Entanglement-Assisted Quantum Error Correction with Linear Optics
EEntanglement-Assisted Quantum Error Correction with Linear Optics
Mark M. Wilde, ∗ Hari Krovi, and Todd A. Brun
Communication Sciences Institute, Department of Electrical Engineering,University of Southern California, Los Angeles, California 90089, USA (Dated: November 11, 2018; Received November 11, 2018; Revised; Accepted; Published)We construct a theory of continuous-variable entanglement-assisted quantum error correction.We present an example of a continuous-variable entanglement-assisted code that corrects for anarbitrary single-mode error. We also show how to implement encoding circuits using passive opticaldevices, homodyne measurements, feedforward classical communication, conditional displacements,and off-line squeezers.
PACS numbers: 03.67.-a, 03.67.Hk, 42.50.DvKeywords: quantum error correction, stabilizer, entanglement assisted, continuous variables
INTRODUCTION
Entanglement is a critical resource for quantum in-formation processing. Shared entanglement between asender and receiver enables several quantum communi-cation protocols such as teleportation [1] and superdensecoding [2]. Brun, Devetak, and Hsieh exploited the re-source of shared entanglement to form a general theoryof quantum error-correcting codes—the entanglement-assisted stabilizer formalism [3, 4].Standard quantum error-correcting codes protect a setof qubits from decoherence by encoding the qubits in asubspace of a larger Hilbert space [5, 6, 7, 8]. These quan-tum codes protect a state against a particular error set.Quantum errors in the error set then either leave the setof qubits invariant or they take the state out of the sub-space into an orthogonal subspace. Measurements candiagnose which subspace the state is in without disturb-ing the state. One can then reverse the effect of the errorby rotating the state back into the original subspace.Calderbank et al. figured out clever ways of importingclassical codes for use in quantum error correction [9].These methods translate the classical code to a quantumcode. The problem is that the classical codes have tosatisfy a dual-containing constraint. The dual-containingconstraint is equivalent to the operators in the quantumcode forming a commuting set. Few classical codes sat-isfy the dual-containing constraint so classical theory wasonly somewhat useful for quantum error correction afterCalderbank et al.’s results.Bowen provided the first clue for extending the stabi-lizer formalism by constructing an example of a quan-tum error-correcting code exploiting shared entangle-ment [10]. Brun, Devetak, and Hsieh then establishedthe entanglement-assisted stabilizer formalism [3, 4].Entanglement-assisted codes have several key benefits.One can construct an entanglement-assisted code froman arbitrary linear classical code. The classical codeneed not be dual-containing because an entanglement-assisted code does not require a commuting stabilizer.We turn anticommuting elements into commuting ones by employing shared entanglement. Thus we can use thewhole of classical coding theory for quantum error correc-tion. Additionally, a source of pre-established entangle-ment boosts the rate of an entanglement-assisted code.The performance of an entanglement-assisted quantumcode follows from that of the imported classical code sothat a good classical code translates to a good quantumcode. Entanglement-assisted codes can also operate ina catalytic manner for quantum computation if a fewqubits are immune to noise [3, 4].Continuous-variable quantum information has becomeincreasingly popular due to the practicality of its ex-perimental implementation [11]. Error correction rou-tines are necessary for proper operation of a continuous-variable quantum communications system. Braunstein[12] and Lloyd and Slotine [13] independently proposedthe first continuous-variable quantum error-correctingcodes. Braunstein’s scheme has the advantage that onlylinear optical devices and squeezed