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Gaussian private quantum channel with squeezed coherent states Kabgyun Jeong1, Jaewan Kim1 & Su-Yong Lee 2

Received: 28 January 2015 Accepted: 12 August 2015 Published: 14 September 2015

While the objective of conventional quantum key distribution (QKD) is to secretly generate and share the classical bits concealed in the form of maximally mixed quantum states, that of private quantum channel (PQC) is to secretly transmit individual quantum states concealed in the form of maximally mixed states using shared one-time pad and it is called Gaussian private quantum channel (GPQC) when the scheme is in the regime of continuous variables. We propose a GPQC enhanced with squeezed coherent states (GPQCwSC), which is a generalization of GPQC with coherent states only (GPQCo) [Phys. Rev. A 72, 042313 (2005)]. We show that GPQCwSC beats the GPQCo for the upper bound on accessible information. As a subsidiary example, it is shown that the squeezed states take an advantage over the coherent states against a beam splitting attack in a continuous variable QKD. It is also shown that a squeezing operation can be approximated as a superposition of two dierent displacement operations in the small squeezing regime.

Te notion o private quantum channel (PQC) or quantum one-time pad 1  is very useul in quantum inormation processing, such as superdense coding 2, quantum data hiding 3, quantum state sharing protocol4 (or improving their efficiency), and the proo o additivity counter-example o the classical capacity on quantum channels 5,6. Te PQC is briefly introduced as ollows. I the two communicating parties, Alice and Bob, share a classical secret key (e.g., via quantum key distribution procedure), then PQC can be used to transmit an arbitrary unknown quantum state rom Alice to Bob securely. Te intermediate state in PQC is close to the maximally mixed state, so the state exhibits almost maximum entropy. Te receiver Bob always decrypts the encoded state by using the unitary inverse operations rom the pre-shared secret key, whereas no third party (not having the key) can obtain the original quantum state. Private quantum channel which belongs to a completely positive and trace preserving-map, represents the transormation o any quantum states into the maximally mixed state. It is different rom the private capacity o quantum channels 7–9 that is the maximally transmitted rate o classical secret inormation on quantum channels. A discrete version o private quantum channel was first proposed by Ambainis et al.1 in 2000, and the optimality o PQC was proved that we need exactly d 2 unitary operations to encrypt 10,11 a d -dimensional -dimensional quantum state 10, . In the case o approximate encryption, it is sufficient to have the 3,12 ,13 number o unitary operations being less than d  log  log d 3,12, . Ten, it is natural to ask how we can realize the PQC in continuous variable (CV) systems. Previously Brádler proposed CV private quantum channel (PQC) using coherent states that are obtained by displacement operations on the vacuum state 14, where he defined a CV maximally mixed state in Gaussian regime and then constructed GPQC via the conformation method conformation method o coherent states. Generally a single-mode Gaussian state is parametrized as a combination o displacement, squeezing operations and a thermal field15. Specifically squeezed states, which were considered in CV quantum key distribution 16–20, are crucial or a security demonstration o quantum key distribution using coherent states 21. Moreover squeezed 22,23 coherent states are useul or enhancing the security o quantum cryptography 22, , and or improving 24 phase sensitivities o intererometers . In this paper, we generalize the Gaussian private quantum channel (GPQC) with a combination o displacement and squeezing  operations.   operations. Explicitly, we construct GPQC in terms o the displacement and 1

School of Computational Sciences, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun, Seoul 130-722, Korea. 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore. Correspondence and requests for materials should be addressed to K.J. (email: [email protected]) [email protected]) or J.K. (email: [email protected] (email: [email protected])) or S.-Y.L. (email: [email protected] [email protected])) SCIENTIFIC

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− tanh r  ⋅ cos (2θpq − φ) }   whereas Brádler’s GPQC is repre sented only by the displacement element, exp (− r  p2 ). Ten, we study a subsidiary example o GPQC with

{



squeezed coherent states (GPQCwSC), especially or an eavesdropping attack. In the limit o small squeezing, urthermore, we show that the squeezed coherent states can approach a non-Gaussian regime by replacing the squeezing operation with a non-Gaussian operation, i.e., a superposition operation o two different displacements.

