aa r X i v : . [ qu a n t - ph ] A ug coherence of quantum Gaussian channels Jianwei Xu ∗ College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China (Dated: August 15, 2019)Coherence is a basic notion for quantum states. Instead of quantum states, in this work, Weestablish a resource theory for quantifying the coherence of Gaussian channels. To do this, wepropose the definitions of incoherent Gaussian channels and incoherent Gaussian superchannels.
PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Aa
I. INTRODUCTION
Coherence is a fundamental ingredient for quantumphysics and an important resource for quantum informa-tion science. In the past few years there have been manyworks focusing on the coherence of quantum states andhave achieved fruitful results both in theories and appli-cations (recent reviews see [1, 2] and references therein).However, the past research for coherence is only for quan-tum states. Quantum information processings inevitablyinvolve quantum channels, then it is necessary to considerthe coherence of quantum channels. From a higher per-spective, quantum states can be regarded as degeneratedquantum channels [3], or we call a resource possessed inquantum states as static resource while a resource pos-sessed in quantum channels as dynamic resource [4]. Re-cently many researchers begin to exploit the coherence ofquantum channels [5–11].All these research about coherence both for quan-tum states or quantum channels mainly aims to thefinite-dimensional systems setting. When we considerthe case of infinite dimensional systems, we will en-counter many difficulties. Firstly, a physical quantumstate in an infinite-dimensional system should be con-vergent in some sense. Furthermore, the expressions forfinite-dimensional systems often become hard to evalu-ate for infinite-dimensional systems even they are conver-gent such as the relative entropy of coherence for quan-tum states [12]. Quantum Gaussian states are a classof infinite-dimensional states which play significant rolein quantum optics and in quantum information theory(reviews see e.g. [13, 14]) with thermal states, coher-ent states and squeezed states as special cases of Gaus-sian states. Till now the coherence of quantum Gaussianstates has been discussed in many works e.g. [15–20].In this work, we turn to study the coherence of Gaus-sian channels under the quantum resource theory (QRT)[21]. This paper is organized as follows. In section II, wegive the definitions of incoherent Gaussian channels andincoherent Gaussian superchannels, we also explore thestructures of them. In section III, we establish a QRTfor quantifying the coherence of Gaussian channels, and ∗ Electronic address: [email protected] explicitly provide such a measure. Section VI is a briefsummary. For clarity of the structure and easy reading,we postpone most necessary proofs for this work to theAppendix part.
II. INCOHERENT GAUSSIAN CHANNELSAND INCOHERENT GAUSSIANSUPERCHANNELS
Let {| j i} ∞ j =0 be an orthonormal basis of Hilbert space H, we call the basis {| j i} ∞ j =0 Fock basis and call eachstate | j i Fock state. Note that the notion of coherence isdependent on the chosen orthonormal basis, so we alwayssuppose the Fock basis is fixed, or call it reference basis.When we consider the n -fold tensor Hilbert space H ⊗ n , we adopt the tensor basis ( {| j i} ∞ j =0 ) ⊗ n as its referencebasis.For a state ρ on H ⊗ n , its characteristic function χ ( ρ, λ )is defined as χ ( ρ, λ ) = tr [ ρD ( λ )] , (1) D ( λ ) = Π nj =1 D ( λ j ) , (2) D ( λ j ) = exp( λ j a † j − λ ∗ j a j ) , (3)where λ = ( λ x , λ y , λ x , λ y , ..., λ nx , λ ny ) t ∈ R n , λ j = λ jx + iλ y , λ jx , λ y ∈ R , a j , a † j are the annihilation andcreation operators acting on the j th Hilbert space H, and*, t , † represent the complex conjugation, transpositionand Hermitian conjugation. Notice that we use both no-tations λ j = ( λ jx , λ jy ) and λ j = λ jx + iλ y according todifferent contexts.A state ρ on H ⊗ n is called a Gaussian state if its char-acteristic function χ ( ρ, λ ) has the form χ ( ρ, λ ) = exp[ − λ t Ω V Ω t λ − i (Ω d ) t λ ] , (4)where d = ( d x , d y , d x , d y , ..., d nx , d ny ) t ∈ R n is called displacement vector, d j = d jx + id jy , d jx , d jy ∈ R , Ω = ⊕ nj =1 ω , ω = (cid:18) − (cid:19) , V is a 2 n × n real symmetric matrix called covariance matrix satisfyingthe uncertainty relation [22] V + i Ω ≥ . (5)We see that a Gaussian state ρ is completely describedby ( V, d ), then we denote ρ = ρ ( V, d ), and also wedenote the set of all Gaussian states on H ⊗ n by GS n . Notice that GS n is not a convex set. Definition 1.