states prepared off-line implement the encoding circuit [12, 14]. The per-formance of the code depends solely on the performanceof the off-line squeezers, beamsplitters, and photodetec-tors. The disadvantage of Braunstein’s scheme is thatsmall errors accumulate as the computation proceeds ifthe performance of squeezers and photodetectors is notsufficient to detect these small errors [15].In this paper, we extend the entanglement-assistedstabilizer formalism to continuous-variable quantum in-formation [11]. Figure 1 illustrates how a continuous-variable entanglement-assisted code operates. Brun, De-vetak, and Hsieh constructed the entanglement-assistedstabilizer formalism in terms of a sympletic space Z n over the field Z . The theory behind continuous-variableentanglement-assisted quantum error-correcting codesexploits a symplectic vector space R n over the field R .We first review the relation between symplectic spaces,unitary operators, and the canonical operators for sin-gle and multiple modes. We present two theorems thatplay a crucial role in constructing continuous-variableentanglement-assisted codes. We then provide a canon-ical code and show how a symplectic transformation re-Typeset by REVTEX a r X i v : . [ qu a n t - ph ] A ug FIG. 1: The above figure demonstrates the operation of acontinuous-variable entanglement-assisted code. Lines withbars through them denote multiple modes. Thin lines denotequantum information and thick lines denote classical infor-mation. Alice possesses states | ϕ (cid:105) , | (cid:105) , and half of the entan-gled modes ˛˛ Φ + ¸ . Bob possesses the other half of entangledmodes ˛˛ Φ + ¸ . The unitary U encodes the multi-mode state | ϕ (cid:105) with the help of several position-quadrature squeezed ancillas | (cid:105) and entangled modes ˛˛ Φ + ¸ . Alice sends her modes overa noisy quantum channel. The entanglement-assisted com-munication paradigm assumes that the noisy channel affectsAlice’s modes only. Bob measures all the modes to diagnosethe errors and corrects them with a recovery operator R . Bobcan perform these measurements with homodyne detection. lates an arbitrary code to the canonical one. Our presen-tation parallels the approach for qubits [4]. The per-formance of our codes depends solely on the level ofsqueezing and photodetector efficiency that is techno-logically feasible. We give an example of a continuous-variable entanglement-assisted quantum error-correctingcode that corrects a arbitrary single-mode error.Our entanglement-assisted quantum error-correctingcodes are vulnerable to finite squeezing effects and in-efficient photodetectors for the same reasons as thosegiven in [12]. Our scheme works well if the errors dueto finite squeezing and inefficiencies in beamsplitters andphotodetectors are smaller than the actual errors.Our second contribution is an algorithm for construct-ing the encoding circuit using linear optics. We refer toany scheme implementing an optical circuit with passiveoptical elements, homodyne measurements, feedforwardcontrol, conditional displacements, and off-line squeez-ers as a linear-optical scheme. The algorithm exploitsand extends previous techniques [16, 17]. The algorithmemploys a symplectic Gaussian elimination technique todecompose an arbitrary encoding circuit into a linear-optical circuit. The transmission amplitudes and phaseshifts of passive beamsplitters encode all the logic ratherthan the interaction strength of nonlinear devices. SYMPLECTIC ALGEBRA FOR CONTINUOUSVARIABLES