Gaussian private quantum channel (GPQC): coherent states Gaussian private quantum channel (GPQC) was introduced by Brádler in 2005, where he defined a maximally mixed state as �b in Gaussian regime 14. Similarly to the discrete case (identity over the dimension: ), the CV maximally mixed state in phase space has a broad Gaussian shape (because equiprob�/d ), able mixture depends only on the radius at some boundary). Brádler’s main proposition is that the Hilbert-Schmidt Hilbert-Schmidt distance d HS between the CV maximally mixed state and PQC-encryption o arbitrary coherent states is very close or sufficiently large N  (N   ( N : number o input displacement operations),

where Γ N  denotes the mixture o all conormations o coherent states that will be defined in Eqs (3) and (4). Also note that d HS (ρ1, ρ 2) : = tr (ρ1 − ρ 2)2   or any matrices ρ 1,2 1,2  and it is symmetric, d HS (ρ1, ρ 2) = d HS (ρ 2, ρ1).  By using the unitary invariance o the distance, we can prove the statement on an arbitrary coherent state β  : or β    and CV private quantum channel   N , where is a displaced CV maximally mixed state rom �b to the position o β  . Te proo is a bit complex but straightorward (See details in Re. 14). 14). Now we review the (Brádler’s) CV maximally mixed state 14. A CV maximally mixed state can be chosen as an integral perormed over all possible single mode states within the boundary circle o radius r ≤ b  in a coherent state α . I r > b, the occurrence probability is 0. Te coherent state is created by  ˆ† ˆ applying the displacement operator ˆ (α) = e αa − α a  to the vacuum state as 0 n 2 α − α / 2 ∞ ˆ ( α) 0 = e n . Ten, we have the CV maximally mixed state α =D ∑ n= 0 n!

where the normalization constant is C = πb2. Te purpose o GPQC is to encrypt an input coherent state into a high entropy state. Tus the encryption should be close to the maximally mixed state in Hilbert-Schmidt distance. In order to do that, we introduce a notion o conormation through vacuum displacements. Note that α pq = r p e iθ pq  or π  p  and q  are positive integers. For some fixed  p,  p, an input coherent state is θ pq  = (2q − 1), where  p   p

m

iθ pq ∞ (r p e ) |m⟩. From the Re. 14, 14, the general and slightly mod∑ m =0 m !

iθ pq

−r p2/ 2

=

1  p |α pq ⟩ ⟨α pq |  p q= 1

=

m +n e−r  p  p ∞ r  p e i θ pq (m−n) m n  p q=1 m,n =0 m! n!

=

m +n  p 2πi e−r  p ∞ r  p q (m− n) − iπ (m −n) e  p e  p m n  p m,n= 0 m! n! q=1

described by |α pq ⟩ = |r p e ⟩ = e ified p ified  p--conformation ( p ∈ +) is given by the ollowing equations ρ p

∑ 2

∑∑

2

= e



−r  p2





m,n =0

= e

−r  p2



r  pm+n m! n! ∞



m,n =0

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r  pm+n m! n !

m −n  p δ  m =n (mod p )

m n

m n δ m,n (mod p ),

,

( 3)

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where  pq= 1 e  p q (m− n) = p or m = n mod p  mod p,, and 0 or otherwise. Te Eq. (4) is ollowed by the absorption o the phase term into m ’s and n ’s. Tis is equivalent to the Brádler’s original  p-conormation.  p-conormation. Te conormation technique provides an equiprobable positioning o vacuum states at some fixed radius r  p, so the uniormity o the distribution o CV quantum states is strengthened. Finally we review the mixture o all  p-conormations  p-conormations ( p = 1, …, N ). ). Suppose that N ≥ 1 and define ( p − 1) b ≤ b, then r p = 2πi



Γ N  =

1 N   pρ  M  p= 1  p



=

1 N   p ˆ ˆ  † (α ) , D (α pq ) 0 0 D pq  M  p= 1 q =1

∑∑

(5 )

where  M = N (N  + 1)/2. As an encrypted state o GPQC, the Γ N   represents the output state o PQC over (uniormly chosen) M  chosen)  M  unitary  unitary operations, where the input state is in vacuum state 0 0 . One o the  M  CV   CV states is fixed by pre-shared classical secret key between Alice and Bob as (classical) one-time pad, and then it is sent to Bob. Te Brádler’s proposition states that Γ N  is sufficiently close to the CV maximally mixed state. Encoding an arbitrary coherent state β   is essentially equivalent to the vacuum state encryption or the unitary invariance o the distance: . Also † note that, or any completely positive and trace-preserving (CP) map   N ,   N ( β β ) = Dˆ (β ) ΓN Dˆ (β )   (See Eq. (9) in Re. 14). 14). Tereore we derive Brádler’s main result as Eq. (1) by combining the above properties o CV maximally mixed state,  p-conormation,  p-conormation, and its mixture.

