Bosonic loss, Kerr shifts and finite-code comparison scope

Assume Countable Choice. A single bosonic mode is represented here by the symmetric Fock space over a one-dimensional complex Hilbert space with a chosen unit vector. Its occupation basis is $\{\mathrm{\mid }n⟩:n=0,1,2,\dots \}$, where $n$ is the occupation number. On the finite linear span of these vectors, define the annihilation operator $a$ by $a\mathrm{\mid }0⟩=0$ and $a\mathrm{\mid }n⟩=\sqrt{n}\mathrm{\mid }n-1⟩$ for $n\ge 1$, and define the number operator by $N\mathrm{\mid }n⟩=n\mathrm{\mid }n⟩$. In this model a single-loss event is represented by $a$ on vectors in its domain; this specifies an error operator, not a normalized channel or a loss rate.

In units with $\mathrm{\hslash }=1$, prescribe the diagonal Kerr Hamiltonian $H=\omega N+\frac{K}{2}N(N-1)$, with real parameters $\omega ,K$, initially on the finite occupation span. Its energy assigned to $\mathrm{\mid }n⟩$ is $E_n=\omega n+\frac{K}{2}n(n-1)$; the adjacent downward transition frequency in this model is $E_n-E_{n-1}=\omega +K(n-1)$ for $n\ge 1$. Here Kerr anharmonicity means the occupation-dependent quadratic term; no sign or stability condition on $K$ is assumed.

For a finite-dimensional logical Hilbert space $L$, an oscillator encoding is a linear isometry $V:L\to {\mathcal{F}}_s(\mathbb{C})$: it preserves inner products and its image is the code subspace. It has finite mean occupation when every normalized $V\psi ={\sum }_{n\ge 0}{c}_n\mathrm{\mid }n⟩$ satisfies ${\sum }_{n\ge 0}n\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }$. Normalizability alone means only ${\sum }_{n\ge 0}\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }$ and does not impose this energy condition. The ideal comb states in the Gottesman–Kitaev–Preskill construction are nonnormalizable; their use as actual encoded states requires an approximation or a separately specified limiting prescription.

Fix the dimensionless position representation by the complex-linear unitary $U:{\mathcal{F}}_s(\mathbb{C})\to L^2(\mathbb{R},dq)$ with $U\mathrm{\mid }n⟩=h_n$. Here $h_0(q)={\pi }^{-1\mathrm{/}4}{e}^{-q^2\mathrm{/}2}$ and $h_{n+1}=(q-{\mathrm{\partial }}_q)h_n\mathrm{/}\sqrt{2(n+1)}$ for $n\ge 0$ are the normalized Hermite functions. Thus $U({\sum }_{n\ge 0}{c}_n\mathrm{\mid }n⟩)={\sum }_{n\ge 0}{c}_n{h}_n$, with convergence in $L^2$. On $L^2$ use $⟨f,g{⟩}_2={\int }_{\mathbb{R}}\overline{f(q)}g(q)dq$, matching the Fock convention. The Hermite suppliers use the conjugate, first-variable-linear pairing; conjugating it leaves norms and completeness unchanged. On the finite Hermite span, $Ua{U}^{-1}=(q+{\mathrm{\partial }}_q)\mathrm{/}\sqrt{2}$ and $(UN{U}^{-1})h_n=n{h}_n$. Only these finite-span differential identities are asserted.

For the GKP qubit family used here, use this dimensionless position coordinate $q\in \mathbb{R}$ and a spacing $\alpha >0$. Its ideal code is the formal span of ${\psi }_j(q)={\sum }_{s\in \mathbb{Z}}\delta (q-\alpha (2s+j))$ for $j\in \{0,1\}$, where $\delta (q-x)$ is the unit point mass at $x$. These combs are distributions, not normalizable oscillator vectors. A Gaussian approximation uses

${\tilde{\psi }}_j(q)=C_j{\sum }_{s\in \mathbb{Z}}\exp \text{\hspace{0.17em}⁣}(-\frac{{\kappa }^2{\alpha }^2(2s+j{)}^2}{2})\exp \text{\hspace{0.17em}⁣}(-\frac{(q-\alpha (2s+j){)}^2}{2{\mathrm{\Delta }}^2}),j\in \{0,1\},\mathrm{\Delta },\kappa >0.$

Here $C_j>0$ normalizes each wavefunction, $\mathrm{\Delta }$ specifies peak width, and ${\kappa }^{-1}$ specifies the width of the envelope weighting the peak centers. The corresponding normalized oscillator vectors are $U^{-1}{\tilde{\psi }}_j$. The two approximate vectors need not be orthogonal; an encoding still must meet the isometry condition above. Approximating an ideal comb refers to narrow peaks and a broad envelope, not Hilbert-norm convergence to a normalizable ideal vector.

