Finite-energy superactivation of Gaussian amplifiers and additive-noise channels
- Field
- Topics
Problem
Is there a single-mode phase-insensitive Gaussian amplifier or additive-noise channel of zero finite-energy quantum capacity that acquires positive finite-energy quantum capacity when used in parallel with a two-mode positive-partial-transpose Gaussian channel of zero finite-energy quantum capacity? Consider single-mode, phase-insensitive bosonic Gaussian channels \(\mathcal G_{\tau,y}\) with finite real parameters \(\tau\geq1\) and \(y\geq|1-\tau|\). In the convention where the vacuum covariance matrix is \(I_2\), their displacement vectors and covariance matrices transform as
Here \(\tau>1\) describes an amplifier, while \(\tau=1\) and \(y>0\) describes an additive-noise channel. Restrict to the non-entanglement-breaking parameter range \(y<1+\tau\). Let \(\mathcal H\) be any Gaussian channel with two input modes and two output modes, acting on covariance matrices as \(V\mapsto X_HVX_H^T+Y_H\) for real \(4\times4\) matrices \(X_H\) and \(Y_H=Y_H^T\). Write
Require both complete positivity and the positive-partial-transpose (PPT) property:
For a bosonic channel \(\mathcal N\), let \(Q_E(\mathcal N)\) be its unassisted quantum capacity when the total mean input photon number over \(n\) uses is at most \(nE\), and define
allowing arbitrary, including non-Gaussian, codes. The question is whether some pair \((\mathcal G_{\tau,y},\mathcal H)\) allowed by Eqs. (1)–(3) and the stated parameter ranges satisfies
with capacities as in Eq. (4).
Source
Lim, Takagi, Adesso, and Lee studied activation of all single-mode phase-insensitive Gaussian channels by the two-mode PPT channel family of Smith, Smolin, and Yard. After finding no activation of amplifiers by their methods, they conjectured in Section III that single-mode Gaussian amplifiers cannot be activated, and in Section IV that additive-noise channels and amplifiers cannot exhibit activation or superactivation [LTA+19]. The present formulation isolates the superactivation case of that conjecture for the class of all two-mode PPT Gaussian helpers, which is broader than the helper family tested there, and allows arbitrary finite-energy codes. It is set in the Gaussian superactivation framework of Smith, Smolin, and Yard [SSY11].
Progress
Smith, Smolin, and Yard established Gaussian quantum-capacity superactivation for a different single-mode channel, the 50% pure-loss channel \(\mathcal L_{1/2}:V\mapsto\frac12V+\frac12I_2\). They constructed a two-mode Gaussian channel \(\mathcal H_0\) for which both matrices in Eq. (3) have eigenvalues \(0,0,2\sqrt2,2\sqrt2\), and a three-mode Gaussian input with about \(0.05\) bits of coherent information at about \(60\) photons per channel use. Hence
\begin{equation} Q_{\mathrm{fin}}(\mathcal L_{1/2})=Q_{\mathrm{fin}}(\mathcal H_0)=0, \qquad Q_{\mathrm{fin}}(\mathcal L_{1/2}\otimes\mathcal H_0)>0, \tag{6} \end{equation}where the first factor is antidegradable and the second is PPT. The pure-loss channel has transmissivity \(1/2\) and is not a member of the families in Eq. (1) [SSY11].
Lim, Takagi, Adesso, and Lee extended activation and superactivation to a broad range of thermal attenuators, including low transmissivities. They used the two-parameter PPT channel family of Smith, Smolin, and Yard and a three-parameter family of Gaussian inputs, and compared coherent information with upper bounds on the attenuator capacity. They found no amplifier activation by these methods, which underlies the conjectures in Sections III and IV. Their calculations do not optimize over all helper channels, all Gaussian input states, non-Gaussian inputs, or blocklengths, and they identify tighter capacity upper bounds for amplifiers and additive-noise channels as needed for a conclusive test [LTA+19].
In the covariance convention of Eq. (1), a phase-insensitive channel with \(\tau\geq1\) is entanglement breaking exactly when
\begin{equation} y\geq1+\tau, \tag{7} \end{equation}as shown in Fig. 1 of the same paper. Its Theorem 9 in Appendix B proves that for an entanglement-breaking channel \(\Phi\) and an arbitrary channel \(\Psi\), the coherent information of \(\Phi\otimes\Psi\) with finite mean input energy on each factor equals the corresponding coherent information of \(\Psi\). The paper concludes that bosonic entanglement-breaking channels, such as those in the region of Eq. (7), cannot be helped by another zero-capacity channel when inputs have finite energy. This is a genuine no-go result, unlike the unsuccessful restricted searches outside that region [LTA+19].
Comment
No resolution of the stated finite-energy question was located in the literature checked through 15 September 2026. Allowing every two-mode PPT Gaussian helper is a formulation motivated by the 2019 conjecture, not a claim that its numerical study covered that whole class. An affirmative example must certify both individual regularized finite-energy capacities as zero and a positive joint rate. A negative answer must cover all admissible helpers and unrestricted codes, rather than only Gaussian single-use coherent information. Certifying the hypothesis \(Q_{\mathrm{fin}}(\mathcal G_{\tau,y})=0\) is itself unresolved for some parameters. The capacities of the two families are the subjects of the records on the unassisted quantum capacity of a thermal amplifier and on the quantum capacity of the Gaussian random-displacement channel; the latter is the additive-noise case \(\tau=1\), whose positivity is undecided on part of the range \(y<1\).