Results Gaussian private quantum channel: squeezed coherent states. o construct a GPQCwSC, we examine a single-mode squeezed vacuum state. A single-mode squeezing operation is defined by  ξ aˆ 2 − ξ aˆ †2 , where ξ = reiφ. When we apply the squeezing operator to the vacuum state, we Sˆ (ξ ) = exp  



2



produce a squeezed vacuum state such that ξ , 0 :

=



1

∑ r 

cosh

n =0

(2n) ! ( −e iφ tanh r )n 2n . n 2 n!

ˆ (α) Sˆ (ξ )  0 ) Ten, applying a displacement operation, we obtain a coherent squeezed state ( α , ξ = D 1 2 25 which orms an overcomplete set, i.e., ∫ d  α α, ξ α, ξ =   � . It is a main ingredient o the squeezed π CV conormation. For simplicity, we consider a squeezed coherent state, instead o the coherent squeezed state. It is reasonable that squeezed coherent states are transormed into coherent squeezed states by the relation, ˆ (α) 0 = D ˆ (α cosh r − αe iφ sinh r ) Sˆ (ξ ) 0 25 . Generally, a squeezed coherent state represents Sˆ (ξ ) D squeezing o a coherent state 26,

ξ, α

= Sˆ ( ξ ) Dˆ (α) 0  1 2 (ν /2coshr )m/ 2 ν α 2     =  exp  −  α − cosh r   cosh r ⋅ m!  2    α × H m  m,  2ν  cosh r  

( 6)

where ν  : = e iφ sinh r   and φ = arg(ξ ), ), the argument o the squeezing parameter ξ . H m(·) denotes the mth-degree complex Hermite polynomials. By exploiting the Eq. (6), we can derive a squeezed conormations and its mixture in the ollowing section. Ten we prove that, or sufficiently large N   and or any squeezing o a coherent state β  , there exists a CP map     such that (See ollowing second section.)

General squeezed conformations. Now, we show the explicit calculation o the squeezed  p-conormation.  p-conormation. Let us apply the squeezing operation to the coherent state (|α pq ⟩ ), Sˆ (ξ ) |α pq ⟩

=





m= 0

(

m/ 2 ν  2coshr 

)

cosh r ⋅ m!

 ν α 2  2 − 12  α pq −  pq   cosh r  e 

α pq   × H m  m,  2ν  cosh r  

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(a)

(b)

(c)

(d )

Figure 1. For some fixed r  and r  p, the argument 0 ≤ φ < 2π) of  ξ  depends on θ pq ∀ q ∈ +. Tis figure represents the squeezed 16th-conormation (r  ( r  p = r 16) in the actor K = 1 − tanh r  ⋅ cos (2 θ pq − φ):  π (a) non-squeezed (r  ( r = 0), (b) φ = 0, (c) φ = ± , and (d) φ  = − π  cases, respectively. 2 4

sinhr  ( p   p  and q  are positive integers). Using the ollowing relations: = r pe iθ pq  and v = eiφsinhr   2 2 2 i (2θ pq− φ) + e−i (2θ pq−φ) ) = 2r p2 sinh r cos (2 θ pq   − φ ) and 2 sinh r · cosh r  ν α pq + να pq = r p sinh r (e = sinh (2r ), then we derive the ormula where α pq



Sˆ (ξ ) |α pq ⟩ ⟨α pq |Sˆ ( ξ )  =

(

∑ cosh m,n

(m+ n) / 2

)

tanh r  2

2

r m ! n!

e iφ (m−n)/ 2 ⋅ e−Kr p

 r e i (θ pq− φ2 )   p  H (c . c .) m n , × H m   n  sinh (2r ) 

 

