Efficiency of Entanglement Concentration by Photon Subtraction
EEfficiency of Entanglement Concentration by Photon Subtraction
Wulayimu Maimaiti ∗ and Stefano Mancini School of Science and Technology, University of Camerino, I-62032 Camerino, Italy
We introduce a measure of efficiency for the photon subtraction protocol aimed at entanglementconcentration on a single copy of bipartite continuous variable state. We then show that iteratingthe protocol does not lead to higher efficiency than a single application. In order to overcome thislimit we present an adaptive version of the protocol able to greatly enhance its efficiency.
PACS numbers: 03.65.Bg, 42.50.Dv
I. INTRODUCTION
The most used states in continuous variable quantum information processing are Gaussian states, such as coherentstates, squeezed states, or Einstein-Podolsky-Rosen like states [1]. The latter type of entangled states are two modesqueezed vacuum and represents a common source for most quantum communication protocols, such as quantumteleportation [2], quantum key distribution [3], quantum dense coding [4], etc. However due to the exponentialdecay of the entanglement over the length of quantum communication channels [5], protocols such as entanglementconcentration are needed to ensure faithful quantum communication [6]. They deserve to enhance the amount ofentanglement by using local operations (exploiting ancillary systems) and eventually classical communication.Entanglement concentration involves several copies of bipartite states each having a low amount of entanglementand aims at producing a smaller number of copies possessing a higher amount of entanglement [6]. However, it couldalso work on a single copy of a bipartite entangled state [7], but unfortunately it cannot be realized within continuousvariable by Gaussian operations [8]. Hence methods like “photon subtraction” have been proposed [9, 10], where non-Gaussian operations are realized by photon number measurements after beam splitter transforms with ancillary modes.However the photon subtraction protocol is probabilistic since its success depends on the probability of getting ‘good’measurement outcomes. Hence in characterizing it one should take into account both entanglement enhancement andprobability of success. To this end we introduce here a measure of efficiency for the photon subtraction protocol andshow that iterating it does not lead to higher efficiency than a single application. In order to overcome this limit wepresent an adaptive version of the protocol able to greatly enhance its efficiency. We restrict our analysis to the caseof weak fields so to consider only few relevant measurements outcomes.
II. PHOTON SUBTRACTION PROTOCOL
The standard entanglement concentration scheme with photon subtraction is shown in figure 1 (Left). The initialstate | ψ (cid:105) AB is a two-mode squeezed vacuum state | ψ (cid:105) AB = (cid:112) − λ ∞ (cid:88) n =0 λ n | n (cid:105) A | n (cid:105) B , λ ∈ R + . (1)The non-Gaussian operation in photon subtraction scheme is induced by photon number measurement on ancillarymodes C and D that interact with entangled modes A and B through beam splitters of transmittance T . In Fockspace the effect of the beam splitter unitary transformation is [10] | n (cid:105) A | (cid:105) C ˆ U AC (cid:55)−→ n (cid:88) k =0 ξ nk | n − k (cid:105) A | k (cid:105) C , (2)where ξ nk = ( − k (cid:114)(cid:16) nk (cid:17) ( T ) ( n − k ) / (1 − T ) k/ , (3) ∗ [email protected] a r X i v : . [ qu a n t - ph ] A ug FIG. 1: Entanglement concentration scheme with photon subtraction. (i) The standard protocol. (ii) The standard protocoliterated N times. with (cid:16) nk (cid:17) the binomial coefficient. Analogous considerations hold true for BD modes. Now consider the photonnumber detection of ancillary modes C, D . Suppose the number of photons detected are c , d respectively, then thecorresponding probability operator value elements [11] areˆ M cd = | c (cid:105) C (cid:104) c | ⊗ | d (cid:105) D (cid:104) d | . (4)As a consequence, the a posteriori state to this measurement reads1 √ P cd ˆ M cd ˆ U AC ˆ U BD | ψ (cid:105) AB | (cid:105) C | (cid:105) D , (5)where P cd is the joint probability of detecting c (resp. d ) photons in C (resp. D ) mode P cd = ∞ (cid:88) n =max[ c,d ] ( α n ξ nc ξ nd ) , (6)with α n = √ − λ λ n . Tracing the state of Eq.