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On quantum operations of photon subtraction and photon addition

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Pith's one-line read Beam-splitter photon addition and subtraction are uniformly valid on two energy-moment classes, and these classes are optimal.

desk verdict A clean, mostly self-contained paper that pins down exactly when beam-splitter photon subtraction/addition converges uniformly to the ideal maps; worth refereeing after fixing typos and expanding the mixed-state proofs. read the letter →

arxiv 1908.02207 v1 pith:6YYQ5R57 submitted 2019-08-06 quant-ph

S. N. Filippov This is my paper · ORCID
classification quant-ph
keywords photonsubtractionadditionquantumoperationenergyconstraintuniformconvergencetracedistancebeamsplittermoments
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks when the textbook photon subtraction and addition maps, $\varrho \mapsto t a \varrho a^\dagger$ and $\varrho \mapsto t a^\dagger \varrho a$, can be replaced by real quantum operations. These maps are completely positive but not trace nonincreasing, so no experiment can implement them exactly; a beam splitter provides fair conditional operations $\mathcal{N}_-$ and $\mathcal{N}_+$ that approximate them. The main result is a sharp certificate for uniform approximation: photon addition converges on states with finite second energy moment $\operatorname{tr}[\varrho H^2] \leq E_2$, while photon subtraction converges on states that also have energy bounded below by $E_1 > 0$. Both restrictions are proved necessary, and the result extends to multiple-photon subtraction and addition. This matters because it tells a user of a fixed beam-splitter setup exactly which input states can be trusted to behave like the idealized operation.

What carries the argument

The load-bearing objects are the two fair operations of Eqs. (4) and (5), beam-splitter conditional maps whose exponential damping $e^{-\gamma H}$ with $H=a^\dagger a$ makes the maps trace nonincreasing by cutting off high-Fock-number tails. The proof shows that on the relevant moment classes the damping error is controlled by simple inequalities of the form $\sqrt{8\gamma E_2/E_1}$ for subtraction and $\sqrt{8\gamma(3E_2+1)}$ for addition, so choosing $\gamma$ small enough gives uniform $\varepsilon$-closeness. A purification step then lifts the pure-state estimates to mixed states by contractivity of the trace distance under partial trace.

What would settle it

Compute, for fixed $\gamma,E$ and the family $|\psi_N\rangle=\sqrt{1-E/N^2}|0\rangle+\sqrt{E/(2N^2)}|1\rangle+\sqrt{E/(2N^2)}|N\rangle$ from Proposition 4, the trace norm $\|\cdot\|_1$ between the normalized ideal subtraction output and the normalized $\mathcal{N}_-(\gamma)$ output as $N \to \infty$. The paper predicts it approaches 2; if it stays below 1, the claimed impossibility fails. For photon addition, a single state in $\mathcal{S}^{(2)}_{E_2}$ exceeding the stated $\sqrt{8\gamma(3E_2+1)}$ bound would disprove the uniform estimate.

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Extended reading notes

Core claim

The central discovery is that the physically realisable approximate operations $\mathcal{N}_+(\gamma)[\varrho]=(e^{2\gamma}-1)e^{-\gamma a^\dagger a}a^\dagger \varrho a e^{-\gamma a^\dagger a}$ and $\mathcal{N}_-(\gamma)[\varrho]=(e^{2\gamma}-1)a e^{-\gamma a^\dagger a}\varrho e^{-\gamma a^\dagger a}a^\dagger$, after normalization, converge uniformly to the ideal conditional outputs $a^\dagger \varrho a/\operatorname{tr}[a^\dagger \varrho a]$ and $a\varrho a^\dagger/\operatorname{tr}[a\varrho a^\dagger]$ exactly on the stated energy-moment classes. For addition, the class is $\mathcal{S}^{(2)}_{E_2}(\mathcal{H})$, states with $\operatorname{tr}[\varrho H^2] \leq E_2 < \infty$; for subtraction, the class is $\mathcal{S}^{(1;2)}_{E_1;E_2}(\mathcal{H})$, states with $\operatorname{tr}[\varrho H] \geq E_1 > 0$ and $\operatorname{tr}[\varrho H^2] \leq E_2 < \infty$. The paper proves that finite energy alone cannot give uniform convergence for either operation, and that the second-moment bound alone cannot give it for subtraction, because families of states with energy tending to zero force the trace distance to its maximum possible value. The same mechanism, applied to $k$-fold operations, replaces the second moment by the $(k+1)$-th energy moment for multiple photon subtraction and addition.