(8 )

where K : = 1 − tanh r  ⋅ cos (2θ pq − φ) and c.c. denotes the complex conjugate o the argument o H m. Te definition o K   determines the position o squeezed coherent states and the squeezing angles, as shown in Fig. 1. 1. Ten, we can find the squeezed  p-conormation  p-conormation ρ pξ 

=

† 1  p ˆ S (ξ ) |α pq ⟩ ⟨α pq |Sˆ (ξ )    p q =1

=





∑ κm,n ⋅ e−r {1− tanh r ⋅ cos(2 2  p

θ pq− φ)

}m n,

m,n =0

where θ pq 

=  pπ (2q − 1) and the constant κm,n is defined by  κm,n : =

(m+ n )/ )/ 2 φ 1  p (tanh r / 2) exp i (m  2  p q =1 cosh r m!n!



 r e i (θ pq− φ2 )   p × H m   sinh (2r )

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  H (c. c.).  n 

 − n)  

(10)

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For some fixed squeezing r  and   and the argument φ, the (complex) Hermite polynomials are orthogonal to each other or m ≠ n such that the value o κm,n becomes a constant. Te κm,n converges to r pm +n/ m !n! as r → 0. Tereore, the actor or some  p,  p, exp  − r p2 1 − tanh r  ⋅ cos (2θ pq − φ) , is the main compo-

{



}

nent in Eq. (9). We finally consider the mixture o all squeezed  p-conormations  p-conormations or 0 ≤ p ≤ N . Suppose N ≥ 1 and ( p − 1) b define r p = ≤ b, then N 

Γ ξ N  =

1 N   p ˆ ˆ (α ) 0 0 D ˆ † (α ) Sˆ † (ξ ) ,   S (ξ ) D  pq pq  M  p=1 q = 1

∑∑

(11)

where  M = N (N + 1 )/ )/2. Alice equiprobably chooses one rom the set o  M   displacement operators †  ˆ ˆ α a α a − ˆ (α ) = e  pq pq  and the squeezing parameter r > 0. (Once again note that, or some fixed  p and  p and r , D  pq the squeezing argument φ depends on θ pq or all q.) Alice sends the encrypted state through a quantum channel towards Bob who perorms the inverse operations to decrypt the state. Te point is that we encrypt an arbitrary input state, i.e., an arbitrary coherent state ( β  ). Ten, we can write down a general encryption CP map    with M   with  M  unitary  unitary elements as in Re. 14   N (ξ, β

β

) =

1 N   p ˆ ˆ (α ) D ˆ (β ) 0 0 D ˆ † (β )D ˆ † (α ) Sˆ † (ξ )   S (ξ ) D  pq pq  M  p=1 q = 1

=

1 N   p ˆ ˆ (β ) D ˆ (α ) 0 0 D ˆ † (α ) D ˆ † (β )Sˆ † (ξ )   S (ξ ) D  pq pq  M  p=1 q = 1

∑∑

∑∑





= Sˆ (ξ ) Dˆ ( β ) ΓN Dˆ (β ) Sˆ (ξ ) .   From the above equation, we propose that the corresponding Hilbert-Schmidt (HS) distance is equivalent to one o Eq. (11), o the HS distance, whereas the states are not the same as   N (ξ, β β  )

by the unitary invariance

≠Γ

ξ  . N 

The proof of the main proposition and the number of secret bits. Here we prove our main proposition. Te proposition is as ollow: For sufficiently large N  in   in any squeezing o an arbitrary coherent state β  , there exists CP map   N  such that



N −2

+ O (N −4) ,

(13)

where the HS distance between �b  and Γ ξ N   becomes quite close in sufficiently large N . Te Eq. (12) is obtained rom the unitary invariance o the HS distance. Te Eq. (13) is derived via the unitary invariance o squeezing operations in the HS distance (Eq. (14) below) and it is ollowed by the norm convexity (Eq. (15)). Explicitly speaking, in the case o ξ > 0, we assert that (N   1) dHS (Γ ξ N ,

( (Sˆ (ξ) 0



ΓN ) = dHS Sˆ (ξ ) ΓN Sˆ (ξ ), ΓN = dHS



)

 

0 Sˆ (ξ ), 0 0

)