(5) together with its dual over modes C and D we get the followingstill pure, though non-Gaussian, conditional state | ψ cd (cid:105) AB = 1 √ P cd ∞ (cid:88) n =max[ c,d ] α n ξ nc ξ nd | n − c (cid:105) A | n − d (cid:105) B . (7) III. EFFICIENCY OF THE PROTOCOL
In order to evaluate the effectiveness of the entanglement concentration protocol we need to compare the prior andposterior amount of entanglement. A useful entanglement measure is the so called negativity defined as [12] N ( ρ AB ) := || ρ T B || − , (8)where || ˆ O || = Tr (cid:112) ˆ O † ˆ O is the trace norm of the operator ˆ O and T B denotes the partial transposition with respectto subsystem B . For pure states with Schmidt decomposition as in Eq.(7) the negativity reads N ( | ψ cd (cid:105) AB ) = 12 ∞ (cid:88) n =max[ c,d ] α n ξ nc ξ nd √ P cd − . (9)Analogously, the negativity of the initial two mode squeezed vacuum state of Eq.(1) reads N ≡ N ( | ψ (cid:105) AB ) = λ − λ . (10)Thus, for a specific measurement outcome cd we can calculate the negativity difference N − N and check if it ispositive thus witnessing entanglement enhancement. However this is not enough to evaluate the performance of thescheme. We should also account for the outcome probability. Hence we define the efficiency as E := (cid:88) m P m ∆ N ( | ψ m (cid:105) ) N ( | ψ m (cid:105) ) , (11)where the index m runs overall possible measurement outcomes and ∆ N ( | ψ m (cid:105) ) is defined as∆ N ( | ψ m (cid:105) ) := (cid:26) N ( | ψ m (cid:105) ) − N if N ( | ψ m (cid:105) ) − N >
00 if N ( | ψ m (cid:105) ) − N ≤ . (12)Notice that only the measurement outcomes with increased negativities contribute to the efficiency. Furthermore itis 0 ≤ E ≤ IV. ITERATION OF THE PROTOCOL
In this Section we consider the iteration of the standard photon subtraction scheme as shown in Figure 1 (Right).This means to have N ancillary modes C i , i = 1 , . . . , N (resp. D j , j = 1 , . . . , N ) on arm A (resp. B ) and couple eachof them sequentially with A mode (resp. B mode).Suppose in the A (resp. B) arm c i (resp. d j ) photons are detected in the i -th (resp. j -th) ancilla and zero everywhereelse, then the resultant state can be straightforwardly computed following Eq.(7) | ψ c i d j (cid:105) AB = 1 (cid:112) P c i ,d j ∞ (cid:88) n =max[ c i ,d j ] α n ξ i + j − n, ξ n,c i ξ n,d j ξ N − in − c i , ξ N − jn − d j , ×| n − c i (cid:105) A | n − d j (cid:105) B , (13)where P c i ,d j is the probability of detecting c i photons in i -th ancilla in A arm and d j photons in j -th ancilla in B arm P c i ,d j = ∞ (cid:88) n =max[ c i ,d j ] ( α n ξ i + j − n, ξ n,c i ξ n,d j ξ N − in − c i , ξ N − jn − d j , ) . (14)Of course we should consider not only the photons detected in the i -th ancilla of A arm and j -th ancilla of B arm,but on all ancillary modes of the A arm as well as of the B arm. For the sake of convenience we put some limitsin our consideration of the number of detected photons. Actually we restrict our attention to λ ∈ (0 , . λ ) is quite low.Therefore from now on we will consider single photon detection in each arm. Then, we can divide the possibleoutcomes into two classes: symmetric having single photon detection in both arms ( (cid:80) Ni =1 c i = (cid:80) Nj =1 d j = 1) and asym-metric having single photon detection only in one arm (either (cid:80) Ni =1 c i = 0 , (cid:80) Nj =1 d j = 1 or (cid:80) Ni =1 c i = 1 , (cid:80) Nj =1 d j = 0).
