OOscillator-to-oscillator codes do not have a threshold
Lisa H¨anggli and Robert K¨onigZentrum Mathematik, Technical University of Munich, GermanyFebruary 11, 2021
Abstract
It is known that continuous variable quantum information cannot be protected againstnaturally occurring noise using Gaussian states and operations only. Noh et al. (PRL125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussianresource states as an alternative, and showed that these encodings can lead to a reductionof the effective error strength at the logical level as measured by the variance of theclassical displacement noise channel. An oscillator-to-oscillator code embeds K logicalbosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N -mode unitary and N − K auxiliary one-mode Gottesman-Kitaev-Preskill-states.Here we ask if – in analogy to qubit error-correcting codes – there are families ofoscillator-to-oscillator codes with the following threshold property: They allow to convertphysical displacement noise with variance below some threshold value to logical noise withvariance upper bounded by any (arbitrary) constant. We find that this is not the caseif encoding unitaries involving a constant amount of squeezing and maximum likelihooderror decoding are used. We show a general lower bound on the logical error probabilitywhich is only a function of the amount of squeezing and independent of the number ofmodes. As a consequence, any physically implementable family of oscillator-to-oscillatorcodes combined with maximum likelihood error decoding does not admit a threshold. The theory of fault-tolerance builds on the fact that information can be protected by introduc-ing redudancy combined with suitable recovery procedures. A prime example is the classicalrepetition code encoding one logical bit into three physical bits according to0 (cid:55)→
000 and 1 (cid:55)→ . Assuming that each physical bit undergoes independent bit flip noise with probability (cid:15) , thisencoding improves protection as long as (cid:15) < /
2: The probability of erroneous correction is oforder (cid:15) L = O ( (cid:15) ) . a r X i v : . [ qu a n t - ph ] F e b igher-order error suppression of the form (cid:15) L = O ( (cid:15) N ) can be achieved by the use of error-correcting codes with larger distance, obtained e.g., by concatenation. Such error supressionmechanisms are a cornerstone of John von Neumann’s pioneering theory of fault-tolerant classi-cal computation presented in a paper titled “Probabilistic Logics and the Synthesis of ReliableOrganisms from Unreliable Components” [27]. For present-day efforts to synthesize reliablequantum computers it is imperative to identify similar error suppression methods for quantuminformation in its various forms. Great strides have been made in this direction. For encod-ing qubits into qubits, the theory of stabilizer codes is highly developed. On the other hand,when encoding qubits into bosonic modes, Gottesman-Kitaev-Preskill (GKP) codes [8], catcodes [5, 14], and binomial codes [15] are prominent candidates for experimental realizations.More recently, Noh, Girvin, and Jiang [17, 18] proposed new encodings of some number K ofbosonic modes into N > K bosonic modes along with suitable recovery maps. Such oscillator-to-oscillator codes can indeed achieve error suppression, as exemplified by the so-called GKPtwo-mode squeezing code for which (
N, K ) = (2 , σ is reduced todisplacement noise with standard deviation σ L = O ( σ · polylog (1 /σ )) (1)at the logical level provided that the initial noise strength σ is below some constant. Fur-thermore, higher-level suppression can be achieved using N > K -mode bosonic state | Ψ (cid:105) is encoded with the map | Ψ (cid:105) (cid:55)→ U ( N ) ( | Ψ (cid:105) ⊗ | GKP (cid:105) ⊗ N − K ) (2)by means of an N -mode Gaussian unitary U ( N ) and N − K copies of the canonical one-modeGKP state | GKP (cid:105) . The class of these codes will be referred to as oscillator-to-oscillator codes ofGKP type here, or simply oscillator-to-oscillator codes. For the GKP two-mode squeezing code,the unitary U (2) is a two-mode squeezing operation, whereas for the GKP-squeezed repetitioncode, a recursively defined unitary U ( N ) rep is used.Here we investigate to what extent the error suppression achieved by oscillator-to-oscillatorcodes can be exploited towards establishing a fault-tolerance threshold theorem for quantumcomputation with bosonic modes. Roughly, a threshold theorem amounts to the claim that forsufficiently weak noise (i.e., noise strength below some threshold value), errors can be recoveredfrom with any desired accuracy at the cost of introducing a sufficient amount of redundancy.In order to formalize and motivate this problem, it is useful to review the case of qubits: Herea rigorous threshold theorem for fault-tolerant quantum computation has been established ina series of pioneering works [23, 11, 12, 13, 10, 1]. Fault-tolerance threshold with qubit error correcting codes.
For simplicity, let us fo-cus on the question of fault-tolerant memories. To be concrete, let us assume that each physical2ubit is affected by a probabilistic Pauli noise channel N : B ( C ) → B ( C ). The strength δ ofthe noise can be measured e.g., by the diamond norm distance δ = (cid:107)N − id B ( C ) (cid:107) (cid:5) of N to thesingle-qubit identity channel id B ( C ) . A quantum code encoding K logical qubits into N physicalqubits is a 2 K -dimensional subspace C N ⊂ ( C ) ⊗ N . It can be understood as the image of anisometric encoding map from C to ( C ) ⊗ N which takes the form (2) for a qubit state | Ψ (cid:105) , an N -qubit encoding unitary U ( N ) , and | GKP (cid:105) replaced by the computational basis state | (cid:105) . Anassociated recovery operation is a quantum channel R N : B (cid:0) ( C ) ⊗ N (cid:1) → B (cid:0) ( C ) ⊗ N (cid:1) , typicallycomposed of syndrome measurement and a subsequent (conditional) unitary correction opera-tion. We say that the pair ( C N , R N ) recovers with accuracy (cid:15) > δ iffor any 1-qubit noise channel N with strength δ , the composition R N ◦ N ⊗ N takes the set ofstates supported on C N to itself and satisfies (cid:107) R N ◦ N ⊗ N (cid:12)(cid:12) B ( C N ) − id B ( C N ) (cid:107) (cid:5) ≤ (cid:15) .With these definitions, the concept of a threshold can be formulated as follows: An (infinite)family { ( C N , R N ) } N ∈ Γ of codes (where C N ⊂ ( C ) ⊗ N ) with associated recovery maps has athreshold if there is some value δ > δ < δ , and anyaccuracy (cid:15) >
0, there is some N = N ( δ, (cid:15) ) ∈ Γ such that ( C N , R N ) recovers with accuracyat least (cid:15) from noise of strength δ . This property formalizes the notion that essentially idealquantum memories can be emulated using a sufficient number of noisy qubits, provided thatthese are not too noisy (i.e., have a noise strength below some threshold). We note that incontrast to typical settings in fault-tolerant quantum computing, where any physical operationis assumed to be imperfect, here we consider a simplified notion of quantum memories byassuming that the recovery operation can be implemented perfectly. In fact, this assumptionstrenghtens the no-go result we find.One of the crowning achievements of the theory of quantum error correction was to showthat quantum code families for qubits exhibiting a threshold exist. In fact, not only do theyexist, but we have explicit constructions at hand. Indeed, all that is needed is a family ofquantum error-correcting codes with a code distance scaling extensively with the number N ofphysical qubits. Such a code achieves error suppression which is exponential in N . By choosing N to be sufficiently large, this means that any (sufficiently weak) preexisting physical noisecan be converted to noise below any desired strength at the logical level. Code families withmacroscopic distance can be obtained by a variety of construction techniques.To deploy such codes in the real world, both the threshold value δ and the actual de-pendence of N = N ( δ, (cid:15) ) on the parameters δ and (cid:15) are of key interest: They determine therequired resources. For typical codes and decoders considered in the literature, N scales as N ( δ, (cid:15) ) = poly (log 1 /δ, log 1 /(cid:15) ). The parameter N also determines the required amount ofcomputational resources: For most code families (in particular stabilizer codes), the encodingunitary U ( N ) involves poly ( N ) elementary gates (e.g., one- and two-qubit gates). Similarly, forpractical relevance, we typically are interested in recovery operations R N that require poly ( N )measurements and poly ( N )-time classical computation. In summary, we have qubit codes thatallow for efficient encoding and recovery/decoding as measured in terms of basic operations(one- and two-qubit gates, one-qubit measurements, and classical arithmetic). Most impor-tantly, the nature of these operations does not change with increasing N . In this sense, quan-tum fault-tolerance with qubit codes can be considered strictly as a way of adding redudancy:Simply using more of the same (noisy) resources yields the desired effect.3 o fault-tolerance threshold for oscillator-to-oscillator codes. Motivated by the fa-vorable properties of qubit quantum error-correcting codes, we ask if an analogous fault-tolerance threshold statement holds for oscillator-to-oscillator codes. A candidate formula-tion is immediately obtained by substituting single-qubit noise by a bosonic noise channel N : B ( L ( R )) → B ( L ( R )), which we will assume to be a classical noise channel with vari-ance σ . The question is then whether there is a family of subspaces C N ⊂ L ( R ) ⊗ N thatare images of encoding maps as in (2), each defined by a Gaussian unitary U ( N ) on N modes,together with suitable recovery operations R N : B ( L ( R ) ⊗ N ) → B ( L ( R ) ⊗ N ), such that the fol-lowing holds: Provided that σ < σ for some constant threshold σ , there is some N = N ( σ, σ (cid:48) )for any σ (cid:48) > R N ◦ N ⊗ N is given by displacement noise of varianceless than σ (cid:48) when restricted to the code space C N ∼ = L ( R ) ⊗ K . In view of error suppressionproperties such as (1) achieved by certain oscillator-to-oscillator codes, such a fault-toleranceproperty may appear to be potentially feasible using these codes. A corresponding result wouldthen suggest that the idea that “more is better” when it comes to fault-tolerance extends tothe setting of oscillator-to-oscillator codes.We show here that this is not the case if maximum likelihood error decoding is used: nofamily of oscillator-to-oscillator codes with physically meaningful encoding unitaries exhibitsa threshold in the above sense. The key difference to the finite-dimensional setting is thatthe feasibility of implementing a bosonic unitary U does not depend on measures such as itsgate complexity only: In addition, the amount of squeezing it introduces is a key quantity forexperimental purposes. This quantity, expressed by a “squeezing measure” sq ( U ) we introducein Section 2.1, should realistically be bounded by a constant.For a two-mode squeezing unitary U (2) as used in the GKP two-mode squeezing code, thesqueezing measure sq ( U (2) ) is in one-to-one correspondence with the so-called gain g ( U (2) ) morecommonly used in physics. In [18], the authors argue that the error suppression (1) results for again which scales as g ∗ ∼ O (( σ log(1 /σ )) − ) + 1 /
2. The fact that this diverges as σ → N -mode GKP-squeezed repetition code proposed in [17], the amount ofsqueezing sq ( U ( N ) rep ) of the encoding unitary U ( N ) rep diverges as N → ∞ . Thus this code family isnot a candidate for giving a threshold theorem if we restrict our attention to constant-squeezingencoding unitaries.Our no-go-result is not restricted to the N -mode repetition code only, however. Instead, weshow that for any family of N -mode Gaussian unitaries { U ( N ) } N with the property that sq ( U ( N ) )is bounded by a constant independent of N , the associated family of oscillator-to-oscillatorcodes does not exhibit a threshold. In particular, simply using more modes does not solve thefault-tolerance threshold issue unless increasingly stronger squeezing is used.As indicated above, this result is established for the maximum likelihood error decoder basedon GKP-type syndrome information. The strategy of this decoder is to identify the most likelyerror yielding the observed syndrome, and to subsequently correct for this error. It performsat least as well as the decoder used in [17, 18] – the latter is a heuristic approximation ofmaximum likelihood error decoding. While maximum likelihood error decoding is the naturalchoice when aiming to identify the physical error applied to the system, our work does notimmediately provide insight on the performance of other decoding strategies. In particular, itdoes not exclude the possibility that maximum likelihood (error coset) decoding may outperform4he decoder considered here. Such a decoder proceeds by identifying the most likely coset oferrors (modulo the stabilizer group). This involves coset probabilities that typically cannot beevaluated in closed form. Outline
The remainder of the paper is organized as follows: We first introduce oscillator-to-oscillatorcodes and the physical setup considered here in detail in Section 2. In Section 3, we reformu-late the problem of analysing the ability of oscillator-to-oscillator codes to recover from errorsas a purely classical estimation problem. In the same section, we also present our no-go re-sult for oscillator-to-oscillator codes. The underlying technical results concerning the classicalestimation problem are established in Section 4.
