using a mode sorter <cit.>.
  Recently, a controlled-X̂ gate between the radial degree of freedom of light and its OAM has been shown <cit.>, providing another perfect testbed for our coupling.
  
  
  In a recent work, a high-finesse cavity has been used to couple a ^87Rb atom to the coherent state of a light field reflected at the cavity for creating Schrödinger cat states <cit.>. 
  This technique may also allow to couple the state of the light field to the atom using the CROT operation as given in Eq. (<ref>).
  Our proposal could hence facilitate probing for entanglement of two light fields.
  
  Furthermore, our proposal can also be used if both systems A and B are actually qudits.
  For example, entanglement of a system of two high-dimensional trapped ions could be probed by properly designing the interaction with two other ions using a suitable modified CROT operation such that the qudits of system B make use of only a two-level submanifold of the ion. This greatly simplifies their read-out as they can now be treated as qubits.
  This procedure is applicable also to other high-dimensional systems such as, say, superconducting transmon qudits.
  
  Finally, we would like to mention that our approach also works for multiqubit systems, in which the entanglement between two specific subsets of particles is to be analyzed. The entanglement between a set A_1 of qubits and a set A_2 of qubits can be studied by first compressing the multiqubit states ρ_A_1 and ρ_A_2 into the single qubits B_1 and B_2, respectively, using a CROT operation. Afterwards, the verification of entanglement of those two single qubit systems implies entanglement between the initial multiqubit systems.
  
  
  
  
  
  § CONCLUSIONS
  
  
  In this paper we address the problem of detecting entanglement properties of a complex system by analyzing an auxiliary system coupled to the original one. In order to do this we define a coupling operator which transforms the auxiliary system so that after the operation the measured properties of the coupled system provide relevant information about the nature of the original one. Since the auxiliary system is chosen to be of lower dimensionality than the original one, the transfer of information through the coupling operator cannot be exact, hence we can consider the operation a lossy compression. In the process, however, we are being offset by the reduction of the number of measurements required to analyze the entanglement properties of the measured system. Moreover, the scheme works also when we intend to detect entanglement between two continuous-variable systems in a Gaussian state, which in principle can be partially encoded in a simple two-qubit state.  
  
  
  
  § ACKNOWLEDGMENTS
  
  
  W.K., P.K., L.K. and W.L. acknowledge the support by DFG (Germany)
  and NCN (Poland) within the joint funding initiative
  “Beethoven2” (Grant No. 2016/23/G/ST2/04273).
  W.L. acknowledges partial support by the Foundation
  for Polish Science (IRAP project, ICTQT, Contract
  No. 2018/MAB/5, cofinanced by EU via Smart Growth
  Operational Programme).
  L.K. is grateful to Marcus Huber and Stephan Welte for helpful comments.
  
  
  
  
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