Load-bearing premise

The result assumes the beam-splitter operations (4) and (5) are the faithful physical approximations of photon subtraction and addition, with no additional loss, detector inefficiency, or noise; the purification bound stated as 'one can readily see' in Propositions 3 and 5 is also used without full derivation.

Editorial extensions

If this is right

  • For any beam-splitter transmittance, a user can certify photon addition on all states with $\operatorname{tr}[\varrho H^2] \leq E_2$ by choosing $\gamma$ below an explicit threshold; no state-by-state verification is needed.
  • Photon subtraction cannot be certified on any class containing states of arbitrarily small mean energy, so experiments with very low photon number should not treat the beam-splitter output as ideal subtraction.
  • Multiphoton subtraction and addition inherit the same dichotomy: the $(k+1)$-th energy moment controls $k$-photon addition, while $k$-photon subtraction additionally needs a positive lower bound on energy.
  • The impossibility results give explicit stress-test states: the families in Propositions 2 and 4 are ready-made worst cases for any approximate implementation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof structure suggests the same uniform-convergence certificate should hold for any damping profile $e^{-f(H)}$ with $f$ growing at least linearly, not only the beam-splitter form; replacing $\gamma n$ by $f(n)$ in the estimates would yield analogous moment conditions.
  • The sharp failure at vanishing energy points to a resource-theoretic reading: ideal photon subtraction is faithfully implementable exactly when the state carries at least one photon on average, so mean photon number acts as a necessary resource for the operation.
  • A direct experimental check could use the families in Propositions 2 and 4 as adversarial inputs: if a fixed setup is claimed to realize approximate subtraction over a broad class, measuring the trace distance on those families should reveal the divergence the paper predicts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the ideal photon subtraction and addition transformations ρ → t a ρ a† and ρ → t a† ρ a, which are not trace-nonincreasing and therefore are not valid quantum operations. The author considers physically motivated approximate operations N_-(γ) and N_+(γ) derived from beam-splitter models and asks when the conditional output states of these approximate operations converge uniformly, in trace norm, to the ideal normalized outputs. The main results are: (Prop. 3) for photon addition, uniform convergence holds on the class of states with bounded second energy moment tr[ρ H^2] ≤ E; (Prop. 4) for photon subtraction, this condition alone is not sufficient, because states with vanishing energy cause failure; and (Prop. 5) adding a lower bound tr[ρ H] ≥ E1 > 0 restores uniform convergence for subtraction. Proposition 2 shows that first-moment bounds alone are insufficient for either operation, and the paper also generalizes the results to multiple photon addition and subtraction. The proofs use explicit trace-norm bounds via inner products for pure states, purification for mixed states, and explicit counterexample families for the necessity claims.

Significance. If the results are correct, the paper provides a clean and essentially complete characterization of the moment constraints under which the standard approximate photon operations faithfully reproduce the ideal transformations in trace norm. This fills a genuine gap in the literature: previous work addressed approximations for individual states but not uniform convergence over state classes. The proofs are elementary, constructive, and parameter-free; the error bounds are explicit and the counterexamples are concrete. The paper also correctly identifies a qualitative asymmetry between photon addition and subtraction, namely the need for a nonvanishing energy lower bound in the subtraction case. The result is likely to be useful for quantum optics and for rigorous treatments of continuous-variable quantum operations. The main weaknesses are presentational: several typographical errors, a terse mixed-state step, and a misstated limit value in Prop. 2; none of these appears to affect the validity of the central conclusions.

minor comments (6)
  1. [Section III, proof of Prop. 2] The displayed limit in the proof of Prop. 2 should read 2 sqrt(E/(E+1)) rather than 2 sqrt(E)/(E+1); as printed, the stated limit is smaller than the target bound sqrt(E/(E+1)) for E > 4, so the proof of inequality (8) is incomplete unless the correct value is used.
  2. [Section IV, Prop. 3, mixed-state paragraph] The step 'One can readily see that ⟨Φ|X⟩ ≥ (F1+1-γ(1+2F1+F2))/(F1+1)' is an omitted derivation that is load-bearing for extending the theorem to mixed states; please include the short argument, for instance by applying the pure-state bound to each spectral component and using Cauchy-Schwarz for the denominator.
  3. [Section V, Prop. 5, mixed-state paragraph] There are copy-paste errors in this paragraph: the states should be the approximate and ideal subtraction outputs, not ~ρ_+ and N_+, and the denominator in the definition of |X⟩ should involve e^{-2γ a†a} (equivalently e^{-2γH}) rather than e^{-2γ aa†}.
  4. [Section II, proof of Prop. 1, first case] The sentence 'there exists s1 > 2 such that [expression] = ...' is imprecise because the expression depends continuously on s and only tends to the displayed limit as s → 2+0; the argument should say that the expression can be made arbitrarily close to that limit, hence larger than (e-1)/(2e).
  5. [Abstract and Introduction] The phrase 'these conditions cannot be relaxed' should be understood relative to the moment-constrained families considered here; Fock states have unbounded tr[ρH^2] yet the approximate operations are exact on them, so the statement is not one of absolute maximality of the state set.
  6. [Throughout] There are several typographical errors that should be corrected: 'nonicreasing' → 'nonincreasing', 'Simirlary' → 'Similarly', 'transmittence' → 'transmittance', and the title page has 'addi tion'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniform-convergence theorems are derived from explicit inequalities and counterexamples, with no fitted parameter or self-referential premise.