A comparison of logical lifetimes as mean occupation varies must specify an encoding family, a time-indexed physical noise family, a decoding or recovery procedure, a logical fidelity functional and a target value. Given these, take the time domain to be $[0,\mathrm{\infty })$, write $F(t)$ for the chosen real-valued logical fidelity and $f_{\ast }$ for its real target, and put $S=\{t\in [0,\mathrm{\infty }):F(t)<f_{\ast }\}$. Define the lifetime by $\tau =\inf S$ when $S\mathrm{\ne }\mathrm{\varnothing }$, and by $\tau =+\mathrm{\infty }$ when $S=\mathrm{\varnothing }$; here $+\mathrm{\infty }$ denotes the no-crossing convention. The infimum need not belong to $S$, and no continuity or monotonicity of $F$ is required. This contract by itself asserts no monotonic relation between occupation and lifetime.

Given: The stated definitions, hypotheses and cited interfaces.

Goal: Justify the stated construction and its scope.

1.1 The finite occupation span is invariant under the displayed ladder and diagonal maps, so the formulas define linear maps there. [given, F1, F2, F3]

2.1 By [F3] the occupation coordinates are square-summable, and by [F5] the Hermite functions are an orthonormal basis of $L^2$. The finite-sum map ${\sum }_n{c}_n\mathrm{\mid }n⟩\mapsto {\sum }_n{c}_n{h}_n$ preserves inner products; square-summable coordinates give Cauchy partial sums, so completeness of $L^2$ extends it uniquely to an isometry $U$ on the Fock space. Every $f\in L^2$ has a Hermite expansion with square-summable coefficients by [F5]; those coefficients define a Fock vector by [F3] mapping to $f$, so $U$ is onto. Conjugating the supplier's pairing gives exactly the second-variable-linear pairing specified above, and preserves this calculation. The lowering relation in [F5] agrees with [F2] on each occupation vector, while the number identity follows from $N\mathrm{\mid }n⟩=n\mathrm{\mid }n⟩$; linearity proves both stated identities on the finite Hermite span. [given, F1, F2, F3, F5, step 1.1]

2.2 The assigned energies follow by substitution, and subtracting consecutive values gives the stated gap. The Fock construction supplies the occupation basis and the domain of the loss operator; no unbounded-operator identity is asserted outside the specified domains. For example, the isometry from $\mathbb{C}$ sending $1$ to a normalized multiple of ${\sum }_{n\ge 1}{n}^{-1}\mathrm{\mid }n⟩$ is normalizable but has divergent mean occupation, so the additional finite-occupation requirement is substantive. Indeed, [F6] gives $0<Z:={\sum }_{n\ge 1}{n}^{-2}<\mathrm{\infty }$, so $v=Z^{-1\mathrm{/}2}{\sum }_{n\ge 1}{n}^{-1}\mathrm{\mid }n⟩$ has norm one by [F3], whereas its mean occupation is $Z^{-1}{\sum }_{n\ge 1}{n}^{-1}=+\mathrm{\infty }$ by [F6]. Thus $z\mapsto zv$ is the claimed isometry. [F1, F2, F3, F6, step 1.1]

3.1 If $S\mathrm{\ne }\mathrm{\varnothing }$, it is a subset of $\mathbb{R}$ bounded below by $0$, so [F4] gives its real infimum. This infimum is nonnegative because $0$ is a lower bound. If $S=\mathrm{\varnothing }$, the explicit value $+\mathrm{\infty }$ applies instead. Thus the lifetime is defined even when there is no least below-target time, without asserting a crossing or a monotonic relation between occupation and lifetime. [given, F4] ∎

The GKP combs and Gaussian approximate wavefunctions above are imported from Gottesman, Kitaev and Preskill, Section IV, Eq. (27) with $n=2$, and Section V, Eqs. (30)–(31). The ideal combs are nonnormalizable, whereas the normalized approximate vectors can overlap, as reported there. This construction is an external definition, not a local proof of approximation quality or a lifetime law. The preceding local justification concerns the Fock/Kerr model, the position-representation identification and the lifetime convention only.