We call a Gaussian state incoherentGaussian state if it is a thermal state.This definition comes from the fact that the incoher-ent state is defined as the diagonal state in reference basis[12], and the diagonal Gaussian states are just the ther-mal states (an explicit proof see [16]). We call the set ofall incoherent Gaussian states as
IGS n , that is IGS n = {⊗ nj =1 ρ th ( ν j ) | ν j ≥ ∀ j } , (6) ρ th ( ν j ) = 2 ν j + 1 ∞ X k =0 ( ν j − ν j + 1 ) k | k ih k | . (7)A quantum channel is defined as a linear map fromquantum states into quantum states with the conditionsof complete positivity and trace preservation [23]. Nowwe say that a quantum channel is Gaussian when it trans-forms Gaussian states into Gaussian states. A Gaussianchannel φ on GS n can be described by φ ( T, N, d ) it actson ρ ( V, d ) ∈ GS n as [24] d → T d + d, (8) V → T V T t + N, (9)where d ∈ R n is a displacement vector, while T and N = N t are 2 n × n real matrices, which must satisfythe complete positivity condition N + i Ω − iT Ω T t ≥ . (10)When N = 0, and T is a symplectic matrix, i.e. T Ω T t = Ω , (11)we call such channel unitary Gaussian channel.We denote the set of all Gaussian channels φ ( T, N, d )on GS n by GC n . Definition 2.
A Gaussian channel φ ∈ GC n is calledincoherent, if φ ( ρ th ) ∈ IGS n for ∀ ρ th ∈ IGS n . We denote the set of all incoherent Gaussian channelsby
IGC n . We now determine the structure of
IGC n . Theorem 1.
A Gaussian channel φ ( T, N, d ) ∈ GC n isincoherent iff (if and only if) d = 0 , (12) T = { t j T j } nj =1 ∈ T n , (13) N = ⊕ nj =1 ω j I , (14) ω j ≥ | − X k : r ( k )= j t k det T k | , ∀ j, (15) where t j , ω j ∈ R , T j is a 2 × ∀ j ; I isthe 2 × T n denotes the set of all 2 n × n realmatrices such that for any matrix T ∈ T n , the (2 j − , j )two columns have just one 2 × t j T j with T j T tj = I , i.e., orthogonal, located in (2 r ( j ) − , r ( j ))rows for ∀ j , r ( j ) ∈ { k } nk =1 , and other elements are allzero.Note that the identity Gaussian channel φ ( T = I n , N = 0 , d = 0) ∈ IGC n , where I n is the 2 n × n identity.Superchannel is a completely positive linear map trans-forming channels into channels [3, 25]. Gaussian super-channel is hence defined as superchannel transformingGaussian channels into Gaussian channels. Let GSC n denote the set of all Gaussian superchannels acting on GC n . For φ ( T, N, d ) ∈ GC n , consider the Gaussian state ρ φ ( V, d ) ∈ GS n with V = (cid:18) T T t ch r + N T Σ n sh r Σ n T t sh r I n ch r (cid:19) , (16) d = (cid:18) d (cid:19) , (17)Σ n = ⊕ nj =1 (cid:18) − (cid:19) , (18)and r ∈ R, ch r = e r + e − r , sh r = e r − e − r . The factthat ρ φ is indeed a Gaussian state see e.g. section 5.5.2in [14]. ρ φ is called the Choi state of Gaussian channel φ. From this correspondence between φ and ρ φ , we provide acharacterization of GSC n . The idea is as follows. SupposeΦ ∈ GSC n , which transforms φ ∈ GC n as Φ( φ ) ∈ GC n ,then we will find a Gaussian channel which transforms ρ φ into ρ Φ( φ ) . The result is Theorem 2 below. Theorem 2.