In this section we introduce oscillator-to-oscillator codes and the necessary corresponding back-ground. In particular, we explain how squeezing in Gaussian unitaries can be quantified, andspecify the noise channel and recovery map considered. We also introduce the figure of meritfor the proof of our no-go theorem for oscillator-to-oscillator codes: the decoding success prob-ability. N -mode Gaussian unitaries An N -mode bosonic system is described by N pairs ( Q j , P j ), j = 1 , . . . , N , of position andmomentum operators associated with the quadratures of the electromagnetic field modes. Thecanonical commutation relations can be compactly described by gathering these mode operatorsin a tuple R := ( Q , P , . . . , Q N , P N ): They take the form[ R j , R k ] = iJ jk I where J = N (cid:77) j =1 (cid:18) − (cid:19) . In its most general meaning, the term Gaussian unitary refers to a unitary generated by aHamiltonian which is at most quadratic in the mode operators. Here we use the term Gaussianunitary exclusively for those generated by purely quadratic Hamiltonians, and treat linear con-tributions and the corresponding unitaries separately in Section 2.2. The action of a Gaussianunitary U is completely determined by an element S ∈ Sp (2 N, R ) of the symplectic linear group Sp (2 N, R ) := (cid:8) S ∈ Mat N × N ( R ) | S T J S = J (cid:9) : When conjugated by the unitary U = U S , thequadrature operators transform linearly as U S R j U † S = N (cid:88) k =1 S j,k R k , j = 1 , . . . , N . (3)A special role is played by the harmonic oscillator Hamiltonian H = (cid:80) Nj =1 ( Q j + P j ), whichdetermines the energy of a state and generates rotations in phase space. A unitary U is calledpassive if it commutes with H , i.e., if it preserves the energy. Passive Gaussian unitaries5re particularly easy to implement: every such unitary is a composition of phaseshifters andbeamsplitters, see [21]. The latter belong to the set of operations referred to as passive linearoptics, and are typically readily available. The set of passive Gaussian unitaries is in one-to-one correspondence (via the map S (cid:55)→ U S ) with the group K ( N ) = Sp (2 N, R ) ∩ O (2 N, R )of orthogonal symplectic matrices (here O (2 N, R ) = (cid:8) O ∈ Mat N × N ( R ) | O T O = I (cid:9) ), as theorthogonality constraint ensures that H is left invariant under conjugation.In contrast to passive Gaussian unitaries, active ones are significantly more challenging toimplement in general. Such evolutions involve squeezing, i.e., a process where e.g., photonpairs are created or annihilated. Realizing such dynamics requires elements from non-linearquantum optics such as birefringent materials. An example is the one-mode squeezing uni-tary U z associated with the symplectic matrix diag ( z, /z ), z ∈ R \{ } . It reduces the quantumnoise associated with one quadrature while increasing that of the other. This is particularlychallenging to implement for large z : With present technology, only squeezing of roughly 10 dB (corresponding to z ∼ √
10) is feasible [25].In fact, one-mode squeezing unitaries together with phaseshifters and beamsplitters generatethe set of all Gaussian unitaries according to a certain normal form for symplectic matrices:For any S ∈ Sp (2 N ) there are O , O ∈ K ( N ) such that S = O ZO , with Z = diag ( z , /z , . . . , z n , /z n ) where z , . . . , z n ∈ (0 , ∞ ) , (4)according to the Euler decomposition (see e.g., [3]). In particular, the associated Gaussianunitary U S can be implemented by realizing the unitaries corresponding to O and O (usingpassive linear optics only), and by applying the one-mode-squeezing unitary U z j to every mode j = 1 , . . . , N .Note that for any passive Gaussian unitary U = U S , we clearly have z j = 1 for all j =1 , . . . , N in the decomposition (4). Moreover, we can also use Eq. (4) more generally (goingbeyond passive unitaries) to quantify the amount of squeezing of any Gaussian unitary U S bythe squeezing measure sq ( U S ) := max j =1 ,...,N { z j , /z j } . For later use, we note that Eq. (4) immediately implies that the matrix S T S is positive definiteand sq ( U S ) = (cid:112) λ max ( S T S ) , (5)where λ max ( M ) denotes the maximal eigenvalue of a positive definite matrix M . The quan-tity sq ( U ) describes the maximal amount of squeezing required for any single mode whenimplementing the unitary U . Hamiltonians that are linear combinations of the mode operators generate phase space dis-placements. The corresponding unitaries are parameterized by vectors ξ ∈ R N : For ξ ∈ R N ,we denote by D ( ξ ) the (unitary) displacement operator D ( ξ ) := e iξ T JR .
6t transforms the mode operators as D ( ξ ) R j D ( ξ ) † = R j + ξ j I for j = 1 , . . . , N .
The displacement operators satisfy the Weyl relations D ( ξ ) D ( η ) = e − i ξ T Jη D ( ξ + η ) for ξ, η ∈ R N , (6)and thus D ( ξ ) D ( η ) = e − iξ T Jη D ( η ) D ( ξ ) for ξ, η ∈ R N . (7)An often-studied single-mode error model in linear optics is the Gaussian classical noisechannel N σ : B ( L ( R )) → B ( L ( R )) which randomly displaces a state in phase space accordingto a centered normal distribution with variance σ . It is given by N σ ( ρ ) = 12 πσ · (cid:90) R e − (cid:107) ξ (cid:107) σ D ( ξ ) ρD ( ξ ) † d ξ . (8)The channel (8) is referred to as classical noise since we can think of it as first drawing aclassical random variable Z ∼ N (0 , σ I ) according to a centered normal distribution, andthen displacing by Z .In the following, we will assume that the physical-level noise on each mode is given by N σ .In other words, for an N -mode system, the corresponding noise map is N ⊗ Nσ . Gottesman-Kitaev-Preskill codes [8] are designed to protect against displacement errors. Todefine oscillator-to-oscillator codes, we need a particular instance of this code family. In fact,we only need a certain bosonic stabilizer state (spanning a 1-dimensional subspace) instead of astabilizer code. For a single bosonic mode, the canonical GKP state | GKP (cid:105) is the simultaneous+1-eigenstate of the two commuting displacement operators S Q := e i √ πQ and S P := e i √ πP . (9)(The state | GKP (cid:105) is uniquely defined up to a global phase which is irrelevant here.) We remarkthat | GKP (cid:105) is not an element of the Hilbert space L ( R ) but a distribution (see [4] for a rigoroustreatment). In practice, | GKP (cid:105) needs to be replaced by an approximate GKP state with a finiteaverage photon number.It is instructive to consider how an unknown displacement error applied to the state | GKP (cid:105) can be recovered from. Here we follow the same procedure as for a proper stabilizer code: Ameasurement of the stabilizer operators – with outcome called the syndrome – is followed bythe (conditional) application of a unitary correction operation.
The fact that the operators (9) stabilize the state | GKP (cid:105) means that | GKP (cid:105) has definite valuesof the position and momentum modulo √ π . More precisely, measuring the stabilizer operatorsamounts to a joint measurement of the pair of “modular” operators( Q mod √ π, P mod √ π ) , Q and P themselves, commute). This joint measurement can be realized byconsuming two canonical GKP states and using Gaussian operations and quadrature mea-surements (cf. Fig. 1), see [8]. When measuring the state | GKP (cid:105) , the measurement yields(0 , ∈ [ − (cid:112) π/ , (cid:112) π/ D ( ξ ) | GKP (cid:105) for an unknown displacement (error) vector ξ ∈ R . Applying the modularmeasurement to the state D ( ξ ) | GKP (cid:105) yields the pair s = s ( ξ ) = ( s , s ) ∈ [ − (cid:112) π/ , (cid:112) π/ ofsyndromes where s = ξ mod √ π ,s = ξ mod √ π . (10)Indeed, it follows from the commutation relations (7) and the fact that the operators (9)stabilize | GKP (cid:105) that D ( ξ ) | GKP (cid:105) is an eigenstate of S Q with eigenvalue e i √ πξ and of S P witheigenvalue e i √ πξ . This implies (10). Q mod √ π | Ψ (cid:105) Q,P = Q | Ψ (cid:105) Q ,P | GKP (cid:105) Q ,P (a) Measurement Q mod √ π . P mod √ π | Ψ (cid:105) Q,P = P | Ψ (cid:105) Q ,P | GKP (cid:105) Q ,P (b) Measurement P mod √ π . Figure 1: Measurement of the quadrature operators modulo √ π . The left hand sides of theequalities in Figures 1(a) and 1(b) describe the measurement of Q and P modulo √ π of amode in the state | Ψ (cid:105) ≡ | Ψ (cid:105) Q,P respectively. This is realized by the circuits on the right handside of the corresponding equality: In Figure 1(a), the input state with associated quadratures Q , P is coupled to an auxiliary mode in the state | GKP (cid:105) with associated quadratures Q , P via the SUM , -gate (= e − iQ P ) depicted by the controlled- ⊕ . This transforms the quadraturesas Q (cid:55)→ Q , P (cid:55)→ P − P , Q (cid:55)→ Q + Q , and P (cid:55)→ P . As the second mode is initiallyin the state | GKP (cid:105) we have Q + Q mod √ π = Q mod √ π , and thus the output of themeasurement of the Q -quadrature of the second mode modulo √ π gives us the desired quantity.The measurement circuit for the P -quadrature follows the same strategy. When trying to use the syndrome information (10) to figure out what the error ξ was (inorder to then recover by applying the inverse displacement), one should take into accountthe prior distribution over errors. For example, if we are dealing with the classical noisechannel (8), that is, if we are in fact measuring the noise-corrupted state N σ ( | GKP (cid:105)(cid:104)
GKP | ),then the error ξ ∈ R is distributed according to N (0 , σ I ). The maximum likelihood errordecoding problem then seeks to find the most likely error ˆ ξ ( s ) consistent with the syndrome s =( s , s ), i.e., satisfying ˆ ξ mod √ π = s and ˆ ξ mod √ π = s . The corresponding recoverysuccess probability Pr ξ [ ˆ ξ ( s ( ξ )) = ξ ] is well-studied: It is known as the informed unwrappingproblem of modulo reduced Gaussian vectors (see Section 4.1). The recovery success probabilitydirectly gives the probability that the unitary correction operation D ( ˆ ξ ( s ( ξ )) − applied afterthe error D ( ξ ) gives the identity and therefore the action of the latter is reversed.8he recovery procedure described here is analogous as the one used with GKP-codes en-coding a finite-dimensional subspace. We note that in the special case considered here, where1-dimensional C | GKP (cid:105) is being protected, there is a simpler recovery strategy than the one de-scribed above: just inverting the displacement described by the syndrome vector does the job.This is due to the fact that we are considering a 1-dimensional space: reverting the syndromedisplacement results in a net displacement belonging to the stabilizer group and thus leaves thestate | GKP (cid:105) invariant.