We first review some mathematical preliminaries. Thenotation we develop is useful for stating Theorems 1 and 2 precisely. Theorems 1 and 2 are relevant for construct-ing an entanglement-assisted quantum code and are anal-ogous to the theorems in [3, 4] for discrete variables.We relate the n -mode phase-free Heisenberg-Weylgroup ([ W n ] , ∗ ) to the additive group (cid:0) R n , + (cid:1) . Let X ( x ) be a single-mode position translation by x and let Z ( p ) be a single-mode momentum kick by p where X ( x ) ≡ exp {− iπx ˆ p } ,Z ( p ) ≡ exp { iπp ˆ x } , (1)and ˆ x and ˆ p are the position-quadrature and momentum-quadrature operators respectively. The canonical com-mutation relations are [ˆ x, ˆ p ] = i . Denote the single-modeHeisenberg-Weyl group by W where W ≡ { X ( x ) Z ( p ) | x, p ∈ R } . (2)Let W n be the set of all n -mode operators of the form A ≡ A ⊗ · · · ⊗ A n where A j ∈ W ∀ j ∈ { , . . . , n } .Define the equivalence class[ A ] ≡ { β A | β ∈ C , | β | = 1 } (3)with representative operator having β = 1. The aboveequivalence class is useful because global phases are notrelevant in the formulation of our codes. The group op-eration ∗ for the above equivalence class is as follows[ A ] ∗ [ B ] ≡ [ A ] ∗ [ B ] ⊗ · · · ⊗ [ A n ] ∗ [ B n ]= [ A B ] ⊗ · · · ⊗ [ A n B n ] = [ AB ] . (4)The equivalence class [ W n ] = { [ A ] : A ∈ W n } forms acommutative group ([ W n ] , ∗ ). We name ([ W n ] , ∗ ) the phase-free Heisenberg-Weyl group .Consider the 2 n -dimensional real vector space R n .It forms the commutative group (cid:0) R n , + (cid:1) with opera-tion + defined as vector addition. We employ the no-tation u = ( p | x ) , v = ( p (cid:48) | x (cid:48) ) to represent any vectors u , v ∈ R n respectively. Each vector p and x has ele-ments ( p , . . . , p n ) and ( x , . . . , x n ) respectively with sim-ilar representations for p (cid:48) and x (cid:48) . The symplectic product (cid:12) of u and v is u (cid:12) v ≡ p · x (cid:48) − x · p (cid:48) = n (cid:88) i =1 p i x (cid:48) i − x i p (cid:48) i , (5)where · is the standard inner product. Define a map D : R n → W n as follows: D ( u ) ≡ exp (cid:26) i √ π n (cid:80) i =1 ( p i ˆ x i − x i ˆ p i ) (cid:27) . (6)Let X ( x ) ≡ X ( x ) ⊗ · · · ⊗ X ( x n ) , Z ( p ) ≡ Z ( p ) ⊗ · · · ⊗ Z ( p n ) , (7)so that D ( p | x ) and Z ( p ) X ( x ) belong to the same equiv-alence class: [ D ( p | x )] = [ Z ( p ) X ( x )] . (8)The map [ D ] : R n → [ W n ] is an isomorphism[ D ( u + v )] = [ D ( u )] [ D ( v )] , (9)where u , v ∈ R n . We use the BCH theorem e A e B = e B e A e [ A,B ] and the symplectic product to capture thecommutation relations of any operators D ( u ) and D ( v ): D ( u ) D ( v ) = exp { iπ ( u (cid:12) v ) } D ( v ) D ( u ) . (10)The operators D ( u ) and D ( v ) commute if u (cid:12) v = 2 n and anticommute if u (cid:12) v = 2 n + 1 for any n ∈ Z . Theset of canonical operators ˆ x i , ˆ p i for all i ∈ { , . . . , n } havethe canonical commutation relations:[ˆ x i , ˆ x j ] = 0 , [ˆ p i , ˆ p j ] = 0 , [ˆ x i , ˆ p j ] = iδ ij . Let T n be the set of all linear combinations of the canon-ical operators: T n ≡ (cid:40) n (cid:88) i =1 α i ˆ x i + β i ˆ p i : ∀ i, α i , β i ∈ R (cid:41) . (11)Define the map M : R n → T n as M ( u ) ≡ u · ˆR n , (12)where u = ( p | x ) ∈ R n , ˆR n = (cid:2) ˆ x · · · ˆ x n (cid:12)(cid:12) ˆ p · · · ˆ p n (cid:3) T , (13)and · is the inner product. We can now write T n ≡ (cid:8) M ( u ) : u ∈ R n (cid:9) . The symplectic product gives thecommutation relations of elements of T n :[ M ( u ) , M ( v )] = ( u (cid:12) v ) i. (14)The definitions given below provide terminology used inthe statements of Theorems 1 and 2 and used in theconstruction of our continuous-variable entanglement-assisted codes. Definition 1
A subspace V of a space W is symplecticif there is no v ∈ V such that ∀ u ∈ V : u (cid:12) v = 0 . Definition 2
A subspace V of a space W is isotropic if ∀ u ∈ W , v ∈ V : u (cid:12) v = 0 . Definition 3
Two vectors u , v ∈ R n form a hyperbolicpair ( u , v ) if u (cid:12) v = 1 . Definition 4
The symplectic dual V ⊥ of a subspace V is V ⊥ ≡ { w : w (cid:12) u = 0 , ∀ u ∈ V } . Definition 5
A symplectic matrix Υ : R n → R n pre-serves the symplectic product: Υu (cid:12) Υv = u (cid:12) v ∀ u , v ∈ R n . (15) It satisfies the condition Υ T JΥ = J where J = (cid:20) n × n I n × n − I n × n n × n (cid:21) . (16) THEOREMS FOR ENTANGLEMENT-ASSISTEDQUANTUM ERROR CORRECTION FORCONTINUOUS-VARIABLE SYSTEMS