full rationale

The paper's central results are self-contained mathematical theorems about uniform convergence of approximate photon subtraction and addition. The approximate operations N±(γ) are imported from a standard beam-splitter model, cited to external literature, not assumed to equal the target ideal operations; the paper then proves quantitative trace-norm bounds. The sufficiency proofs in Proposition 3 and Proposition 5 derive explicit choices of γ from elementary inequalities involving F1 = tr[ρH] and F2 = tr[ρH^2], and the necessity proofs in Propositions 1, 2, and 4 construct explicit state families that violate any uniform bound under weaker constraints. None of these steps fits a parameter to the predicted quantity or uses the conclusion as an input. The mixed-state extension summarized as 'one can readily see' is a direct purification estimate plus contractivity of the partial trace, and it does not presuppose the theorem. The self-citations in the references are contextual and not load-bearing for the main argument. Accordingly, no circular step is identifiable and the paper should receive a score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; E, E1, E2, gamma, and epsilon are theorem variables, not numbers fitted to data. The only domain assumption is the standard beam splitter model, which is not ad hoc. No new particles, forces, dimensions, or other entities are introduced.

assumptions (5)
  • standard math A quantum operation is a linear, completely positive, trace nonincreasing map; conditional output states are normalized by the success probability.
    Used in Section I to define fair operations and to state that t a rho a-dagger and t a-dagger rho a are not operations.
  • domain assumption The beam splitter conditional maps N_-(gamma) and N_+(gamma) in Eqs. (4) and (5) faithfully represent photon subtraction and addition experiments.
    The whole comparison between idealized and approximate operations depends on these imported physical models.
  • standard math Trace norm is contractive under partial trace, Theorem 9.2 of Nielsen and Chuang.
    Used in Props 3 and 5 to pass from pure-state bounds to mixed states via purification.
  • standard math For every state, 0 <= tr[rho H] <= tr[rho H^2] and e^(-x) >= 1 - x for x >= 0.
    These inequalities drive the uniform error bounds in Props 3 and 5.
  • standard math The zeta and polylogarithm identities used in Prop 1 hold as stated.
    Used to construct finite-energy states for which the approximate and ideal operations remain far apart.

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Pith. "Pith review of On quantum operations of photon subtraction and photon addition." pith.science (2026). https://pith.science/paper/6YYQ5R57

@misc{pith2026190802207,
  author       = {Pith},
  title        = {Pith review of: On quantum operations of photon subtraction and photon addition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YYQ5R57}},
  note         = {Machine review of arXiv:1908.02207}
}
abstract

The conventional photon subtraction and photon addition transformations, $\varrho \rightarrow t a \varrho a^{\dag}$ and $\varrho \rightarrow t a^{\dag} \varrho a$, are not valid quantum operations for any constant $t>0$ since these transformations are not trace nonincreasing. For a fixed density operator $\varrho$ there exist fair quantum operations, ${\cal N}_{-}$ and ${\cal N}_{+}$, whose conditional output states approximate the normalized outputs of former transformations with an arbitrary accuracy. However, the uniform convergence for some classes of density operators $\varrho$ has remained essentially unknown. Here we show that, in the case of photon addition operation, the uniform convergence takes place for the energy-second-moment-constrained states such that ${\rm tr}[\varrho H^2] \leq E_2 < \infty$, $H = a^{\dag}a$. In the case of photon subtraction, the uniform convergence takes place for the energy-second-moment-constrained states with nonvanishing energy, i.e., the states $\varrho$ such that ${\rm tr}[\varrho H] \geq E_1 >0$ and ${\rm tr}[\varrho H^2] \leq E_2 < \infty$. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.

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