Here we briefly review the definition of oscillator-to-oscillator codes introduced by Noh, Girvinand Jiang in [17, 18]. An oscillator-to-oscillator code encodes K bosonic modes into N bosonicmodes by means of a Gaussian N -mode unitary U ( N ) according to the map (2). The code is theimage of this map and is isomorphic to L ( R ) ⊗ K . Any code state | ¯Ψ (cid:105) := U ( N ) ( | Ψ (cid:105)⊗| GKP (cid:105) ⊗ N − K )is clearly stabilized by the operators U ( N ) S Q j ( U ( N ) ) † and U ( N ) S P j ( U ( N ) ) † where j = K + 1 , . . . , N , (11)and where S Q j := e i √ πQ j and S P j := e i √ πP j . We note that the term GKP-stabilizer code issometimes used for these codes, although this is somewhat ambiguous since the original GKPcode by Gottesman, Kitaev, and Preskill [8] is itself a kind of stabilizer code. We will stick tothe term oscillator-to-oscillator code here.In the following, we assume that U ( N ) = U S is the Gaussian operation associated with asymplectic matrix S ∈ Sp (2 N, R ), cf. (3). encoding noise process recovery | Ψ (cid:105) U ≡ U ( N ) N σ R N | GKP (cid:105) ⊗ N − K N σ N σ N σ | ¯Ψ (cid:105) D ( ξ ) | ¯Ψ (cid:105) D (ˆ ξ ( s ( ξ ))) − D ( ξ ) | Ψ (cid:105) Figure 2: Error correction circuit for oscillator-to-oscillator codes. The encoding consists of theapplication of the N -mode unitary U ( N ) to the K -mode input state | Ψ (cid:105) and N − K modes in thecanonical GKP state | GKP (cid:105) , yielding the encoded state | ¯Ψ (cid:105) . In the subsequent noise process,the Gaussian classical noise channel N σ , which applies a random displacement according toa centered normal distribution with variance σ , is applied to each mode. This process canequivalently be described as drawing a 2 N -dimensional real vector ξ ∼ N (0 , σ I N ) accordingto the centered normal distribution with variance σ , and subsequently applying the displace-ment D ( ξ ) to the encoded state | ¯Ψ (cid:105) . The last step, i.e., the recovery, is described in more detailin Fig. 3. 9 tep 1 step 2 step 3 step 4 s ( ξ ) U − U − D (ˆ ξ ) − U ˆ ξ ≡ ˆ ξ ( s ( ξ )) U ( N ) D ( ξ ) | ¯Ψ (cid:105) U − D ( ξ ) | ¯Ψ (cid:105) U − D ( ξ ) | ¯Ψ (cid:105) U − D (ˆ ξ ) − D ( ξ ) | ¯Ψ (cid:105) D (ˆ ξ ) − D ( ξ ) | Ψ (cid:105) Figure 3: Recovery step of the error correction circuit for oscillator-to-oscillator codes (cf.Fig. 2). The input is the corrupted state D ( ξ ) | ¯Ψ (cid:105) after the noise process. In step 1 theencoding is reversed, resulting in the state ( U ( N ) ) − D ( ξ ) | ¯Ψ (cid:105) . In step 2, the syndrome s ( ξ )is measured, i.e., for every mode j = K + 1 , . . . , N the quadrature operators Q j = R j − , P j = R j are measured modulo √ π (cf. Fig. 1) yielding the entries s j − − K , s j − K of the2( N − K )-dimensional vector s respectively. The measurement result is a deterministic func-tion s = s ( ξ ) of the error ξ (see Eq. (13)). In particular, the measurement does not changethe state. The classical syndrome measurement outcome is then used to compute the cor-rection operation ( U ( N ) ) − D ( ˆ ξ ( s ( ξ ))) − U ( N ) , which is applied in step 3 and yields the state( U ( N ) ) − D ( ˆ ξ ( s ( ξ ))) − D ( ξ ) | ¯Ψ (cid:105) . Finally, in step 4, the corrected state is encoded again. Theoutput is the state D ( ˆ ξ ( s ( ξ ))) − D ( ξ ) | Ψ (cid:105) . A (simultaneous) measurement of the family of commuting operators (11) is equivalent to thejoint measurement of the commuting set of the 2( N − K ) modular operators (cid:32) N (cid:88) k =1 S j,k R k (cid:33) mod √ π where j = 2 K + 1 , . . . , N , (12)according to Eq. (3). Measuring a corrupted code state D ( ξ ) | ¯Ψ (cid:105) , where ξ ∈ R N , yields thesyndrome s = ( s , . . . , s N − K ) ) ∈ [ − (cid:112) π/ , (cid:112) π/ N − K ) with s j = (cid:32) N (cid:88) k =1 S K + j,k ξ k (cid:33) mod √ π for j = 1 , . . . , N − K ) . (13)(This again follows from the commutation relations (7).) We will write s = s ( ξ ) to emphasizethat the syndrome is a function of the displacement vector ξ . This joint measurement can forexample be implemented by applying ( U ( N ) ) − to the state to be measured using the measure-ment circuits mentioned in Section 2.3 to measure the one-mode modular operators ( S Q j , S P j )for j = K + 1 , . . . , N , and applying U ( N ) again to the post-measurement state (cf. Fig. 3).10 .4.2 Error recovery and logical error Recovery from a displacement error D ( ξ ) proceeds as in any stabilizer code by first extractingthe syndrome s ∈ [ − (cid:112) π/ , (cid:112) π/ N − K ) by measurement, and subsequently applying a unitarycorrection operation to the post-measurement state. In the case under consideration, a recoverystrategy is specified by an estimator (function)ˆ ξ : [ − (cid:112) π/ , (cid:112) π/ N − K ) → R N s (cid:55)→ ˆ ξ ( s ) (14)for the actual error ξ , based on the syndrome s . When observing the syndrome s , the correctionoperation D ( ˆ ξ ( s )) − is applied. The function ˆ ξ is typically chosen such that the syndrome forthe error D ( ˆ ξ ( s )) is identical to s (i.e., s ( ˆ ξ ( s )) = s ), to ensure that this operation returns thestate to the code space.The resulting state when starting from the corrupted state D ( ξ ) | ¯Ψ (cid:105) is thus D ( ˆ ξ ( s ( ξ ))) − D ( ξ ) | ¯Ψ (cid:105) . Up to an irrelevant global phase (cf. (6)), the resulting effect on the physical modes is adisplacement by the vector η = η ( ξ ) := − ˆ ξ ( s ( ξ )) + ξ ∈ R N . (15) Maximum likelihood error decoding.
We will assume that we are dealing with (inde-pendent and identical) classical noise with variance σ on each mode, see Eq. (8). That is,the noise-corrupted encoded state is N ⊗ Nσ ( | ¯Ψ (cid:105)(cid:104) ¯Ψ | ). In this case, the displacement error vec-tor ξ ∈ R N has a centered normal prior distribution, i.e., ξ ∼ N (0 , σ I N ). Maximum likeli-hood error decoding then amounts to choosing the error that is most likely given the observedsyndrome s ∈ [ − (cid:112) π/ , (cid:112) π/ N − K ) . In other words, the estimator function is given byˆ ξ ML ( s ) := argmax ξ ∈ R N f Z | s ( Z )= s ( ξ ) , (16)where f Z | s ( Z )= s is the conditional probability density function when Z ∼ N (0 , σ I N ) is condi-tioned on s ( Z ) = s . Here and below, ties in expressions such as (16) when taking argmax arebroken arbitrarily. By Bayes’ rule, this definition is equivalent toˆ ξ ML ( s ) = argmax ξ ∈{ z ∈ R N | s ( z )= s } f Z ( ξ ) . (17) Logical error and figure of merit.