Theorem 1 applies to parity check matrices for ourcontinuous-variable entanglement-assisted codes. Thetheorem gives an optimal way of decomposing an arbi-trary subspace of R n into a purely isotropic subspaceand a purely symplectic subspace. Thus we can decom-pose the rows of an arbitrary parity check matrix in thisfashion. We later see that this theorem determines howmuch entanglement is necessary for the code. Theorem 1
Let V be a subspace of R n . Suppose dim ( V ) = m . There exists a symplectic subspace symp ( V ) = span { u , . . . , u c , v , . . . , v c } of R n where dim (symp ( V )) = 2 c . The hyperbolic pairs ( u i , v i ) where i = 1 , . . . , c span symp ( V ) . There exists anisotropic subspace iso ( V ) = span { u c +1 , . . . , u c + l } where dim (iso ( V )) = l . Subspace V has dimension m = 2 c + l and is the direct sum of its isotropic and symplectic sub-spaces: V = iso ( V ) ⊕ symp ( V ) . A constructive proof of the above theorem is in [18].The set of basis vectors for iso ( V ) corresponds to a com-muting set of observables in both W n and T n usingthe maps D and M respectively. Each hyperbolic pair( u i , v i ) in symp ( V ) corresponds via D to a pair of ob-servables in W n that anticommute and corresponds via M to a pair in T n with commutator [ M ( u i ) , M ( v i )] = i .Theorem 2 is useful in relating a general continuous-variable entanglement-assisted quantum error-correctingcode to a canonical one (described below) by a unitaryoperator. The unitary operator corresponds to an encod-ing circuit for the code. Theorem 2
There exists a unitary operator U Υ corre-sponding to a symplectic matrix Υ so that the followingtwo conditions hold ∀ u ∈ R n : [ D ( Υu )] = (cid:2) U Υ D ( u ) U − Υ (cid:3) , M ( Υu ) = U Υ M ( u ) U − Υ . (17)Theorem 2 is a consequence of the Stone-von Neumanntheorem [19].The unitary U − Υ for the encoding circuit relates a gen-eral continuous-variable entanglement-assisted quantumerror-correcting code to the canonical one. CANONICAL ENTANGLEMENT-ASSISTEDQUANTUM ERROR-CORRECTING CODE
We first consider a code protecting against a canonicalerror set S ⊂ R n with errors D ( u ) where u ∈ R n .We later extend to a more general error set by applyingTheorem 2.Continuous-variable errors are equivalent to transla-tions in position and kicks in momentum [12, 15]. Theseerrors correspond to vectors in R n via the inverse map D − .Suppose Alice wishes to protect a k -mode quantumstate | ϕ (cid:105) : | ϕ (cid:105) = (cid:82) ··· (cid:82) dx · · · dx k ϕ ( x , . . . , x k ) | x (cid:105) · · · | x k (cid:105) . (18)Alice and Bob possess c sets of infinitely-squeezed, per-fectly entangled states | Φ (cid:105) ⊗ c where | Φ (cid:105) ≡ (cid:18)(cid:90) dx | x (cid:105) | x (cid:105) (cid:19) / √ π. The state | Φ (cid:105) is a zero-valued eigenstate of the relativeposition observable ˆ x A − ˆ x B and total momentum observ-able ˆ p A + ˆ p B . Alice possesses l = n − k − c ancilla registersinitialized to infinitely-squeezed zero-position eigenstatesof the position observables ˆ x k +1 , . . . , ˆ x k + l : | (cid:105) = | (cid:105) ⊗ l .She encodes the state | ϕ (cid:105) with the canonical isometricencoder U as follows: U : | ϕ (cid:105) | Φ (cid:105) ⊗ c → | ϕ (cid:105) | (cid:105) | Φ (cid:105) ⊗ c . (19)The canonical code corrects the error set S = (cid:26) ( α ( a , a , a ) , b , a | β ( a , a , a ) , a , a ): b , a ∈ R l , a , a ∈ R c (cid:27) , (20)for some known functions α, β : R l × R c × R c → R k .Suppose an error D ( u ) occurs where u = ( α ( a , a , a ) , b , a | β ( a , a , a ) , a , a ) . (21)The state | ϕ (cid:105) | (cid:105) | Φ (cid:105) ⊗ c becomes (up to a global phase) Z ( α ) X ( β ) | ϕ (cid:105) ⊗ | a (cid:105) ⊗ | a , a (cid:105) , (22)where | a (cid:105) = X ( a ) | (cid:105) and | a , a (cid:105) = X ( a ) Z ( a ) | Φ (cid:105) ⊗ c .Bob measures the position observables of the ancillas | a (cid:105) and the relative position and total momentum observ-ables of the state | a , a (cid:105) . He obtains