A Gaussian superchannel Φ ∈ GSC n can be represented by Φ( A, O, Y, d ) , and for φ ( T, N, d ) ∈GC n , we have Φ[ φ ( T, N, d )] = ψ ( T ′ , N ′ , d ′ ) with T ′ = AT Σ n O t Σ n , (19) N ′ = AN A t + Y, (20) d ′ = Ad + d, (21)where A, O, Y are all 2 n × n real matrices, Y = Y t ,OO t = I n , d ∈ R n , and Y + i Ω − iA Ω A t ≥ , (22) i Ω − iO Ω O t ≥ . (23)We can also express any Gaussian superchannel interms of compositions of Gaussian channels, this is The-orem 3 below. Theorem 3.
A Gaussian superchannel Φ(
A, O, Y, d )can be represented by Φ( φ ) = φ ◦ φ ◦ φ for ∀ φ ∈ GC n with fixed φ ( T , N , d ), φ ( T , N , d ) ∈ GC n . One suchrepresentation is T = Σ n O t Σ n , N = 0 , d = 0; (24) T = A, N = Y, d = d. (25)To establish a QRT for coherence of Gaussian channels,we need to specify the definition of incoherent Gaussiansuperchannel. Definition 3.
A Gaussian superchannel Φ ∈ GSC n iscalled incoherent if Φ( φ ) ∈ IGC n for ∀ φ ∈ IGC n . We denote the set of all incoherent Gaussian super-channels by
IGSC n . The structure of
IGSC n is as fol-lowing Theorem 4. Theorem 4.
For Gaussian superchannelΦ(
A, O, Y, d ) , the following (1), (2) and (3) areequivalent.(1). Φ( A, O, Y, d ) is incoherent.(2). d = 0, A ∈ T n , O ∈ T n , Y = ⊕ nj =1 η j I , η j ∈ R. (3). Φ( φ ) = χ ◦ φ ◦ χ for ∀ φ ∈ GC n , with fixed χ , χ ∈ IGC n . III. FRAMEWORK FOR QUANTIFYINGCOHERENCE OF GAUSSIAN CHANNELS.
With the definitions of incoherent Gaussian channelsand incoherent Gaussian superchannels, we now establisha quantum resource theory for quantifying the coherenceof Gaussian channels.We propose the following conditions that any coher-ence measure for Gaussian channels should satisfy.(C1). C ( φ ) ≥ ∀ φ ∈ GC n , and C ( φ ) = 0 iff φ ∈ IGC n . (C2). C [Ψ( φ )] ≤ C ( φ ) , ∀ φ ∈ GC n , ∀ Ψ ∈ IGSC n . Note that, (C2) is equivalent to (C3a)+(C3b) below.This can be seen by Theorem 4 and the fact that theidentity Gaussian channel φ ( T = I n , N = 0 , d = 0) ∈IGC n . (C3a). C ( φ ◦ χ ) ≤ C ( φ ) , ∀ φ ∈ GC n , χ ∈ IGC n . (C3b). C ( χ ◦ φ ) ≤ C ( φ ) , ∀ φ ∈ GC n , χ ∈ IGC n . Theorem 5 below provides a way to construct a classof coherence measures for Gaussian channels, the proofis simple.
Theorem 5.