Recall that the overall effect of a displacement errorand subsequent correction is given by a displacement vector η as in (15). To see what the effecton the logical information is, let us assume that U ( N ) = U S is given by the symplectic matrix S .Observe that for x = Sη we have D ( η ) | Ψ (cid:105) = U S D ( x ) (cid:0) | Ψ (cid:105) ⊗ | GKP (cid:105) ⊗ N − K (cid:1) = U S (cid:0) D ( x A ) | Ψ (cid:105) ⊗ D ( x B ) | GKP (cid:105) ⊗ N − K (cid:1) , where we used the fact that D ( η ) U S = U S D ( Sη ), and where we write x = ( x A , x B ) ∈ R K × R N − K ) ∼ = R N . Under the assumption that D ( ˆ ξ ( s )) causes syndrome s , the overall11isplacement D ( η ) maps the code space to itself, and thus D ( x B ) stabilizes | GKP (cid:105) ⊗ N − K . Inparticular, we conclude that D ( η ) U S (cid:0) | Ψ (cid:105) ⊗ | GKP (cid:105) ⊗ N − K (cid:1) = U S (cid:0) D ( x A ) | Ψ (cid:105) ⊗ | GKP (cid:105) ⊗ N − K (cid:1) , which shows that the logical error is a displacement by x A , i.e., by the vector x A ∈ R K consisting of the first 2 K components of S ( − ˆ ξ ( s ( ξ )) + ξ ).Note that x A = x A ( ξ ) is a deterministic function of the displacement error vector ξ de-pending on the choice of the estimator function (14). In the case where ξ ∼ N (0 , σ I N ), x A is a random variable supported on R K . To quantify error suppression, we may use e.g., thevariance of this random variable (as in [17, 18] for example). Here we use a slightly differentmeasure: for any (cid:15) >
0, we set P succ ( (cid:15) ) := Pr ξ ∼N (0 ,σ I N ) [ (cid:107) x A ( ξ ) (cid:107) ≤ (cid:15) ] , (18)where (cid:107) y (cid:107) := (cid:16)(cid:80) Kj =1 | y j | (cid:17) / is the Euclidean norm of y ∈ R K . In other words, we areinterested in the probability that the resulting logical error after recovery belongs to an (cid:15) -ballin R K centered at the origin. We call P succ ( (cid:15) ) the decoding success probability. In this section, we reformulate the recovery success probability (18) with respect to maximumlikelihood error decoding of an oscillator-to-oscillator code subject to Gaussian classical noise interms of a purely classical estimation problem in Section 3.1. In Section 3.2, we then summarizeour main upper bound on this quantity.
Our goal is to establish an upper bound on the quantity (18) when maximum likelihood esti-mation of the (physical) error is used. To this end, we first reformulate this quantity in termsof an estimation problem of purely classical information-theoretic nature.This reformulation is guided by the measurement procedure for the operators (12). Supposethe code state | Ψ (cid:105) undergoes a displacement D ( ξ ). Using that U † S D ( ξ ) = D ( Sξ ) U † S , we have(similarly as before) U † S D ( ξ ) | Ψ (cid:105) = D ( x )( | Ψ (cid:105) ⊗ | GKP (cid:105) ⊗ N − K ) where x = Sξ ∈ R N . Writing x = ( x A , x B ) ∈ R K × R N − K ) , we have D ( x ) = D ( x A ) ⊗ D ( x B ), and thus measurementof GKP-state stabilizers S Q j , S P j on the modes j ∈ { K + 1 , . . . , N } provides the syndrome s = x B mod √ π ∈ [ − (cid:112) π/ , (cid:112) π/ N − K ) , where the modulo-operation is applied to each entry of x B .If the original error is distributed according to ξ ∼ N (0 , σ I N ), then x ∼ N (0 , σ ( S T S ) − )by definition of x . Furthermore, if ˆ ξ ML is the maximum likelihood estimator for ξ (based on the12yndrome s (cf. (17))), then ˆ x ML = S ˆ ξ ML is the maximum likelihood estimator for x . It can bewritten as ˆ x ML ( s ) = argmax x ∈{ x ∈ R N | x B mod √ π = s } f X ( x ) , where f X is the distribution function of the random variable X ∼ N (0 , σ ( S T S ) − ). In partic-ular, the residual logical error after error correction is given by the first 2 K components of thevector − ˆ x ML ( s ) + x . We have thus arrived at the following reformulation: Let Π A : R N → R K be the projection map taking a vector x ∈ R N to a vector with only its first 2 K components.Similarly, let Π B : R N → R N − K ) denote the projection map onto the last 2( N − K ) com-ponents. Then the decoding success probability (18) (when using maximum likelihood errordecoding) can be written as P succ ( (cid:15) ) = Pr X ∼N (0 ,σ ( S T S ) − ) (cid:104)(cid:13)(cid:13)(cid:13) Π A ( X ) − Π A (cid:16) ˆ x ML (Π B X mod √ π ) (cid:17)(cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:105) . In other words, P succ ( (cid:15) ) can be understood as the probability of (cid:15) -approximately estimating thefirst 2 K components of a random Gaussian vector X , given the remaining 2( N − K ) modulo- √ π -reduced components. In Section 4 we will analyze this estimation problem more generally. We consider a randomvector X ∈ N (0 , Σ) on R n drawn according to a n -variate centered normal distribution withcovariance matrix Σ, and study the probability of (cid:15) -correctly estimating the first k componentsof X , given the modulo-∆-reduced values of the remaining n − k components (using maximumlikelihood estimation of the entire vector). One of our main results (Corollary 4.13) concerningthis problem is the upper bound P succ ( (cid:15) ) ≤ µ Z (cid:16) B (cid:15)/ √ λ min (Σ) (0) (cid:17) , where µ Z is the probability measure of a centered normal Gaussian vector Z ∼ N (0 , I k ), andwhere B δ (0) ⊂ R k is the closed δ -ball with respect to the Euclidean norm. For an oscillator-to-oscillator code encoding K logical modes into N physical modes, we have ( n, k ) = (2 N, K )and (∆ , Σ) = ( √ π, σ ( S T S ) − ) . Since 1 /λ min (Σ) = λ max (Σ − ) we obtain (cf. (5)) the following: Theorem 3.1.
Let (cid:15) > . The decoding success probability P succ ( (cid:15) ) when encoding K into N modes using an oscillator-to-oscillator code with a Gaussian encoding unitary U satisfies P succ ( (cid:15) ) ≤ µ Z (cid:0) B (cid:15) · sq ( U ) /σ (0) (cid:1) , where µ Z is the probability measure of a centered normal Gaussian vector Z ∼ N (0 , I K ) . Since the norm of a 2 K -dimensional centered normal vector is close to √ K with highprobability (see e.g., [26, Theorem 3.1.1]) this implies that the amount of squeezing sq ( U ) hasto grow roughly as √ Kσ/(cid:15) in the decoding error (cid:15) in order to achieve a success probabilityclose to 1. This establishes our no-go result: with a bounded amount of squeezing, an arbitrarilysmall decoding error cannot be achieved. 13
Unwrapping Modulo Reduced Gaussian Vectors
In this section, we derive our main result concerning a classical estimation problem. We referto it as partial unwrapping of modulo reduced Gaussian vectors.In Section 4.1, we first review the informed unwrapping problem of modulo reduced Gaussianvectors, which has been studied before. In Section 4.2, we then introduce the partial unwrappingproblem. In Section 4.3, we give a high-level overview of how we obtain an upper bound on thecorresponding figure of merit, the decoding success probability.We then derive our technical results: In Section 4.4, we show how the decoding success prob-ability can be written as the probability mass of a countable union of displaced and thickeneddegenerate Voronoi cells under a centered unit-variance Gaussian normal distribution. Herethe relevant displacements are given by a lattice. In Section 4.5, we describe a polytope that isan outer bound on the degenerate Voronoi cell. A parametrization of this polytope (exploitingthe fact that it is lower-dimensional) is described in Section 4.6. In Section 4.7, we show how toparametrize the union of translates of this polytope (up to zero measure sets). In Section 4.8,we show how to integrate over thickened versions of these polytopes, and give a general upperbound on the probability mass of the union of thickened and displaced polytopes. Finally, inSection 4.9, we combine these results to obtain a general upper bound on the decoding successprobability for the partial unwrapping problem.
Let n > > ∈ Mat n × n ( R ) the covariance matrix ofa normal distribution on R n . We note that such matrices are positive definite and symmetric,a fact we will use below. Let X = ( X , . . . , X n ) ∼ N (0 , Σ) be a random n -dimensional vectordrawn according to the centered normal distribution with covariance matrix Σ. Let X ∗ bethe n -dimensional vector in [ − ∆ / , ∆ / n whose j -th entry X ∗ j is obtained by reducing thecoordinate X j modulo ∆.The informed unwrapping problem of modulo reduced Gaussian vectors asks to reconstruct X from X ∗ . This problem arises naturally in signal processing since the modulo operation appliedto x ∈ R amounts to discarding the most significant bits in the binary representation of x . Thetask is thus to reconstruct the Gaussian vector from the least significant bits of its coordinates.The expression informed refers to the fact that the covariance matrix Σ is known to the decoder.The MAP (for maximum a posteriori likelihood) estimator ˆ x MAP : [ − ∆ / , ∆ / n → R n forthis problem is the function that maximizes the decoding probability Pr [ˆ x ( X ∗ ) = X ], i.e.,ˆ x MAP ( z ) = argmax y ∈ R n : y ∗ = z f Σ ( y ) , z ∈ [ − ∆ / , ∆ / n , (19)where f Σ ( x ) = 1(2 π ) n/ (det Σ) / e − x T Σ − x (20)is the probability density function of X . It can be shown (see [22, Section III]) that thecorresponding success probability is related to the probability that the random variable Z =Σ − / X ∼ N (0 , I n ) belongs to the Voronoi region V := { z ∈ R n | (cid:107) z (cid:107) ≤ (cid:107) z − ∆Σ − / b (cid:107) for all b ∈ Z n }
14f the lattice ∆Σ − / Z n , i.e., Pr (cid:2) ˆ x MAP ( X ∗ ) = X (cid:3) = Pr [ Z ∈ V ] . (21)Here (cid:107) x (cid:107) := (cid:113)(cid:80) nj =1 x j denotes the Euclidean norm of x ∈ R n . With (21), the decoding errorprobability can be related to the quantity ∆ / (det Σ) n/ , see [19, 7, 20]. Here we consider a variant of the above problem which we refer to as partial informed unwrap-ping of modulo reduced Gaussian vectors. In this variant, one is given n − k modulo reducedentries of a random vector X = ( X , . . . , X n ) ∼ N (0 , Σ) (where 0 < k < n ). The task is toreconstruct the k remaining coordinates of X .To describe this in more detail, let us write n -dimensional vectors as x = ( x , . . . , x n ) = (cid:0) x A x B (cid:1) ∈ R k × R n − k where x A = ( x , . . . , x k ) and x B = ( x k +1 , . . . , x n ). It will also be useful tointroduce corresponding projection mapsΠ A : R n → R k (cid:0) x A x B (cid:1) (cid:55)→ x A and Π B : R n → R n − k (cid:0) x A x B (cid:1) (cid:55)→ x B . For a realization x = (cid:0) x A x B (cid:1) ∈ R n of X , let x ∗ B := ((Π B x ) mod ∆) ∈ [ − ∆ / , ∆ / n − k , (22)where the modulo operation is applied entrywise.Let X = (cid:0) X A X B (cid:1) ∼ N (0 , Σ). The partial informed unwrapping problem asks to reconstruct X A from X ∗ B . For a given estimator h : [ − ∆ / , ∆ / n − k → R k and an error tolerance (cid:15) >