the reduced errorsyndrome r = ( a , a , a ). The reduced error syndromespecifies the error up to an irrelevant value of b in (21).Bob reverses the error u by applying the map D ( − u (cid:48) )where u (cid:48) = ( α ( a , a , a ) , , a | β ( a , a , a ) , a , a ) . (23)The canonical code is degenerate because the Z ( b ) errorsdo not affect the encoded state and Bob does not needto know b to correct the errors. We can describe the operation of the canonical code us-ing binary matrix algebra. This technique gives a corre-spondence between the canonical code and classical cod-ing theory. The following parity check matrix F charac-terizes the errors that the canonical code can correct: F ≡ l × k I l × l l × c c × k c × l I c × c c × k c × l c × c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l × k l × l l × c c × k c × l c × c c × k c × l I c × c . (24)The rows in the above matrix F correspond to observ-ables via the map M in (12). Bob can measure theseobservables to diagnose the error. However, a problemexists. Suppose Bob naively attempts to learn the errorby measuring the observables M ( f ) for all rows f in F .Bob disturbs the state because these observables do notcommute. We remedy this situation later by supposingthat Alice and Bob share entanglement as in the aboveconstruction in (19).Let us define the canonical symplectic code C corre-sponding to F to be all the real vectors symplecticallyorthogonal to the rows of F : C ≡ rowspace ( F ) ⊥ . (25)Let S be the set of correctable errors. All pairs of errorsin S obey one of the following constraints: ∀ u , u (cid:48) ∈ S with u (cid:54) = u (cid:48) either u − u (cid:48) / ∈ C or u − u (cid:48) ∈ iso (cid:0) C ⊥ (cid:1) .The condition u − u (cid:48) / ∈ C states that an error is cor-rectable if it has a unique error syndrome. The lattercondition applies if any two errors have the same effecton the encoded state.The rowspace of F is a (2 c + l )-dimensional subspaceof R n . Therefore it decomposes as a direct sum of anisotropic and symplectic subspace according to Theo-rem 1. The first l rows of F are a basis for the isotropicsubspace and the last 2 c rows are a basis for the sym-plectic subspace.We can remedy the problems with the parity checkmatrix in (24) by constructing an augmented parity checkmatrix F aug as l × k I l × l l × c l × c c × k c × l I c × c − I c × c c × k c × l c × c c × c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l × k l × l l × c l × c c × k c × l c × c c × c c × k c × l I c × c I c × c . The error-correcting properties of the code are the sameas before. The extra entries correspond to Bob’s half ofentangled modes shared with Alice. These extra modesare noiseless because they are on the receiving end of thechannel. The isotropic subspace of rowspace( F ) remainsthe same in the above construction. The symplectic sub-space of rowspace( F ) becomes isotropic in the higher di-mensional space rowspace( F aug ). Each row f of F aug corresponds to an element of the set M ≡ { M ( f ) : f is a row of F aug } . (26)Observables in M commute because rowspace( F aug ) ispurely isotropic. Bob can then measure these observ-ables to learn the error without disturbing the state.The canonical codespace C is the simultaneous zeroeigenspace of operators in M —the encoding in (19) sat-isfies this constraint. Measurement of the observablescorresponding to the first l rows of F aug gives Bob theerror vector a . The next c measurements give Bob theerror vector a and the last c measurements give Bob theerror vector a . This reduced syndrome ( a , a , a ) spec-ifies the error up to an irrelevant value of b . Bob canreverse the error u by applying the map D ( − u (cid:48) ) with u (cid:48) defined in (23). The number of entangled modes used inthe code is c = dim (iso (rowspace ( F ))) / , and the number of encoded modes is k = n − dim (symp (rowspace ( F ))) − c. Thus Alice and Bob can use the above canonical codewith entanglement assistance to correct for a canonicalerror set.