If the functional C : GS n → R satisfies(B1). C ( ρ ) ≥ ∀ ρ ∈ GS n , and C ( ρ ) = 0 iff ρ ∈ IGS n ;(B2). C [ χ ( ρ )] ≤ C ( ρ ) , ∀ ρ ∈ GS n , ∀ χ ∈ IGC n , then C ( φ ) = sup ρ th ∈IGS n C [ φ ( ρ th )] (26)is a coherence measure for Gaussian channels. Consider the functional C r : GS n → R , C r ( ρ ) = inf σ ∈IGS n S ( ρ || σ ) , ∀ ρ ∈ GS n , (27)with S ( ρ || σ ) = tr ( ρ log ρ ) − tr ( ρ log σ ) the relative en-tropy. It is easy to check that C r satisfies (B1) and (B2),and also C r ( ρ ) has the analytical expression as [16] C r [ ρ ( V, d )] = n X j =1 f ( n j ) − n X j =1 f ( ν j −
12 ) , (28) f ( x ) = ( x + 1) log ( x + 1) − x log x, x ∈ R, (29) n j = 14 [ V ( j )11 + V ( j )22 + d jx + d jy − , (30)where S ( ρ ) = P nj =1 f ( ν j − ) is the entropy of ρ [24], { ν j } nj =1 are the symplectic eigenvalues of V [13], n j isdetermined by the j th-mode covariance matrix V ( j ) anddisplacement vector d j .Consequently, C r ( φ ) = sup ρ th ∈IGS n C r [ φ ( ρ th )] (31)is a coherence measure for Gaussian channels.It is straightforward to check that C r satisfies (B3) and(B4) below.(B3). C r ( ρ ⊗ ρ ) = C r ( ρ ) + C r ( ρ ) , ∀ ρ ∈ GS n , ρ ∈GS n . (C4). C r ( φ ⊗ φ ) = C r ( φ ) + C r ( φ ) , ∀ φ ∈ GC n , φ ∈GC n . We give two concrete examples to show the calculationof coherence C r ( φ ). Definition 4.
We call a Gaussian channel φ ( T, N, d ) ∈GC n a constant Gaussian channel if φ ( ρ ) = ρ ′ for ∀ ρ ( V, d ) ∈ GS n where ρ ′ ( V ′ , d ′ ) ∈ GS n is fixed.Such constant Gaussian channel can be represented as φ ( T, N, d ) = φ ( T = 0 , N = V ′ , d = d ′ ) . Exmple 1.
For a constant Gaussian channel, accordingto Eq. (31), C r [ φ ( T = 0 , N = V ′ , d = d ′ )] = C r [ ρ ′ ( V ′ , d ′ )] . (32) Exmple 2.
Coherence of displacement channels.The displacement operator D ( λ ) = Π nj =1 D ( λ j ) inEqs. (2, 3) is a unitary Gaussian channel acting on thestate ρ as D ( λ ) ρD ( − λ ). For ρ ( V, d ) ∈ GS n , we have D ( λ ) ρ ( V, d ) D ( − λ ) ∈ GS n , and its characteristic func-tion is χ ( D ( λ ) ρ ( V, d ) D ( − λ ) , µ )= tr [ D ( λ ) ρ ( V, d ) D ( − λ ) D ( µ )]= tr [ ρ ( V, d ) D ( − λ ) D ( µ ) D ( λ )]= exp( µλ ∗ − µ ∗ λ ) tr [ ρ ( V, d ) D ( µ )]= tr [ ρ ( V, d + 2 λ ) D ( µ )]= χ ( ρ ( V, d + 2 λ ) , µ ) , (33)where we have used D ( − λ ) D ( µ ) D ( λ ) = D ( µ ) exp( µλ ∗ − µ ∗ λ ) . (34)and Eq. (4). Thus the Gaussian channel of displacementoperator D ( λ ) can be written as φ ( T = I n , N = 0 , d = 2 λ )= ⊗ nj =1 φ ( T = I , N = 0 , d j = 2 λ j ) . (35)From (C4) and Eqs. (31, 28) we have C r [ D ( λ )]= n X j =1 C r [ D ( λ j )]= n X j =1 sup ν ≥ C r [ ρ ( νI , λ j )]= n X j =1 sup ν ≥ [ f ( ν −
12 + | λ j | ) − f ( ν −
12 ) ]= n X j =1 f ( | λ j | ) , (36)the last step is because f ( x ) is a concave function.Besides C r , we can also define the coherence mea-sures for Gaussian channels via the coherence measuresfor Gaussian states based on the Bures metric and theHellinger metric [18] under Theorem 5.We need to point out that there could be a divergenceproblem when we define a coherence measure for Gaus-sian channels under Theorem 5 since the supremum istaken over the unbounded set { ν j | ν j ≥ } nj =1 . IV. SUMMARY
In this work, we proposed the definitions of incoherentGaussian channel and incoherent Gaussian superchannel,established a resource theory for quantifying the coher-ence of Gaussian channels. We proposed two representa-tions for Gaussian superchannels and two representationsfor incoherent Gaussian superchannels. We provided away to construct a class of coherence measures for Gaus-sian channels via coherence measures for Gaussian states.It is worth emphasizing that the definitions of incoherentGaussian channels and incoherent Gaussian superchan-nel in this work are all resource-nongenerating operations[21]. Two concrete examples are given to exemplify thecalculation of coherence measure C r for Gaussian chan-nels. ACKNOWLEDGMENTS
This work is supported by the China Scholarship Coun-cil (CSC, No. 201806305050).