0, we call p hS ( (cid:15) ) = Pr [ (cid:107) h ( X ∗ B ) − X A (cid:107) ≤ (cid:15) ]the decoding success probability of h . A natural choice of decoder for this problem relies onthe estimator (analogous to Eq. (19)) ˆ x MAP : [ − ∆ / , ∆ / n − k → R n for x defined byˆ x MAP ( z ) = argmax y ∈ R n : y ∗ B = z f Σ ( y ) (23)and uses the concatenation h MAP := Π A ◦ ˆ x MAP of this map with the projection map Π A . Wecall this the MAP-decoder. In the following, we will study the associated decoding successprobability denoted by p S ( (cid:15) ). Here we give a high-level overview of our derivation of an upper bound for the decoding successprobability p S ( (cid:15) ) in Sections 4.4-4.9.To find the upper bound, we first establish a connection to the probability mass (under thecentered standard normal distribution) of certain subsets of R n analogous to Eq. (21). To statethis, let V Σ − / := (cid:26) z ∈ R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13) z − Σ − / (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k (cid:27) (24)15enote the Voronoi cell of the (degenerate) lattice Σ − / ( R k × ∆ Z n − k ). Let V Σ − / ( (cid:15) ) := V Σ − / + Σ − / (cid:0) B (cid:15) (0) × { } n − k (cid:1) . be an (cid:15) -“thickening” of V Σ − / . In this expression, X + Y := { x + y | x ∈ X , y ∈ Y} is theMinkowski sum of two subsets X , Y ⊂ R n , and B (cid:15) ( y ) := { h ∈ R k | (cid:107) h − y (cid:107) ≤ (cid:15) } is the closed (cid:15) -ball around y ∈ R k . The relevant subset is a union of translates of the set V Σ − / ( (cid:15) )of the form Σ − / (cid:18) b (cid:19) + V Σ − / ( (cid:15) ) with b ∈ Z n − k . That is, we have the following: The decoding success probability p S ( (cid:15) ) is given by p S ( (cid:15) ) = Pr (cid:2) Z ∈ Σ − / ( { } k × ∆ Z n − k ) + V Σ − / ( (cid:15) ) (cid:3) where Z ∼ N (0 , I n ) . (25)We will prove (25) below, see Theorem 4.2.To proceed, we find an outer bound on the set Σ − / ( { } k × ∆ Z n − k ) + V Σ − / ( (cid:15) ). This allowsus to obtain an upper bound on the quantity (25) which depends on the marginal distributionof X B = Π B X , and the conditional distribution of X A = Π A X given that X B = x B . It is ofthe form p S ( (cid:15) ) ≤ (cid:88) b ∈ Z n − k (cid:90) R b f X B ( ξ ) · Pr [ Y ∈ B (cid:15) ( c ( ξ, ∆ b ))] dξ , (26)where {R b } b ∈ Z n − k is a collection of measurable subsets of R n − k such that (cid:80) b ∈ Z n − k (cid:82) R b f X B ( ξ ) dξ ≤
1, the ball centers c ( ξ, ∆ b ) ∈ R k are located at c ( ξ, ∆ b ) := Γ ξ + Γ ∆ b for certain matrices Γ , Γ depending on Σ (see Theorem 4.12), and Y ∼ N (0 , Σ ∗ ) is a normaldistributed random variable on R k whose covariance matrix is the Schur complementΣ ∗ = Σ AA − Σ AB (Σ BB ) − Σ BA . Expression (26) depends on both Σ and ∆ in general. With Anderson’s inequality [2], we obtainan upper bound of the form p S ( (cid:15) ) ≤ Pr (cid:104) Z ∈ B (cid:15)/ √ λ min (Σ) (0) (cid:105) where Z ∈ N (0 , I k ) , with λ min (Σ) denoting the smallest eigenvalue of Σ, see Corollary 4.13. This appears rathercrude, especially as it does not depend on ∆. For our purposes, however, this bound turns outto be sufficient. 16 .4 Decoding success probability and displaced Voronoi cells Let X ∼ N (0 , Σ) be as before and consider the function ˆ x MAP : [ − ∆ / , ∆ / n − k → R n definedby (23). Let u ∈ [ − ∆ / , ∆ / n − k . Since every y ∈ R n with y ∗ B = u is of the form y = (cid:0) hu +∆ b (cid:1) for some ( h, b ) ∈ R k × Z n − k , we can writeˆ x MAP ( u ) = (cid:18) h ( u ) u + ∆ b ( u ) (cid:19) where (cid:18) h ( u ) b ( u ) (cid:19) := arg max( hb ) ∈ R k × Z n − k f Σ (cid:18)(cid:18) hu + ∆ b (cid:19)(cid:19) . (27)The decoder h MAP := Π A ◦ ˆ x MAP is then given by the function h and has success probability p S ( (cid:15) ) = Pr [ X ∈ R ( (cid:15) )] with R ( (cid:15) ) := (cid:8) x ∈ R n (cid:12)(cid:12) (cid:107) Π A x − h ( x ∗ B ) (cid:107) ≤ (cid:15) (cid:9) . The set R ( (cid:15) ) of realizations x of the random vector X which lead to decoding success can beexpressed as follows. Lemma 4.1.
Let (cid:15) > . Then x ∈ R ( (cid:15) ) if and only if z := Σ − / x satisfies the following. Thereis some c ∈ B (cid:15) (0) × ∆ Z n − k such that (cid:107) z − Σ − / c (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13) z − Σ − / (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k . (28) Proof.
Suppose that x ∈ R ( (cid:15) ) and x ∗ B is defined by (22). By the definitions (27), (20), we have (cid:18) h ( x ∗ B ) b ( x ∗ B ) (cid:19) = arg max( hb ) ∈ R k × Z n − k f Σ (cid:18)(cid:18) hx ∗ B + ∆ b (cid:19)(cid:19) = arg min( hb ) ∈ R k × Z n − k (cid:18) hx ∗ B + ∆ b (cid:19) T Σ − (cid:18) hx ∗ B + ∆ b (cid:19) = arg min( hb ) ∈ R k × Z n − k (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) hx ∗ B + ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , (29)where the last equation follows by symmetry of the covariance matrix Σ. Eq. (29) and thedefinitions of R ( (cid:15) ) and h ( x ∗ B ) imply that x ∈ R ( (cid:15) ) if and only if there is some h (cid:48) ∈ B (cid:15) (Π A x )and b (cid:48) ∈ Z n − k such that (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) h (cid:48) x ∗ B + ∆ b (cid:48) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) h (cid:48) x ∗ B + ∆ b (cid:48) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h (cid:48) , b (cid:48) ) ∈ R k × Z n − k . By definition of x ∗ B as the modulo-∆ reduced vector Π B x , we have the following: for every b (cid:48) ∈ Z n − k , there is some b ∈ Z n − k such that x ∗ B + ∆ b (cid:48) = Π B x + ∆ b . We can apply thissubstitution to both b (cid:48) and b (cid:48) , yielding the existence of some h (cid:48) ∈ B (cid:15) (Π A x ) and b ∈ Z n − k suchthat (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) h (cid:48) Π B x + ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) h (cid:48) Π B x + ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h (cid:48) , b ) ∈ R k × Z n − k . Let us write h (cid:48) = Π A x + h with h ∈ B (cid:15) (0) and similarly h (cid:48) = Π A x + h . We then concludethat there exists h ∈ B (cid:15) (0) and b ∈ Z n − k such that (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) Π A x + h Π B x + ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Σ − / (cid:18) Π A x + h Π B x + ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k ,
17r equivalently (cid:13)(cid:13)(cid:13)(cid:13) z + Σ − / (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) z + Σ / (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k . The claim follows because (cid:0) h ∆ b (cid:1) ∈ B (cid:15) (0) × ∆ Z n − k by definition, and thus − (cid:0) h ∆ b (cid:1) ∈ B (cid:15) (0) × ∆ Z n − k since this set is symmetric around the origin. Theorem 4.2.
Let V Σ − / be the Voronoi cell of the (degenerate) lattice Σ − / ( R k × ∆ Z n − k ) defined by (24) . Let (cid:15) > . Then p S ( (cid:15) ) = Pr (cid:2) Z ∈ Σ − / ( B (cid:15) (0) × ∆ Z n − k ) + V Σ − / (cid:3) , where Z ∼ N (0 , I n ) .Proof. Lemma 4.1 states that z ∈ Σ − / R ( (cid:15) ) if and only if there is some c ∈ B (cid:15) (0) × ∆ Z n − k such that (28) holds. But since the inequality (28) holds for all ( h, b ) ∈ R k × Z n − k and c is ofthe form c = (cid:0) h ∆ b (cid:1) with ( h , b ) ∈ R k × Z n − k by definition, condition (28) is equivalent to (cid:107) z − Σ − / c (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ( z − Σ − / c ) − Σ − / (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k . This shows that z ∈ Σ − / R ( (cid:15) ) if and only if z − Σ − / c ∈ V Σ − / for some c ∈ B (cid:15) (0) × ∆ Z n − k ,i.e., Σ − / R ( (cid:15) ) = V Σ − / + Σ − / ( B (cid:15) (0) × ∆ Z n − k ) . (30)Since Z := Σ − / X ∼ N (0 , I n ) and p s ( (cid:15) ) = Pr [ X ∈ R ( (cid:15) )], this implies the claim. To derive an upper bound on p S ( (cid:15) ), we need detailed information about the Voronoi cell V Σ − / and the union of its thickened displaced versions (cf. (30)). For brevity, we use the shorthandΛ := Σ − / in the following. Recall that this matrix is positive definite and symmetric. The setof interest is V Λ := (cid:26) z ∈ R n (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) z (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13) z − Λ (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all ( h, b ) ∈ R k × Z n − k (cid:27) . (31)This is the Voronoi cell of the degenerate lattice Λ( R k × ∆ Z n − k ). It can be thought of in severaldifferent ways: on the one hand, it is the intersection of an uncountable number of half-spaces H ( h,b ) := (cid:40) z ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) Λ (cid:18) h ∆ b (cid:19) , z (cid:29) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Λ (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) (cid:41) with ( h, b ) ∈ R k × Z n − k . (32)This is illustrated in Fig. 4(a). 18 - - - - - (a) Illustration of an uncountable number of half-spaces defining V Λ . Every pair ( h, b ) ∈ R k × Z n − k defines a half-space H ( h,b ) (cf. (32)). Half-spaces as-sociated to pairs ( h,
0) with b = 0 are shown in red. - - - - - - (b) A countable number of surfaces defined byquadratic constraints. Every b ∈ Z n − k \{ } defines aset E b bounded by a surface defined by a quadraticform (cf. (33)). The set E is a hyperplane (red). Figure 4: Different interpretations for the degenerate Voronoi cell V Λ . The set V Λ is theintersection of the yellow lemon-shaped region and the red line in Fig. 4(b).Alternatively, the set V Λ can be understood as the intersection of a countable number ofsets of the form E b := (cid:40) z ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) Λ (cid:18) h ∆ b (cid:19) , z (cid:29) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Λ (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all h ∈ R k (cid:41) with b ∈ Z n − k , (33)see Fig. 4(b) for an illustration. Each set E b with b (cid:54) = 0 is of the form E b := { z ∈ R n | q b ( z ) ≥ } for a quadratic form q b : R n → R . The explicit form of q b is given in Lemma A.1 in Appendix A.The set E is specified by linear equality constraints, see Lemma 4.3 below.Our goal here is not to provide a full characterization of this set. Instead, we establisha few necessary conditions for v ∈ R n to belong to this set. To state these conditions, let (cid:104) x, y (cid:105) = (cid:80) nj =1 x j y j denote the standard inner product on R n , and let { e j } nj =1 be the canonicalorthonormal basis of R n .Let ˆΛ ∈ Mat k × ( n − k ) ( R ) be an arbitrary matrix (to be chosen later). Consider the lattice L generated by the matrix L (Λ , ˆΛ , ∆) := Λ · (cid:18) I ∆ ˆΛ0 ∆ I (cid:19) , (34)i.e., L = L Z n . (Note that L has non-zero determinant, whence L is well-defined.) Let V ( L )denote the Voronoi cell of L . By definition, v ∈ V ( L ) if and only if |(cid:104) x, v (cid:105)| ≤ (cid:107) x (cid:107) x ∈ L . (35)19he following shows that the Voronoi cell V ( L ) of L contains the set V Λ of interest. Lemma 4.3.