0,

(14)

where the second equality also holds by the unitary invariance in the HS distance, i.e., or all unitary † † † ˆˆ   ˆ ˆ, Uˆ : = SD Uˆ ′ : = DS and dHS (Uˆ 0 0 Uˆ , Dˆ 0 0 Dˆ ) = dHS (Uˆ ′ 0 0 Uˆ ′ † , Dˆ 0 0 Dˆ ) = dHS † (Sˆ 0 0 Sˆ , 0 0 ). In general, the last equality is not exactly equal to zero, but, asymptotically converges † 2sinh (r / 2) 27 to 0, i.e., d HS (Sˆ 0 0 Sˆ , 0 0 ) =  0 . cosh r  Tereore, by using the norm convexity and the above equations (within the symmetric property o the HS distance) we derive

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Tus, it implies that �b approximately equals to the sum o the squeezed coherent states, and thereore completes the proo. In addition, we mention the total number o unitary operations L and corresponding secret bits. Te number o total displacement is  M = N (N + 1 )/ )/2  and just one (pre-fixed) squeezing operation is required. From this reason, L = M + 1. Tus, we have the number o secret bits o  = log L ~ 2logN  or  1. It is interesting to note that i we use the approximate random unitary channels such as in Res. 3,12, 12,13, 13, then it is expected to construct PQC with only about  -bits o secret keys. Tere is no 2 advantage in the key efficiency, but the accessible inormation can be slightly improved as ollow.

Holevo bound on the GPQCwSC. One o important principles o the von Neumann entropy states that quantum operations never increase the quantum mutual inormation. By using this property, we propose that our GPQCwSC is stronger (i.e., tight upper bound) than the Brádler’s GPQC in the language o accessible inormation. Formally, Brádler’s protocol14  with coherent states consists o a set o where   denotes   denotes the set o all coherent states β  ,  p  is the probability distribution o  ,   N ( ⋅ ) is the CP map with N  displacement operations, and is the (displaced) (displaced ) CV maximally mixed state. Similarly, let us express our GPQCwSC as a set o where squeezing elements are added and the set    emphasizes the squeezing with displacement operations. Ten we assert that (β : = β β  )

where the Holevo inormation χ : = max   I acc (B : E ). Te B and E are corresponding to input and output distributions o the channel   N ( ⋅ ) between Bob and Eve, and the maximum o the accessible inormation (by Eve) is taken over all input ensemble   in the channel. Note that the quantum mutual inormation is defined by I (A : B) = S (A) + S (B) − S (AB)  or any quantum system  A   A  and B, where S(σ ) = − trσ log σ  is the von Neumann entropy. Tis act directly comes rom ‘the principle o quantum operation’ about the entropy: For any quantum operation Q, Iacc (Q (B) : Q (E ) ) ≤ Iacc (B : E ). I we ˆ (  ) and E : = D ˆ (  ) or some ensembles   substitute Q to a squeezing operation Sˆ , and define : = D 1 2 1 and  2, respectively, then we have ˆ ( ) : D ˆ ( ) ) ) I acc (Sˆ (D 1 2

≤ I acc (Dˆ ( 1 ): Dˆ ( 2 ) ).  

(17)

Tis provides a better upper bound on the accessible inormation χ  than the Brádler’s analysis. In other words, the amount o eavesdropping inormation on the encrypted state via the GPQCwSC is less than that by the Brádler’s GPQC.

Subsidiary example of GPQC. We introduce a simple example that squeezed coherent states can take an advantage over coherent states in CV quantum key distribution, where the scheme is in a preliminary procedure o GPQC. o distribute quantum keys, we consider the BB84 protocol 28. In discrete  variable systems, Alice and Bob share keys with single photon states in mutually unbiased bases. In continuous variable (CV) systems, correspondingly, Alice and Bob share keys with Gaussian states in uncertainty relation o field quadratures. Ten, in the limit o small squeezing, we show that the squeezed coherent state scheme can approach even a non-Gaussian regime by replacing a squeezing operation with a superposition operation o two different displacements. Simple eavesdropping attack in CV quantum key distribution. As a simple eavesdropping attack, we assume that Eve perorms a beam splitting attack. As an input state, we compare a squeezed coherent state with a coherent one. For an input squeezed coherent state, Eve transorms the input state by a 50:50 beam splitter, ˆ (α) 0 Bˆ BE Sˆ B (ξ ) D B