AppendixA. Proof of Theorem 1
Suppose φ ( T, N, d ) ∈ IGC n , then for any ρ th ( V = ⊕ j ν j I , d = 0) we have φ ( ρ th ) ∈ IGS n . From Eqs. (8, 9)it follows that d = 0 and for any { ν j | ν j ≥ } nj =1 , thereexist { ν ′ j | ν ′ j ≥ } nj =1 such that T ( ⊕ j ν j I ) T t + N = ⊕ j ν ′ j I . (A1)Write T as n × n block matrix T = ( T jk ) jk with each T jk a 2 × j − , j ) rows and(2 k − , k ) columns, and N = ( N jk ) jk similarly. Then,Eq. (A1) yields n X l =1 ν l T jl T tkl + N jk = 0 , j = k, (A2) n X l =1 ν l T jl T tjl + N jj = ν ′ j I , (A3)where T tkl = ( T kl ) t . Varying { ν j | ν j ≥ } nj =1 in Eqs. (A2,A3) leads to T jl T tkl = 0 , j = k, (A4) N jk = 0 , j = k, (A5) T jl T tjl = t jl I , t jl ∈ R, (A6) N jj = ω j I , ω j ∈ R. (A7)Eqs. (A4, A6) together imply that in { T jl } nj =1 there is atmost one nonzero matrix, we denote this matrix by t l T l with t l ∈ R, T l T tl = I . Taking these conditions into Eq. (10) and using thefact that for any 2 × M,M ωM t = ω det M, (A8)we get ω j I + iω (1 − X k : r ( k )= j t k det T k ) ≥ , ∀ j. (A9)this evidently leads to Eq. (15). B. Proof of Theorem 2
For φ ( T, N, d ) ∈ GC n , ρ φ ( V, d ) ∈ GS n , consider anyΦ( X, Y , e d ) ∈ GC n with e d = (cid:18) d (cid:19) . (A10)Write X = (cid:18) A BC D (cid:19) , Y = (cid:18) A ′ B ′ ( B ′ ) t D ′ (cid:19) , (A11) A , B , C , D , A ′ , B ′ , D ′ , are all 2 n × n real matrices.According to Eq. (9) we have XV X t + Y = (cid:18) V ′ V ′ V ′ V ′ (cid:19) + Y (A12)with V ′ , V ′ , V ′ , V ′ are all 2 n × n real matrices asrespectively( AT T t A t + BB t ) ch r + ( B Σ n T t A t + AT Σ n B t ) sh r + AN A t , (A13)( AT T t C t + BD t ) ch r + ( B Σ n T t C t + AT Σ n D t ) sh r + AN C t , (A14)( CT T t A t + DB t ) ch r + ( D Σ n T t A t + CT Σ n B t ) sh r + CN A t , (A15)( CT T t C t + DD t ) ch r + ( D Σ n T t C t + CT Σ n D t ) sh r + CN C t . (A16)If XV X t + Y has the form of Eq. (16), noticing thatr is varying, then there must be AT T t A t + BB t = T ′ T ′ t , (A17) B Σ n T t A t + AT Σ n B t = 0 , (A18) AT T t C t + BD t = 0 , (A19) B Σ n T t C t + AT Σ n D t = T ′ Σ n , (A20) CT T t C t + DD t = I n , (A21) D Σ n T t C t + CT Σ n D t = 0 , (A22) AN A t + A ′ = N ′ , (A23) AN C t + B ′ = 0 , (A24) CN C t + D ′ = 0 . (A25)Let T = 0, N + i Ω ≥
0, Eq. (A21) and Eq. (A19) yield DD t = I n , (A26) B = 0 . (A27)Let N = 0, T = I n , it is easy to check that such T , N satisfy Eq. (10). For such case, Eq. (A21) yields C = 0 , (A28)and then Eqs. (A24,A25) lead to B ′ = D ′ = 0 . (A29)Let A ′ = Y , D = O , and notice that for Φ( X, Y , e d ) ∈GC n , Eq. (8) leads to Eq. (21), Eq. (10) leads to Eqs.(22, 23), we then end this proof. C. Proof of Theorem 3