1. Symmetric case
We can use Eq.(14) to evaluate the probability of detecting one photon in the i -th ancilla of A arm and one photonin the j -th ancilla of B arm. It results P i , j = (1 − T ) T − i + j λ (1 − λ )(1 + T N λ )(1 − T N λ ) . (15)Clearly the probability of detecting one photon in first ancillary modes ( i = j = 1) is the highest. The evaluation ofnegativity for single photon detected on each arm turns out to be equivalent to the evaluation of negativity for thestate of Eq.(13). Actually we have N ( | ψ i , j (cid:105) AB ) = T N λ (2 + T N λ + T N λ )(1 − T N λ )(1 + T N λ ) , (16)no matter on which ancilla the photon is detected. Notice that this negativity decreases by increasing N .
2. Asymmetric case
Using Eq.(14) we can get the probability of detecting one photon in i -th ancilla in one arm P i , = P , i = T − i + N (1 − T ) λ (1 − λ )(1 − T N λ ) . (17)The negativity of the state (13) corresponding to a photon detection in i -th ancilla in one arm results N ( | ψ i , (cid:105) AB ) = ( T N λ − λ T N (cid:104) Li − (cid:0) λT N (cid:1)(cid:105) − , (18)where Li − ( • ) is the polylogarithm function of order − . Also in this case the negativity does not depend on theoutcome and decreases with N .
3. Overall efficiency
According to the definition of Eq.(11), we write the overall efficiency of iteration protocol as E = N (cid:88) i,j =1 P i , j ∆ N ( | ψ i , j (cid:105) AB ) N ( | ψ i , j (cid:105) AB ) + 2 N (cid:88) i =1 P i , ∆ N ( | ψ i , (cid:105) AB N ( | ψ i , (cid:105) AB ) . (19)where ∆ N ( | ψ i , j (cid:105) AB and ∆ N ( | ψ i , (cid:105) AB are as defined in Eq.(12). The factor 2 in front of the second term comefrom the fact that | ψ i , (cid:105) AB = | ψ , i (cid:105) AB and P i , = P , i .Using Eqs.(15), (16) and Eqs.(17), (18) we get E as function of ( λ, N, T ) E = (1 − T N ) λ (1 − λ )1 − λ T N (cid:20) (1 − T N )(1 + λ T N )1 − λ T N E + 2 T N E (cid:21) , (20)with E = max (cid:104) , − (1 − λT N )(1+ λ T N ) T N (1 − λ )(2+ λT N + λ T N ) (cid:105) , (21) E = max , − λ − λ λ T N (1 − λ T N ) (cid:20) Li − ( λT N ) (cid:21) − T N λ . (22)In Figure 2 (Left) we show the loci in the N − T plane where E reaches its maximum for given values of λ . Interestinglythe maximum values taken by E are the same for fixed λ . This means that for a given value of λ the same efficiencycan be reached for many pair values of N − T including a pair with N = 1. FIG. 2: Loci of the points in the N − T plane giving maximum efficiency for iteration of the standard protocol (Left) andadaptive protocol (Right) when λ = 0 .
15 and 0 .