B

0

E

 ξ   ξ   ξ   α   α  = Sˆ B   Sˆ E   Sˆ BE   Dˆ B   Dˆ E   0 B 0 E ,  2   2   2   2   2 

(18)

where the subscript B (E) represents Bob (Eve), and the transormation o the squeezing operation is given by Re. 29. 29. When Eve perorms a measurement to get an inormation o the input state (| ⋅ ⟩E ), the state | ⋅ ⟩B ) sent to Bob is disturbed by the non-local effect o the two-mode squeezing operation Sˆ BE ξ  , 2 except the uncertainty o the field quadrature. For an input coherent state, there is no non-local effect afer the beam splitting attack. For the beam splitting attack, thus, Alice and Bob detect the existence o Eve much easier with the input squeezed coherent state than the input coherent one.

()

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Figure 2. Optical implementation for a superposition operation of two different displacements. Te beam splitter is highly reflective.

Non-Gaussian regime. We show that the squeezed coherent state can approach even a non-Gaussian regime. In the limit o small squeezing, we describe a non-Gaussian regime by a truncation o the squeezξ ξ  ing parameter, Sˆ (ξ ) ≈ 1 − aˆ † 2 + aˆ 2. However the truncation operation is not implemented by 2 2 reducing the squeezing parameter in experiment. In order to apply the truncation operation to coherent states, we consider a superposition operation o two different displacements. Since an even coherent state is quite similar to a squeezed vacuum state, we derive the corresponding parameters in the limit o r , β  2  1, 2

β

+ −β  2

+ e−2 β  )

2 (1

ξ, 0

≈1−

β 2 r  cos (2φ

− ϕ), (19)

where β = β  e iϕ  and ξ = reiφ. When ϕ = 2φ ± π /2, the even coherent state is approximated to the squeezed vacuum state. Note that, or β  2  1, the even coherent state is close to a Gaussian state but it is a non-Gaussian state 30. Tereore, the variables r  and   and φ in the squeezing parameter can be replaced by the ones β  and   and ϕ in the even coherent state. We need to know i the uncertainties o the field quadratures are maintained by replacing the squeezed  vacuum state with the even coherent coherent state. Because CV quantum quantum key distribution distribution is secured via uncer16–18 ˆ ˆ −iθ + aˆ †e iθ )/ 2, we derive the tainties o field quadratures . Using the quadrature operator  X θ =  ( ae quadrature variance o the squeezed vacuum state as

∆ X θ2

sv 

= ≈

1 [cosh (2r ) − sinh (2r ) cos (2θ 4 1 [1 − 2r  cos (2θ − φ) ] , 4

− φ )] (20)

where the quadrature variance is approximated or r   1. According to the phase parameter φ, the quadrature variance oscillates between (1 − 2r )/ 4 and 1 + 2r )/ 4. Te quadrature variance o the even coherent state is given by 



 X θ2

ec

=



1 1 4 

+βe

1 [1 4

+2

2

−2iθ

+ β  e

 2 2iθ

+





β 2 cos (2θ

2 − e−2 β  )   2  1 + e−2 β 

2 β  2 (1

− 2ϕ ) ] ,

(21)

where the quadrature variance is approximated or β  2  1. According to the phase parameter ϕ, the quadrature variance oscillates between (1 − 2 β  2 )/ 4 and (1 + 2 β  2 )/ 4. For the quadrature variances, thus, β  2  corresponds to r . Tereore, we find that the uncertainties o the field quadratures are maintained in the substitution o the even coherent state or the squeezed vacuum state. Note that, or the beam splitting attack, the even coherent state also plays a role o a squeezing operator by generating an entangled state with a beam splitter. Now we see how to realize the non-Gaussian operation with an optical implementation o a superposition operation o two different displacements, as shown in Fig. 2. 2. Previously the displacement operation was implemented by a beam splitter with high reflectivity 31, where the displacement amplitude is described with the multiplication (γ   )  o an amplitude o coherent lights ( γ ) and the transmission coefficient o the beam splitter (  ). ). In Fig. 2, 2, the superposition operation o two different displacements is implemented by a beam splitter with high reflectivity (  → 0), where  γ   represents β  in   in the superposition operation o two different displacements. Note that the input even coherent state can be generated by a nonlinear Kerr medium 32–34 in all-optical systems.