GENERAL ENTANGLEMENT-ASSISTEDQUANTUM ERROR-CORRECTING CODES
We now show how to construct an entanglement-assisted quantum error-correcting code from an arbitrarysubspace C of R n . We give an example of this construc-tion as we develop the theory. Suppose that subspace C is (2 n − m )-dimensional where m = 2 c + l for some c, l ≥ c + l < n . Think of subspace C as anarbitrary symplectic code. We can find a symplectic ba-sis { u i , v i } ni =1 for R n by Theorem 1 with the followingtwo constraints. First, it has hyperbolic pairs ( u i , v i ) i = 1 , . . . , n . Second, 2 n − m vectors in { u i , v i } ni =1 cor-respond to a basis for C and the other m vectors are abasis for the m -dimensional subspace C ⊥ . Let us definethe set R ≡ { u , . . . , u c + l , v , . . . , v c } (27)as a basis for the m -dimensional subspace C ⊥ . Definethe set R ≡ { e , . . . , e c + l , e n +1 , . . . , e n + c } (28)as a basis for the canonical subspace C ⊥ .How do we find the symplectic basis for R n ? Wecan employ a symplectic Gram-Schmidt orthogonaliza-tion procedure similar to that outlined in Ref. [4]. Sup-pose we have an initial arbitrary set of vectors that forma basis for C . We can multiply and add the vectors to-gether without changing the error-correcting propertiesof the eventual code that we formulate. These operations are “row operations.” Row operations are useful for de-termining an alternate set of vectors that determine abasis for C ⊥ . This alternate set then decomposes intopurely symplectic and purely isotropic parts.We turn to an example to highlight the above theory.Consider the following four vectors: (cid:0) (cid:12)(cid:12) (cid:1) , (cid:0) (cid:12)(cid:12) (cid:1) , (cid:0) (cid:12)(cid:12) (cid:1) , (cid:0) (cid:12)(cid:12) (cid:1) . (29)Suppose they span the dual C ⊥ of an arbitrary sub-space C . C ⊥ is then a four-dimensional vector space.This subspace is similar to one for a discrete-variableentanglement-assisted quantum error-correcting code [3].We use it to develop a continuous-variable entanglement-assisted code. We perform row operations on the aboveset of vectors and obtain the following four vectors: u = (cid:0) (cid:12)(cid:12) (cid:1) , u = (cid:16) − (cid:113) √ −√ (cid:113) (cid:12)(cid:12)(cid:12) (cid:113) − (cid:113) (cid:113) (cid:17) , v = (cid:0) (cid:12)(cid:12) (cid:1) , v = (cid:16) −√ (cid:113) − (cid:113) (cid:113) (cid:12)(cid:12)(cid:12) (cid:113) −√ (cid:113) (cid:17) . (30)The above vectors define a symplectic basis for C ⊥ andare in the set R . The above vectors have the same sym-plectic relations as the following four standard basis vec-tors: e = (cid:0) (cid:12)(cid:12) (cid:1) , e = (cid:0) (cid:12)(cid:12) (cid:1) , e = (cid:0) (cid:12)(cid:12) (cid:1) , e = (cid:0) (cid:12)(cid:12) (cid:1) . (31)The above standard basis vectors are in the set R .We return to the general theory. A symplectic ma-trix Υ then exists that maps the hyperbolic pairs ( u i , v i )to the standard hyperbolic pairs ( e i , e n + i ) for all i [18].Let H and F be the matrices whose rows consist of ele-ments of R and R respectively. Let H aug and F aug bethe augmented versions of H and F respectively. Then H Υ T = F and H aug P Υ T P T = F aug where P is a per-mutation matrix that makes columns n + 1 through n + c be the last c columns and shifts columns n + c +1 through2 n + c left by c positions.The four vectors in (31) determine a canonicalentanglement-assisted code. We place them as row vec-tors in a parity check matrix F : F = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (32)The four vectors in (30) determine an entanglement-assisted code. We place them as row vectors in a paritycheck matrix H : H = − (cid:113) √ −√ (cid:113) −√ (cid:113) − (cid:113) (cid:113) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:113) − (cid:113) (cid:113)