Let v ∈ V Λ . Then (cid:104) Λ e j , v (cid:105) = 0 for j = 1 , . . . , k (36) and V Λ ⊂ V ( L ) . (37) Proof.
Let v ∈ V Λ and j = 1 , . . . , k . We have (cid:107) v (cid:107) ≤ (cid:107) v − Λ δe j (cid:107) for all δ ∈ R by choosing (cid:0) h ∆ b (cid:1) = (cid:0) h (cid:1) = δe j accordingly in Eq. (31). This implies (36).Similarly, choosing (cid:0) h ∆ b (cid:1) = (cid:0) a +ˆΛ∆ b ∆ b (cid:1) with ( a, b ) ∈ Z k × Z n − k yields the condition (cid:107) v (cid:107) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v − Λ (cid:18) a + ˆΛ∆ b ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) for all ( a, b ) ∈ Z n for any v ∈ V Λ . This is equivalent to (cid:107) v (cid:107) ≤ (cid:107) v − x (cid:107) for all x ∈ L , and thus (37) follows.Let P Λ be the set of v ∈ V ( L ) which satisfy (36). Lemma 4.4. P Λ is an ( n − k ) -dimensional polytope.Proof. The Voronoi cell V ( L ) is a polytope, i.e., the intersection of a finite number of halfspaces.Furthermore, it is an n -dimensional polytope. (For a detailed proof of these and related facts,see e.g., [6, Chapter 4].) The definition of the set P Λ additionally includes the linear equalityconstraints (36) (which are all independent since the matrix Λ is invertible), hence the claimfollows.By Lemma 4.3, we have V Λ ⊂ P Λ , i.e., V Λ is contained in the ( n − k )-dimensional polytope P Λ . We will use P Λ as a proxy for V Λ in the following. P Λ and choice of ˆΛ Lemma 4.4 shows that n − k real parameters suffice to parametrize the set P Λ . We nowmake this parametrization explicit. To do so, it is convenient to make a specific choice ofˆΛ ∈ Mat k × ( n − k ) ( R ) (see Eq. (38) below).It is easy to show (using a proof analogous to that of Lemma 4.5 below) that every element v ∈ P Λ is uniquely determined by its projection Π B v . For our purposes, it is more convenientto derive such a statement not for P Λ , but for the transformed set Λ − P Λ .To state this result, we will henceforth identify linear maps M : R n → R n with theirassociated matrix M i,j = (cid:104) e i , M e j (cid:105) when expressed in the standard basis. We also define blockmatrices corresponding to the partition R n ∼ = R k × R n − k , i.e., we write M = (cid:18) M AA M AB M BA M BB (cid:19) . - - - - - Figure 5: Translates of the Voronoi cell of the lattice L Z n , the degenerate Voronoi cell V Λ (intersection of the yellow lemon-shaped region with the red straight line), and the polytope P Λ (intersection of the rectangular blue region with the red straight line). Lemma 4.5.
Let ˆΛ : R n − k → R k be defined as ˆΛ = − ((Λ ) AA ) − (Λ ) AB . (38) Then (Π A − ˆΛΠ B )Λ − P Λ = { } . (39) Proof.
Let w ∈ Λ − P Λ . Then Λ w ∈ P Λ and thus (cid:104) Λ e j , Λ w (cid:105) = 0 for all j = 1 , . . . , k (40)by definition of P Λ . Using the assumption that Λ T = Λ is symmetric, we have (cid:104) Λ e j , Λ w (cid:105) = (cid:104) e j , Λ w (cid:105) = (cid:104) e j , Π A Λ w (cid:105) = (cid:104) e j , (Λ ) AA Π A w + (Λ ) AB Π B w (cid:105) for all j = 1 , . . . , k . Inserted into (40), we conclude that(Λ ) AA Π A w = − (Λ ) AB Π B w . The matrix (Λ ) AA is a principal submatrix of the positive definite matrix Λ , hence it is alsopositive definite and thus invertible. Since w ∈ Λ − P Λ was arbitrary, this implies thatΠ A w = ˆΛΠ B w for all w ∈ Λ − P Λ , which is the claim. 21q. (39) of Lemma 4.5 shows that the map ϕ : Π B Λ − P Λ → Λ − P Λ z (cid:55)→ (cid:0) ˆΛ zz (cid:1) (41)is bijective. Thus we have obtained a bijective parametrization of the transformed poly-tope Λ − P Λ . The following is well-known (see e.g., [6, Chapter 4]): If V is the Voronoi cell of the lattice Λ Z n ,then the lattice translates Λ c + V , c ∈ Z n are a tiling of R n , i.e., R n = (cid:91) c ∈ Z n (Λ c + V ) , int (Λ c + V ) ∩ int (Λ c (cid:48) + V ) = ∅ if c, c (cid:48) ∈ Z n and c (cid:54) = c (cid:48) , where int ( M ) denotes the interior of the set M . Furthermore, the intersection (Λ c + V ) ∩ (Λ c (cid:48) + V ) (if non-empty) is a face of both Λ c + V and Λ c (cid:48) + V . We show a similar statementfor the polytope P Λ , again considering Λ − P Λ for convenience. In particular, we give a partialcharacterization of the intersection of two translates.In more detail, our goal is to give a parametrization of the union of translates (cid:0) b (cid:1) +Λ − P Λ with b ∈ Z n − k of the transformed polytope Λ − P Λ . Since the latter can be uniquelyparametrized by the projected set Π B Λ − P Λ (see Eq. (41)), we would like to show that pro-jecting the translates also leads to disjoint sets. We show that this is essentially the case up tozero-measure sets, see Lemma 4.7 below.As a preparation, let us first consider translations of the transformed polytope Λ − P Λ alongcertain lattice directions of the lattice Λ − L . Recall that by definition of L (cf. (34)), thelattice Λ − L consists of points (cid:0) a +ˆΛ∆ b ∆ b (cid:1) with ( a, b ) ∈ Z k × Z n − k . Lemma 4.6.
For b ∈ Z n − k set U b := (cid:18) ˆΛ∆ b ∆ b (cid:19) + Λ − P Λ . Let b, b (cid:48) ∈ Z n − k with b (cid:54) = b (cid:48) . Then Π B ( U b ∩ U b (cid:48) ) is contained in a polytope of dimension ( n − k − .Proof. By considering the difference of b and b (cid:48) , we can assume that b (cid:54) = 0 and b (cid:48) = 0 withoutloss of generality. Suppose that w ∈ U b ∩ U . The fact that w ∈ U b implies that there is some w (cid:48) ∈ Λ − P Λ such that w = w (cid:48) + Λ − x where x = Λ (cid:18) ˆΛ∆ b ∆ b (cid:19) ∈ L . (42)22bserve that x (cid:54) = 0 as b (cid:54) = 0 and the (linear) map that takes b to x is invertible. By Eq. (35)and because w, w (cid:48) ∈ Λ − P Λ we have (cid:107) x (cid:107) − · |(cid:104) x, Λ w (cid:105)| ≤ / (cid:107) x (cid:107) − · |(cid:104) x, Λ w (cid:48) (cid:105)| ≤ / . Suppose that (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:48) (cid:105) > − /
2. Then (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:105) = (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:48) + x (cid:105) = (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:48) (cid:105) + 1 (44) > − / / , contradicting (43). It follows that we must have (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:48) (cid:105) = − / , and with Eq. (44) also (cid:107) x (cid:107) − · (cid:104) x, Λ w (cid:105) = 1 / . In particular, every w ∈ U b ∩ U satisfies the linear constraint (cid:96) ( w ) = c where (cid:96) ( w ) := (cid:104) x, Λ w (cid:105) and c := (cid:107) x (cid:107) / , (45)with x defined by (42).Let us now parametrize U b ∩ U ⊂ Λ − P Λ by the map ϕ (cf. (41)), i.e., we write w = ϕ ( z )for some z ∈ Π B Λ − P Λ . Because (cid:96) ( w ) = (cid:42) Λ (cid:18) ˆΛ∆ b ∆ b (cid:19) , Λ w (cid:43) , we have (cid:96) ( ϕ ( z )) = ∆ · (cid:42) Λ (cid:18) ˆΛ bb (cid:19) , Λ (cid:18) ˆΛ zz (cid:19)(cid:43) = ∆ · (cid:104) Kb, Kz (cid:105) , where we introduced the map K : R n − k → R n b (cid:55)→ Λ (cid:0) ˆΛ bb (cid:1) . The map K is invertible with inverse K − = Π B ◦ Λ − and thus has full rank. It follows that K T K is positive definite. With Eq. (45), we have thus obtained a non-trivial linear constraintof the form (cid:104) K T Kb, z (cid:105) = c/ ∆ for any z ∈ ϕ − ( U b ∩ U ) ⊂ Π B Λ − P Λ . (46)In other words, ϕ − ( U b ∩U ) is the intersection of the n − k − B Λ − P Λ . Noting that linear transformations, as well asintersections with hyperplanes, map polytopes to polytopes, we conclude that ϕ − ( U b ∩ U ) iscontained in a polytope of dimension n − k −
1. Since Π B w = Π B ϕ ( z ) = z ∈ ϕ − ( U b ∩ U ) for w ∈ U b ∩ U , the same conclusion holds for Π B ( U b ∩ U ) ⊂ R n − k .23e now consider translations with respect to the standard rectangular grid instead of thelattice L used in the definition of P Λ . Lemma 4.7.