32. The values of efficiency are reported beside the points.
V. ADAPTIVE SCHEME
In this section we introduce a possible way to avoid the reduction of negativity increment by N while keeping highsuccess probability. The idea is to apply on each arm a beam splitter conditionally to no photon detection in theprevious one. Whenever there is a detection no more beam splitters will be applied afterwards. This can be donewith the help of some feed-forward control system.According to this strategy all the terms related to no photon detection will be eliminated from the coefficients of thestate in Eq.(13) and from the detection probability in Eq.(14) after a photon detection, namely the state in Eq.(13)becomes | ψ c i d j (cid:105) AB = 1 (cid:112) P c i ,d j ∞ (cid:88) n =max[ c i ,d j ] α n ξ i + j − n, ξ n,c i ξ n,d j | n − c i (cid:105) A | n − d j (cid:105) B , (23)with P c i ,d j now reading P c i ,d j = ∞ (cid:88) n =max[ c i ,d j ] ( α n ξ i + j − n, ξ n,c i ξ n,d j ) . (24)Limiting the maximum number of detected photons to 1, we have likewise the previous Section symmetric andasymmetric cases.
4. Symmetric case
Setting c i = d j = 1 in Eqs.(23), (24) we get P i , j = (1 − T ) T − i + j λ (1 − λ )(1 + T i + j λ )(1 − T i + j λ ) , (25)and the negativity of symmetric outcomes N (cid:0) | ψ i , j (cid:105) AB (cid:1) = (1 − T i + j λ ) − T ( i + j ) / λ ) (1 + T i + j λ ) − . (26)
5. Asymmetric case
In the same way, setting c i = 1 , d j = 0 in Eqs.(23), (24) yields P i , = P , i = T − i + N (1 − T ) λ (1 − λ )(1 − T i + N λ ) , (27)and the negativity of asymmetric outcomes N ( | ψ i , (cid:105) AB ) = (1 − T i + N λ ) λ T ( i + N ) (cid:104) Li − (cid:16) λT ( i + N ) / (cid:17)(cid:105) − . (28)
6. Overall efficiency
To evaluate the overall efficiency of adaptive scheme we can insert Eqs.(25), (28) and Eqs.(27), (28) into Eq.(19).However in this case is not possible to get a compact expression for it. Results from numerical evaluation are shown inFigure 2 (Right). In this case the maximum values taken by E for fixed value of λ are not the same, and monotonicallyincrease towards E = 1 for N → ∞ and T → N − T plane). VI. CONCLUSION
We have considered the efficiency of entanglement concentration by photon subtraction defined as the average ofentanglement increments weighted by their success probability. While entanglement increment diminishes by iteratingthe protocol, its success probability auguments. It results that these two behaviors compensate each other so thatoptimal efficiency can be already reached in one step of the protocol.In contrast, iteration of photon subtraction becomes meaningful when no photon have been previously detected.Thus adaptive strategy can in principle enhances the efficiency up to one. Implementing it is challenging because lightshould be stored during the feedforward time, however quantum memories [13] are promising for achieving significantvalues of efficiency in this way. [1] C. Weedbrook, et al. , Rev. Mod. Phys. 84, 621 (2012).[2] S. Pirandola, and S. Mancini, Laser Physics , 1418 (2006).[3] M. D. Reid, Phys. Rev. A , 062308 (2000).[4] M. Ban, J. of Opt. B: Quant. and Semiclassical Opt. , L9 (1999).[5] D. G. Welsch, S. Scheel, and A. V. Chizhov, arXiv:quant-ph/0105111 (2001).[6] C. H. Bennett, et al. , Phys. Rev. Lett. , 722 (1996).[7] C. H. Bennett, et al. , Phys. Rev. A , 2046 (1996).[8] J. Eisert, S. Scheel, and M. B. Plenio, Phys. Rev. Lett. , 137903 (2002).[9] A. Kitagawa, et al. , Phys. Rev. A , 042310 (2006).[10] S. L. Zhang, and P. van Loock, Phys. Rev. A , 062316 (2010).[11] A. Peres, Quantum Theory: Concepts and Methods , Kluwer Academic Publishers, New York (2002).[12] G. Vidal, and R. F. Werner, Phys. Rev. A , 032314 (2002).[13] C. Simon, et al. , Eur. Phys. J. D58