00 1 0 0 (cid:113) −√ (cid:113) . (33)A symplectic matrix Υ relates F to H . This symplecticmatrix Υ determines the encoding circuit. We augmentthe above matrices F and H to matrices F aug and H aug respectively. The augmented matrices F aug and H aug have the matrix (cid:2) − I × × (cid:3) T to the left of the verti-cal bar in F and H and the matrix (cid:2) × I × (cid:3) T as thelast columns of F and H respectively. All the rows in theaugmented parity check matrices F aug and H aug are thenorthogonal with respect to the symplectic product andtherefore correpond to a commuting set of observablesvia the map M . We later confirm that this code correctsfor an arbitrary single-mode error.Our main general result is as follows. There exists acontinuous-variable entanglement-assisted code with thefollowing properties. Alice encodes her state with theoperation U − Υ U . The set S of correctable errors obeysthe following constraint: ∀ u , u (cid:48) ∈ S : u (cid:54) = u (cid:48) , u − u (cid:48) / ∈ C ∨ u − u (cid:48) ∈ iso (cid:0) C ⊥ (cid:1) . The codespace C is the simultaneous zero eigenspace ofthe ordered set: M ≡ { M ( h ) : h is a row of H aug } . (34)Performing U Υ , measuring the operators in M is equiva-lent to measuring operators in M followed by performing U Υ . Suppose an error D ( u ) occurs where u ∈ S . Thegeneral error set relates to the canonical set by the map-ping in Theorem 2: (cid:2) U Υ D ( u ) U − Υ (cid:3) = [ D ( Υu )]. Bobmeasures the reduced syndrome r by measuring the ob-servables in the set M . Bob finds the error u correspond-ing to the reduced syndrome r and performs D ( − u ) toundo the error. Figure 1 illustrates the above operationsfor an entanglement-assisted code.The code corresponding to the parity check matrix in(33) corrects for an arbitrary single-mode error. Supposethat an error D ( u ) occurs on the first mode. We set u =( p | x ) and p, x ∈ R so that p is a momentum-quadratureerror and x is a position-quadrature error. Then Bobmeasures the error syndrome to be as follows: (cid:2) x (cid:112) / p − x ) x (cid:112) / p − √ x (cid:3) . Suppose the error D ( u ) occurs on modes two, three, orfour. The error syndromes in respective order are then as follows: (cid:2) x √ x − (cid:112) / p p (cid:112) / x − √ p (cid:3) , (cid:2) −√ x + (cid:112) / p x − (cid:112) / x (cid:3) , (cid:2) x (cid:112) / x (cid:112) / p + x ) (cid:3) . The above error syndromes are unique for any nonzero p and x . Bob can uniquely identify on which mode theerror D ( u ) occurs and correct for it. LINEAR-OPTICAL ENCODING ALGORITHM
We give an algorithm for decomposing an arbitraryencoding circuit into one and two-mode operations usinglinear optics. The algorithm is an alternative to the onegiven in [20]. The unitary U − Υ for the encoding circuitis an element of the group G Spn that preserves the phase-free Heisenberg-Weyl group up to conjugation [15, 21].The symplectic group Sp(2 n, R ) is isomorphic to G Spn .Previous results show that any G Spn transformation ad-mits a decomposition in terms of linear optical elementsand squeezers [20, 22]. Our algorithm is a different tech-nique for determining the encoding unitary. It uses asymplectic Gaussian elimination technique similar to adiscrete-variable algorithm [17].The Fourier transform gate, two-mode quantum non-demolition interactions, a squeezer, and a continuous-variable phase gate generate all transformations in G Spn .A position-quadrature squeezer S i ( a ) on mode i rescalesthe position quadrature by a with reciprocal scaling by1 /a in the momentum quadrature:ˆ x i → a ˆ x i , ˆ p i → ˆ p i /a. A Fourier transform F i on mode i acts asˆ x i → − ˆ p i , ˆ p i → ˆ x i . A two-mode position-quadrature nondemolition interac-tion Q X ( g ) with interaction strength g transforms thequadrature observables asˆ x → ˆ x , ˆ p → ˆ p − g ˆ p , ˆ x → ˆ x + g ˆ x , ˆ p → ˆ p . A two-mode momentum-quadrature nondemolition inter-action Q P ( g ) with interaction strength g transforms thequadrature observables asˆ x → ˆ x − g ˆ x , ˆ p → ˆ p , ˆ x → ˆ x , ˆ p → ˆ p + g ˆ p . A position-quadrature phase gate P X ( g ) with interac-tion strength g transforms the quadrature observables asˆ x → ˆ x, ˆ p → ˆ p + g ˆ x, and a momentum-quadrature phase gate P P ( g ) trans-forms the quadrature observables asˆ x → ˆ x + g ˆ p, ˆ p → ˆ p. Filip et al. implemented S ( a ), Q X ( g ), and Q P ( g ) usinglinear optics [16].We provide an implementation of the continuous-variable phase gate. Begin with two modes—we wish toperform the phase gate on mode one. Suppose mode twois a position-squeezed ancilla mode. Perform a position-quadrature nondemolition interaction Q X ( g ) on modesone and two: ˆ x → ˆ x , ˆ p → ˆ p − g ˆ p , ˆ x → ˆ x + g ˆ x , ˆ p → ˆ p . Fourier transform mode two:ˆ x → ˆ x , ˆ p − g ˆ p → ˆ p − g ˆ p , ˆ x + g ˆ x → − ˆ p , ˆ p → ˆ x + g ˆ x . Perform a momentum-quadrature nondemolition interac-tion Q P ( g ) on modes one and two:ˆ x → ˆ x , ˆ p − g ˆ p → ˆ p − g ˆ p + g (ˆ x + g ˆ x ) , − ˆ p → − ˆ p − g ˆ x , ˆ x + g ˆ x → ˆ x + g ˆ x . Measure the position quadrature of mode two to get re-sult x . Mode one collapses asˆ x → ˆ x , ˆ p − g ˆ p + g (ˆ x + g ˆ x ) → ˆ p + g x + g ˆ x + 2 g g ˆ x . Correct the momentum of mode 2 by displacing by g x so that ˆ x → ˆ x , ˆ p + g x + g ˆ x + 2 g g ˆ x → ˆ p + g ˆ x + 2 g g ˆ x . The Heisenberg-picture quadrature observables for modeone are approximately ˆ x , ˆ p + 2 g g ˆ x because the orig-inal quadrature ˆ x has position-squeezing. So we im-plement a continuous-variable position-quadrature phasegate P X ( g = 2 g g ).We use the above gates to detail a symplectic Gaussianelimination procedure. This procedure decomposes anarbitrary encoding circuit whose symplectic matrix is Υ .1. If Υ , equals zero, permute the first mode with thesecond. Continuing permuting modes until Υ , isnonzero. Normalize Υ , by simulating S (cid:0) Υ − , (cid:1) . 2. Simulate Q X i ( − Υ i, ) for all i ∈ { , . . . , n } . Thefirst column then has the form (cid:2) · · · Υ n +1 , Υ n +2 , · · · Υ n, (cid:3) T .
3. Simulate P X ( − Υ n +1 , ) followed by F .4. Simulate Q P i ( − Υ j, ) for all i ∈ { , . . . , n } and j = i + n . Perform F − . The first column has the form (cid:2) · · · (cid:3) T .5. Name the new matrix Υ (cid:48) . Proceed to decouple col-umn n + 1 of Υ (cid:48) . Matrix element Υ (cid:48) , = 1 because Υ (cid:48) is symplectic. Simulate Q P i ( − Υ i + j,n +1 ) for all i ∈ { , . . . , n } and j = i + n .6. Simulate P P ( − Υ ,n +1 ). Perform F − .7. Simulate Q X i ( − Υ i, ) for all i ∈ { , . . . , n } . Per-form F .The first round of the algorithm is complete and thenew matrix Υ (cid:48)(cid:48) has its first row and column equal to e ,its ( n + 1) st row and column equal to e n +1 , and all otherentries equal to the corresponding entries in Υ . The re-maining rounds of the algorithm consist of applying thesame procedure to the submatrix formed from rows andcolumns 2 , . . . n, n + 2 , . . . , n of Υ . All of the operationsin the algorithm consist of one and two-mode operationsimplementable with linear optics. The encoding circuitis the inverse of all the operations put in reverse order. CONCLUSION
We have constructed a general theory of entanglement-assisted error correction for continuous-variable quantuminformation. The theory of continuous-variable quantumerror correction broadens when Alice and Bob share a setof entangled modes. They begin with a set of noncom-muting observables that have good error-correcting prop-erties. They then employ shared entanglement to resolvethe anticommutativity in the original observables.Our codes suffer from the same vulnerabilities asBraunstein’s earlier codes for continuous variables [12].But the theory should be useful as experimentalists im-prove the quality of squeezing and homodyne detectiontechnology.Our example of a continuous-variable entanglement-assisted code requires two entangled modes and correctsfor an arbitrary single-mode error.We also provided a way to construct encoding circuitsusing passive optical elements, homodyne measurements,feedforward control, conditional displacements, and off-line squeezers. The algorithm decomposes the encodingcircuit in terms of a polynomial number of gates. Thealgorithm requires a large number of squeezers to imple-ment an encoding circuit. But this scheme for encodingshould become feasible as technology improves.
ACKNOWLEDGEMENTS
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