For b ∈ Z n − k define S b := (cid:18) b (cid:19) + Λ − P Λ . (47) Let b, b (cid:48) ∈ Z n − k with b (cid:54) = b (cid:48) . Then Π B ( S b ) ∩ Π B ( S b (cid:48) ) is contained in a polytope of dimension ( n − k − .Proof. Without loss of generality (cf. the proof of Lemma 4.6), assume that b (cid:54) = 0 and b (cid:48) = 0.Suppose z ∈ Π B S b ∩ Π B S . Since z ∈ Π B S we have ϕ ( z ) ∈ S , i.e., (cid:18) ˆΛ zz (cid:19) ∈ Λ − P Λ . Since z ∈ Π B S b we have ϕ ( z − ∆ b ) ∈ Λ − P Λ , that is, (cid:18) ˆΛ( z − ∆ b ) z − ∆ b (cid:19) ∈ Λ − P Λ . With v := (cid:0) ˆΛ zz (cid:1) it follows that v ∈ Λ − P Λ and v − (cid:18) ˆΛ∆ b ∆ b (cid:19) ∈ Λ − P Λ , or v ∈ Λ − P Λ ∩ (cid:32)(cid:18) ˆΛ∆ b ∆ b (cid:19) + Λ − P Λ (cid:33) = U ∩ U b . With Lemma 4.6, we conclude that Π B v = z belongs to a polytope of dimension ( n − k − (cid:83) b ∈ Z n − k S b of translates S b of theset Λ − P Λ . To this end, we are going to express (cid:83) b ∈ Z n − k S b as the disjoint union of a countablenumber of measurable sets for which we have a unique parametrization (due to Lemma 4.7),and some measure-zero set (see Lemma 4.9). We will need the following lemma, which showsthat any translate intersects only with a finite number of other translates.24 emma 4.8. For b ∈ Z n − k , let S b ⊂ R n be defined by (47) . Then the following holds: For any b ∈ Z n − k , there are only finitely many b (cid:48) ∈ Z n − k such that S b ∩ S b (cid:48) (cid:54) = ∅ .Proof. Let b ∈ Z n − k be given. If v ∈ S b ∩ S b (cid:48) for some b (cid:48) ∈ Z n − k then there are w, w (cid:48) ∈ P Λ suchthat (cid:18) b (cid:19) + Λ − w = (cid:18) b (cid:48) (cid:19) + Λ − w (cid:48) . It follows that (cid:107) b − b (cid:48) (cid:107) = (cid:107) ∆ − Λ − ( w − w (cid:48) ) (cid:107)≤ λ max (cid:0) ∆ − Λ − (cid:1) · (cid:107) w − w (cid:48) (cid:107)≤ R ∆ − λ max (cid:0) Λ − (cid:1) where R := sup w ∈P Λ (cid:107) w (cid:107) < ∞ since P Λ is bounded. Thus b (cid:48) ∈ Z n − k is contained in a ball ofconstant radius around b . The claim follows from this.We are interested in the union (cid:83) b ∈ Z n − k S b of the translates S b , b ∈ Z n − k . To bound integralsover this set, the following statement will be used: Lemma 4.9.
There are measurable subsets ˚ S b ⊂ S b , T b ⊂ S b for b ∈ Z n − k , such that (cid:32) (cid:91) b ∈ Z n − k ˚ S b (cid:33) ∪ (cid:32) (cid:91) b ∈ Z n − k T b (cid:33) = (cid:91) b ∈ Z n − k S b , ˚ S b ∩ ˚ S b (cid:48) = ∅ for b (cid:54) = b (cid:48) , ˚ S b ∩ T b (cid:48) = ∅ for all b, b (cid:48) , and each set Π B T b is of measure zero (and thus so is Π B (cid:0)(cid:83) b ∈ Z n − k T b (cid:1) ).Proof. The set Z n − k is countable. Fix an enumeration of Z n − k . Slightly abusing notation, letus write {S N } N ∈ N instead of {S b } b ∈ Z n − k .Now define T = (cid:91) M> ( S ∩ S M ) and ˚ S = S \T , T = (cid:91) M> ( S ∩ S M ) and ˚ S = S \ ( T ∪ T ) , and more generally T J = (cid:91) M>J ( S J ∩ S M ) , ˚ S J = S J \ (cid:32) (cid:91) K ≤ J T K (cid:33) , J ∈ N . By definition, the sets ˚ S J are pairwise disjoint, ˚ S J ∩ T J (cid:48) = ∅ for all J, J (cid:48) ∈ N , and (cid:32) (cid:91) N ∈ N ˚ S N (cid:33) ∪ (cid:32) (cid:91) N ∈ N T N (cid:33) = (cid:91) N ∈ N S N . Furthermore, by Lemma 4.8, each T J is a union of a finite number of sets S J ∩ S M , and therefore (cid:83) N ∈ N T N is a countable union of such sets. Together with Lemma 4.7, this implies that Π B T J (Π B (cid:83) N ∈ N T N ) is a finite (countable) union of sets Π B ( S J ∩ S M ) ⊂ Π B ( S J ) ∩ Π B ( S M ) containedin a polytope of dimension n − k −
1. Since any set of dimension < n − k has Lebesgue measurezero in R n − k , and countable unions of zero-measure sets are also of measure zero, the claimfollows. To obtain a bound on the decoding success probability, we first establish a general upper boundon the probability mass of the union of translated and thickened versions of the transformedpolytope Λ − P Λ . This is expressed in Lemma 4.11 below.Let us first consider integrals over subsets of the set S b . Recall that B δ (0) ⊂ R k is the closed δ -ball with respect to the Euclidean norm, centered at the origin. For the remainder of thispaper, we denote by S ( (cid:15) ) := S + B (cid:15) (0) × { } n − k (48)the (cid:15) -thickening of a set S ⊂ R n , for (cid:15) > Lemma 4.10.
Let b ∈ Z n − k be arbitrary. Let Q b ⊂ S b = (cid:0) b (cid:1) + Λ − P Λ be a measurable subset.Let g : R n → R be integrable. Then (cid:90) Q b ( (cid:15) ) g ( x ) dx = (cid:90) ξ ∈ Π B Q b (cid:90) h ∈B (cid:15) (0) g (cid:32)(cid:18) ˆΛ( ξ − ∆ b ) + hξ (cid:19)(cid:33) dhdξ . Proof.
Define φ b : B (cid:15) (0) × Π B Q b → Q b ( (cid:15) ) (cid:0) hξ (cid:1) (cid:55)→ (cid:0) ˆΛ( ξ − ∆ b )+ hξ (cid:1) . By definition (and the fact that the map ϕ defined by (41) is bijective), this map is surjective:every element v ∈ Q b ( (cid:15) ) has the form v = (cid:0) ˆΛ z + hz +∆ b (cid:1) with h ∈ B (cid:15) (0), z ∈ Π B Λ − P Λ and thus v = φ b (cid:16)(cid:0) hξ (cid:1)(cid:17) with ξ = z + ∆ b ∈ Π B Q b . It is also easy to check that it is injective. Furthermore,the Jacobi-matrix of φ b is Dφ b = (cid:18) I k ˆΛ0 I n − k (cid:19) and satisfiesdet Dφ b = 1 . The claim therefore follows from the change-of-variables formula for Lebesgue integrals.26et f : R n → R be a probability density function of an absolutely continuous probabilitymeasure on R n . Writing f = f X A X B , this factorizes as f X A X B ( x A , x B ) = f X B ( x B ) f X A | X B = x B ( x A ) for ( x A , x B ) ∈ R k × R n − k , where f X B is the density function associated with the second marginal, and f X A | X B = x B denotesthe density associated with the conditional distribution of X A given that X B = x B . Our maintechnical statement is the following: Lemma 4.11.
Let f : R n → R be a probability density function. Then (cid:90) (cid:83) b ∈ Z n − k S b ( (cid:15) ) f ( x ) dx ≤ (cid:88) b ∈ Z n − k (cid:90) ξ ∈ Π B ˚ S b f X B ( ξ ) (cid:18)(cid:90) h ∈ B (cid:15) (0) f X A | X B = ξ (cid:16) ˆΛ( ξ − ∆ b ) + h (cid:17) dh (cid:19) dξ . (49) Furthermore, we have the inequality (cid:88) b ∈ Z n − k (cid:90) ξ ∈ Π B ˚ S b f X B ( ξ ) dξ ≤ . (50) Proof.
Since (cid:91) b ∈ Z n − k S b ( (cid:15) ) = (cid:32) (cid:91) b ∈ Z n − k ˚ S b ( (cid:15) ) (cid:33) ∪ (cid:32) (cid:91) b ∈ Z n − k T b ( (cid:15) ) (cid:33) by Lemma 4.9 and definition (48), we obtain (cid:90) (cid:83) b ∈ Z n − k S b ( (cid:15) ) f ( x ) dx ≤ I + I , with I := (cid:88) b ∈ Z n − k (cid:90) ˚ S b ( (cid:15) ) f ( x ) dx and I := (cid:88) b ∈ Z n − k (cid:90) T b ( (cid:15) ) f ( x ) dx . By Lemma 4.10, the right hand side of Eq. (49) coincides with I . It thus suffices to show that I = 0. But (again by Lemma 4.10), we have (cid:90) T b ( (cid:15) ) f ( x ) dx = (cid:90) ξ ∈ Π B T b f X B ( ξ ) (cid:18)(cid:90) h ∈ B (cid:15) (0) f X A | X B = ξ (cid:16) ˆΛ( ξ − ∆ b ) + h (cid:17) dh (cid:19) dξ ≤ (cid:90) ξ ∈ Π B T b f X B ( ξ ) dξ = 0 , as Π B T b is a set of measure zero, see Lemma 4.9. This establishes (49).Now consider (50). We have 1 ≥ (cid:90) Π B (cid:83) b ∈ Z n − k S b f X B ( ξ ) dξ = (cid:88) b ∈ Z n − k (cid:90) Π B S b f X B ( ξ ) dξ , where we used the fact that the projected sets { Π B S b } b ∈ Z n − k have only zero-measure pairwiseintersection, see Lemma 4.7. The claim follows since ˚ S b ⊂ S b for every b ∈ Z n − k .27 .9 An upper bound on the decoding success probability We now return to the partial informed unwrapping problem (see Section 4.2) and give upperbounds on the success probability p S ( (cid:15) ). Our central result is the following theorem previouslysummarized in Eq. (26). It is obtained by specializing Lemma 4.11 to the normal distributionof interest. Theorem 4.12.