For φ ( T , N , d ), φ ( T , N , d ), φ ( T , N , d ) ∈GC n , ρ ( V, d ) ∈ GS n , denote ρ ( e V , e d ) = φ [ ρ ( V, d )], φ ( T , N , d ) = φ ◦ φ ◦ φ , then repeatedly using Eqs.(8, 9) we get e V = T T T V T t T t T t + T T N T t T t + T N T t + N , (A30) e d = T T T d + T T d + T d + d , (A31) T = T T T , (A32) N = T T N T t T t + T N T t + N , (A33) d = T T d + T d + d . (A34)Now we can check that Eqs. (24, 25) realize Eqs. (19-21).Taking Eq. (25) into Eq. (10) we get Eq. (22). TakingEq. (24) into Eq. (10) we get i Ω − i Σ n O t Σ n ΩΣ n O Σ n ≥ . (A35)Left multiply the left side of this equation by O Σ n andright multiply it by Σ n O t , together with the fact thatΣ n ΩΣ n = − Ω , (A36)we will get Eq. (23). D. Proof of Theorem 4
Step 1. We prove (1) ⇒ (2) . Suppose Gaussian superchannel Φ(
A, O, Y, d ) is in-coherent, then for any φ ( T, N, d ) ∈ IGC n , we haveΦ[ φ ( T, N, d )] = ψ ( T ′ , N ′ , d ′ ) ∈ IGC n . From Theorem1 and Theorem 2, similarly to the proof of Theorem 1,we can get d = 0, A ∈ T n , Y = ⊕ j η j I , η j ∈ R . We needto prove O ∈ T n . Let O ′ = Σ n O t Σ n , (A37)then T ∈ T n , A ∈ T n , and T ′ = AT O ′ ∈ T n . (A38)Note that T n is closed under multiplication, then AT ∈T n . If A = 0, then T ′ = AT O ′ = 0 , we let O = O ′ = I n ∈T n . Suppose T n ∋ A = { s j A j } j = 0 , then there exists atleast one j such that s j = 0, A j A tj = I . Let T = { t j T j } j with T j T tj = I , r ( T j ) = j ∀ j , then T n ∋ AT = { s j t j A j T j } j with r ( A j T j ) = r ( A j ) ∀ j . Now write O ′ as n × n block matrix O ′ = ( O ′ jk ) jk with each O ′ jk a 2 × j − , j ) rows and (2 k − , k )columns, then T n ∋ T ′ = AT O ′ = { s j A j X j t j T j O ′ jk } k . (A39)Varying t j , T j , for all j , and using the facts of Lemma1 and lemma 2 below, we can get O ′ ∈ T n and further O = Σ n ( O ′ ) t Σ n ∈ T n . Lemma 1.
Let B , B be two fixed 2 × t , t ∈ R, there exists t ∈ R such that( t B + t B )( t B + t B ) t = tI , (A40)then there exist s , s , s ∈ R such that B B t = s I , (A41) B B t = s I , (A42) B B t + B B t = s I . (A43)Expand Eq. (A40) and vary t , t ∈ R we will getLemma 1. Note that there are 2 × B , B satisfying Eqs. (A41, A42) but not satisfying Eq. (A43),for example B = I , B = √ (cid:18) − (cid:19) . Lemma 2.
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