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Discussion We have constructed GPQCwSC by an equiprobable combination o squeezed coherent states in a continuous-variable regime generalizing GPQCo and shown that GPQCwSC tightens the upper bound on accessible inormation. We have also presented a simple intuitive understanding o the well-known act that the squeezed state scheme has better security than the coherent state scheme in continuous  variable QKD. QKD. A class o non-Gaussian non-Gaussian operations, operations, superpositions superpositions o two different different displacements, displacements, is shown to be an approximation o small squeezing operations. With these results, we pursue an all-optical implementation o PQC easible with available optical technology. As some non-Gaussian states are more robust against decoherence than Gaussian states 35–37, we look orward to investigating non-Gaussian quantum communications compared to GPQC with decoherence.

References 1. Ambainis, A., Mosca, M., app, A. & de Wol, �. Private quantum channels. IEEE 54th Annual Symposium on Foundations of Computer Science p. Science  p. 547 (2000), �edondo Beach, Caliornia. (2000, November 12–14). 2. Harrow, A., Hayden, P. & Leung, D. Superdense Coding o Quantum States. Phys. �ev. Lett. 92, 187901 (2004). 3. Hayden, P., Leung, D., Shor, P. W. W. & Winter, A. �andomi zing Quantum States: Constructi Con structions ons and Applications. Commun. Math. Phys. 250, 371 (2004). 4. Chi, D. P. & Jeong, �. Approximate Quantum State Sharings via Pair o Private Quantum Channels.  J. Quant. Quant. Info. Info. Sci. 4, 64 (2014). 5. Hayden, P. & Winter, A. Counterexamples Counterex amples to the Maximal p Maximal p-Norm -Norm Multiplicativity Multiplicativity Conjecture or all p all p >1. Commun. Math. Phys. 284, 263 (2008). 6. Hastings, M. B. Superadditivity o communication capacity using entangled inputs. Nature Phys. 5, 255 (2009). 7. Pirandola, S., García-Patrón, �., Braunstein, S. L. & Lloyd, S. Direct and �everse Secret-�ey Capacities o a Quantum Channel. Phys. �ev. Lett. 102,  050503 (2009). 8. a�eo�a, M., Guha, S. & Wilde, M. M. Fundamental rate-loss tradeoff or optical quantum �ey distribution. Nature Commun. 5, 5235 (2014). 9. Li, �., Winter, A., Zou, X. & Guo, G. Private Capacity o Quantum Channe ls is Not Additive. Phys. �ev. Lett. 103, 120501 (2009). 10. Nagaj, D. & �erenidis, I. On the optimality o quantum encryption schemes. J. schemes. J. Math. Phys. Phys. 47,  092102 (2006). 11. Bouda, J. & Ziman, M. Optimality o private quantum channels. J. channels.  J. Phys. A: Math. Teor. Teor. 40, 5415 (2007). 12. Dic�inson, P. A. & Naya�, A. Approximate �andomization o Quantum States With Fewer Bits o �ey. Quantum Computing Bac� Action, II �anpur, AIP �anpur,  AIP Conf. Proc. 864, 18 (2006), Springer, New Yor�. (2006, March 6–12). 13. Aubrun, G. On Almost �andomizing Channels with a Short �raus Decomposition. Commun. Math. Phys. 288, 1103 (2009). 14. Brádler, �. Continuous-variable private quantum channel. Phys. �ev. A 72,  042313 (2005). 15. Marian, P., Marian, . A. & Scutaru, H. Quantiying Nonclassicality o One-Mode Gaussian States o the �adiation Field. Phys. �ev. Lett. 88,  153601 (2002). 16. �alph, . C. Continuous variable quantum cryptography. Phys. �ev. A 61,  010303(�) (2000). 17. �alph, . C. Security o continuous-variable quantum cryptography. cryptography. Phys. �ev. A 62,  062306 (2000). 18. Hillery, M. Quantum cryptography with squeezed states. Phys. �ev. A 61, 022309 (2000). 19. Gottesman, D. & Pres�ill, J. Secure quantum �ey distribution using squeezed states. Phys. �ev. A 63,  022309 (2001). 20. Cer, N. J., Lévy, M. & Van Assche, G. Quantum distribution o Gaussian �eys using squeezed states. Phys. �ev. A 63,  052311 (2001). 