Define Γ := − ((Σ − ) AA ) − (Σ − ) AB − Σ AB (Σ BB ) − Γ := ((Σ − ) AA ) − (Σ − ) AB . and set c ( ξ, ∆ b ) := Γ ξ + Γ ∆ b for ξ ∈ R n − k and b ∈ Z n − k . Let Y ∼ N (0 , Σ ∗ ) be a centered normal random variable on R k with covariance matrix Σ ∗ givenby the Schur complement Σ ∗ = Σ AA − Σ AB (Σ BB ) − Σ BA of Σ . Let f X B denote the probability density function of X B ∼ N (0 , Σ BB ) . Then the fol-lowing holds: there is a family {R b } b ∈ Z n − k of measurable subsets of R n − k such that R b ⊂ ∆ b + Π B Σ / P Σ − / for each b ∈ Z n − k , (cid:88) b ∈ Z n − k (cid:90) R b f X B ( ξ ) dξ ≤ , (51) and the decoding success probability p S ( (cid:15) ) is bounded by p S ( (cid:15) ) ≤ (cid:88) b ∈ Z n − k (cid:90) R b f X B ( ξ ) · Pr [ Y ∈ B (cid:15) ( c ( ξ, ∆ b ))] dξ . (52) Proof.
By Theorem 4.2 we have p S ( (cid:15) ) = Pr (cid:2) X ∈ ( B (cid:15) (0) × ∆ Z n − k ) + Σ / V Σ − / (cid:3) where X ∼ N (0 , Σ). With V Σ − / ⊂ P Σ − / we obtain the upper bound p S ( (cid:15) ) ≤ (cid:90) (cid:83) b ∈ Z n − k S b ( (cid:15) ) f X ( x ) dx where S b = (cid:18) b (cid:19) + Σ / P Σ − / for b ∈ Z n − k . With Lemma 4.11 we conclude that p S ( (cid:15) ) ≤ (cid:88) b ∈ Z n − k (cid:90) ξ ∈ Π B ˚ S b f X B ( ξ ) I ( b, ξ ) dξ , (53)where I ( b, ξ ) := (cid:90) h ∈B (cid:15) (0) f X A | X B = ξ (cid:16) ˆΛ( ξ − ∆ b ) + h (cid:17) dh for ( ξ, b ) ∈ R n − k × Z n − k , (54)28ith ˆΛ = − ((Σ − ) AA ) − (Σ − ) AB (see Eq. (38)). According to (54), I ( b, ξ ) = Pr (cid:104) X A ∈ B (cid:15) (cid:16) ˆΛ( ξ − ∆ b ) (cid:17)(cid:12)(cid:12)(cid:12) X B = ξ (cid:105) is the probability mass of an (cid:15) -ball centered at ˆΛ( ξ − ∆ b ) with respect to the conditionaldistribution f X A | X B = x B . Since we are considering X ∼ N (0 , Σ), the latter is normal withcovariance matrix given by the Schur complement Σ ∗ of Σ and centered at m ∗ = Σ AB (Σ BB ) − ξ . Translating, i.e., considering Y := X A − m ∗ , we can write this as I ( b, ξ ) = Pr (cid:104) Y ∈ B (cid:15) (cid:16) ˆΛ( ξ − ∆ b ) − m ∗ (cid:17)(cid:105) where Y ∼ N (0 , Σ ∗ ) . (55)Inserting (55) into (53) and setting R b := Π B ˚ S b for b ∈ Z n − k gives Eq. (52) since for X =( X A , X B ) ∼ N (0 , Σ), we have X B ∼ N (0 , Σ BB ). Eventually, Eq. (51) follows from Lemma 4.11. Corollary 4.13.
Let λ min ( M ) denote the smallest eigenvalue of a positive definite matrix M .Let µ Z denote the probability measure of a centered normal Gaussian distribution, Z ∼ N (0 , I k ) ,on R k . Then p S ( (cid:15) ) ≤ µ Z (cid:16) B (cid:15)/ √ λ min (Σ ∗ ) (0) (cid:17) . (56) In particular, p S ( (cid:15) ) ≤ µ Z (cid:16) B (cid:15)/ √ λ min (Σ) (0) (cid:17) . (57) Proof.
Let Y ∼ N (0 , Σ ∗ ) and let c ∈ R k be arbitrary. ThenPr [ Y ∈ B (cid:15) ( c )] = Pr (cid:2) Z ∈ Σ − / ∗ B (cid:15) ( c ) (cid:3) . With Σ − / ∗ B (cid:15) ( c ) = Σ − / ∗ ( c + B (cid:15) (0)) = Σ − / ∗ c +Σ − / ∗ B (cid:15) (0) and Σ − / ∗ B (cid:15) (0) ⊂ B (cid:15) · λ max (Σ − / ∗ ) (0) = B (cid:15)/ √ λ min (Σ ∗ ) (0) we obtain Pr [ Y ∈ B (cid:15) ( c )] ≤ µ Z ( B (cid:15) ∗ ( c ∗ )) , with (cid:15) ∗ = (cid:15)/ (cid:112) λ min (Σ ∗ ) and c ∗ = Σ − / ∗ c . We have thus related Pr [ Y ∈ B (cid:15) ( c )] to the probabilitymass of a k -dimensional (cid:15) ∗ -ball centered at c ∗ under the canonical Gaussian measure µ Z . Thelatter is maximal when the ball is centered at the origin, i.e., µ Z ( B (cid:15) ∗ ( c ∗ )) ≤ µ Z ( B (cid:15) ∗ (0)) for all c ∗ ∈ R k , according to Anderson’s inequality [2]. (More generally, the latter implies that for a convexset C ∈ R k which is symmetric about the origin µ Z ( C ) ≥ µ Z ( C + a ) for any a ∈ R k .) We havethus shown that Pr [ Y ∈ B (cid:15) ( c )] ≤ µ Z (cid:16) B (cid:15)/ √ λ min (Σ ∗ ) (0) (cid:17) for alll c ∈ R k . Inserting this into Eq. (52) of Theorem 4.12, and subsequently applying Eq. (51) of the sametheorem, we obtain (56).The claim (57) follows immediately from (56). This is because the eigenvalues of the Schurcomplement Σ ∗ can be related to that of the positive definite matrix Σ using the inequality λ min (Σ ∗ ) ≥ λ min (Σ), see [24, Theorem 5]. For completeness, we give a proof of this inequalityin Appendix B. 29 cknowledgements RK acknowledges support by the DFG cluster of excellence 2111 (Munich Center for QuantumScience and Technology). RK and LH acknowledge support by the German Federal Ministry ofEducation through the program Photonics Research Germany, contract no. 13N14776 (QCDA-QuantERA).
A Quadratic constraints for the degenerate Voronoi cell
Lemma A.1.
Consider the set E b := (cid:40) z ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) Λ (cid:18) h ∆ b (cid:19) , z (cid:29) ≤ (cid:13)(cid:13)(cid:13)(cid:13) Λ (cid:18) h ∆ b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) for all h ∈ R k (cid:41) with b ∈ Z n − k \{ } . Define w b ( z ) := (Λ ) AB ∆ b − Π A Λ z , γ b ( z ) := (cid:107) Λ (cid:0) b (cid:1) (cid:107) − (cid:104) z, Λ (cid:0) b (cid:1) (cid:105) , and Ω := (Λ AA ) +Λ AB Λ BA .Let q b : R n → R be the quadratic form q b ( z ) := γ b ( z ) − (cid:104) Ω − w b ( z ) , w b ( z ) (cid:105) . Then E b = { z ∈ R n | q b ( z ) ≥ } Proof.
We have z ∈ E b if and only if α ( h ) + β b ( h, z ) + γ b ( z ) ≥ h ∈ R k where α ( h ) := (cid:13)(cid:13)(cid:13)(cid:13) Λ (cid:18) h (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ,β b ( h, z ) := 2 (cid:28) Λ (cid:18) h (cid:19) , Λ (cid:18) b (cid:19) − z (cid:29) ,γ b ( z ) := (cid:13)(cid:13)(cid:13)(cid:13) Λ (cid:18) b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:28) z, Λ (cid:18) b (cid:19)(cid:29) . Observe that α ( h ) = (cid:104) h, Ω h (cid:105) with Ω := (Λ AA ) + Λ AB Λ BA β b ( h, z ) = 2 (cid:104) h, w b ( z ) (cid:105) with w b ( z ) := (Λ ) AB ∆ b − Π A Λ z . Since Λ is positive definite, the quadratic form Q ( h ) := α ( h ) + β b ( h, z ) + γ b ( z ) is minimal when ∇ h Q ( h ) = 0. This is the case for h ∗ := − Ω − w b ( z ). The corresponding value is Q ( h ∗ ) = γ b ( z ) − (cid:104) Ω − w b ( z ) , w b ( z ) (cid:105) . Since z ∈ E b if and only if Q ( h ∗ ) ≥
0, the claim follows.30
A lower bound on the minimal eigenvalue of the Schurcomplement
In this appendix, we give a proof of the following statement for completeness.
Lemma B.1.
Let
Σ = (cid:18) Σ AA Σ AB Σ BA Σ BB (cid:19) be positive definite, with Σ AA ∈ Mat ( n − r ) × ( n − r ) ( R ) , Σ AB =Σ TBA ∈ Mat ( n − r ) × r ( R ) , and Σ BB ∈ Mat r × r ( R ) . Let Σ ∗ = Σ AA − Σ AB (Σ BB ) − Σ BA . denote the Schur complement. Then λ min (Σ ∗ ) ≥ λ min (Σ) . We refer to [24, Theorem 5] for a more general statement showing that the eigenvalues of theSchur complement of a semidefinite Hermitian matrix interlace the eigenvalues of the matrixitself. The proof relies on the Weyl inequalities. Here we only need the following statement.Let λ min ( A ) = λ ( A ) ≤ λ ( A ) ≤ · · · ≤ λ n − ( A ) ≤ λ n ( A ) = λ max ( A )denote the ordered eigenvalues of a Hermitian n × n -matrix A . Then the following holds: Lemma B.2. (see [9, Corollary 4.3.5]) Let
C, D be Hermitian n × n -matrices. Assume that D has rank at most r . Then λ k ( C + D ) ≤ λ k + r ( C ) for all k = 1 , , . . . , n − r . (58) Proof of Lemma B.1.
For brevity, let us write X := Σ AA , Y = Σ AB and Z := Σ BB . Note thatboth X and Z are positive definite as principal submatrices of the positive definite matrix Σ = (cid:18) X YY T Z (cid:19) , and Σ ∗ = X − Y Z − Y T . We have (cid:18) X YY T Z (cid:19) = (cid:18) X − Y Z − Y T
00 0 (cid:19) + (cid:18) Y Z − Y T YY T Z (cid:19) . That is, Σ = C + D (59)where C := (cid:18) Σ ∗
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