21. Grosshans, F. et al. Quantum al. Quantum �ey distribution using gaussian-modulated coherent states. Nature 421, 238 (2003). 22. Lu, Y. J., Zhu, L. & Ou, Z. Y. Security improvement by using a modified coherent state or quantum cryptography. Phys. �ev. A 71, 032315 (2005). 23. Yin, Z.-Q., Han, Z.-F., Sun, F.-W F.-W.. & Guo, G.-C. Decoy Dec oy state quantum �ey distributi on with modified cohe rent state. Phys. �ev. A 76, 014304 (2007). 24. Caves, C. M. Quantum-mechanical noise in an intererometer. Phys. �ev. D 23, 1693 (1980). 25. Barnett, S. M. & �admore, P. M.  Methods in Teoretical Teoretical Quantum Optics. Optics. Oxord University Press (1997). 26. Vogel, W. & Welsch, D.-G. Quantum Optics. Optics . WILEY-VCH Verlag GmbH & Co. �GaA (2006). 27. Dodonov, V. V., Man’�o, O. V., Man’�o, V. I. & Wünsche, A. Energy-sensitive and “lassical-li�e” Distances between Quantum States. Phys. Scr. 59, 81 (1999). 28. Bennett, C. H. & Brassard, G. Quantum cryptography: Public �ey distribution and coin tossing. Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing , pp. 175–179, 8 (1984), Bangalore, India. IEEE Computer Society Press, New Yor�. (1984, December 10-12). 29. �im, M. S., Son, W., W., Buže�, V. V. & �night, P. P. L. Entanglement by a beam splitter: Nonclassi cality as a prerequisi te or entanglement. Phys. �ev. A 65, 032323 (2002). 30. Genoni, M. G. & Paris, M. G. A. Quantiying non-Gaussianity or quantum inormation. Phys. �ev. A 82, 052341 (2010). 31. Lvovs�y, Lvovs�y, A. I. & Babichev, S. A. Synthesis and tomographic characterization characterization o the displaced Foc� state o light. Phys. �ev. A 66, 011801(�) (2002). 32. Mecozzi, A. & ombesi, P. P. Distinguishable quantum states generated via nonlinear bireringence.Phys. bireringence. Phys. �ev. Lett. 58, 1055 (1987). 33. Yur�e, B. & Stoler, D. Quantum behavior o a our-wave mixer operated in a nonlinear regime. Phys. �ev. A 35, 4846 (1987). 34. Gerry, C. C. Generation o optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder intererometer containing a �err medium. Phys. �ev. A 59, 4095 (1999). 35. Sabapathy, �. �., Ivan, J. S. & Simon, �. �obustness o Non-Gaussian Entanglement against Noisy Amplifier and Attenuator Environments. Phys. �ev. Lett. 107,  130501 (2011). 36. Lee, J., �im, M. S. & Nha, H. Comment on “�ole o Initial Entanglement and Non-Gaussianity in the Decoherence o PhotonNumber Entangled States Evolving in a Noisy Channel”. Phys. �ev. Lett. 107, 238901 (2011). 37. Nha, H., Lee, S.-Y., Ji, S.-W. S.-W. & �im, M. S. Efficient Entanglement Criteria beyond Gaussian Limits Using Gaussian Measurements. Measurements. Phys. �ev. Lett. 108,  030503 (2012).

Acknowledgements We are grateul to K. Brádler or comments. Tis work was partly supported by the I R&D program o MOIE/KEI [10043464]. S.Y.L. acknowledges support rom FQXI and the National Research Foundation and Ministry o Education in Singapore.

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Author Contributions K.J. identified and proposed the study. All the authors carried out the calculations and the analysis o results.

Additional Information Competing financial i nterests: nterests: Te authors declare no competing financial interests. How to cite this article : Jeong, K. et al. Gaussian al. Gaussian private quantum channel with squeezed coherent states. Sci. Rep. 5, 13974; doi: 10.1038/srep13974 10.1038/srep13974 (2015).

Tis work is licensed under a Creative Commons Attribution 4.0 International License. Te images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; i the material is not included under the Creative Commons license, users will need to obtain permission rom the license holder to reproduce the material. o view a copy o this license, visit http://creativecommons.org/li http://creativecommons.org/licenses/by/4.0/ censes/by/4.0/

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