Probabilistic Noiseless Linear Quantum Amplification

• Probabilistic NLA is a quantum process that amplifies coherent states by conditionally heralding successful events, thereby avoiding excess noise.
• It employs sequential photon addition and subtraction to approximate the ideal transformation |α⟩ → |gα⟩ with effective gains close to the target and fidelities exceeding 90%.
• The technique enhances quantum state discrimination and loss recovery in communication and metrology, despite reduced heralding probabilities at higher gains.

Probabilistic noiseless linear quantum amplification (NLA) is a quantum process that enables the amplification of arbitrary quantum states—most notably coherent states—without introducing the excess noise imposed by the fundamental limits of deterministic, phase-insensitive linear amplification. Probabilistic NLAs achieve this by heralding only a subset of successful events, thus circumventing the quantum noise penalty associated with deterministic amplification and the constraints of the no-cloning theorem. The concept was experimentally realized by Fiuràšek, Zavatta, Paris, Bellini, and colleagues using a sequence of photon addition and subtraction to approximate the ideal transformation $|\alpha\rangle \to |g\alpha\rangle$ for weak coherent states, attaining high effective gain and fidelity (Zavatta et al., 2010). NLAs have emerged as essential components in quantum communication, metrology, and information processing, particularly for discriminating overlapping quantum states, recovering quantum information over lossy channels, and enabling advanced quantum repeater technologies.

1. Theoretical Foundations and Mathematical Description

NLAs operate by implementing a nonunitary, trace-decreasing transformation that approximates an ideal, noiseless amplification of the form

$|\alpha\rangle \to |g\alpha\rangle,$

for coherent states $|\alpha\rangle$, with nominal (amplitude) gain $g > 1$. The ideal (but unphysical) amplifier is described by the operator $g^{\hat{n}}$, where $\hat{n} = a^\dagger a$ is the photon number operator:

$$g^{\hat{n}}|\alpha\rangle = \exp\left[\tfrac{1}{2}(|\alpha|^2)(g^2 - 1)\right] |g\alpha\rangle.$$

This action amplifies the amplitude by $g$ but does not increase the quadrature variances, thereby preserving the signal-to-noise ratio—an impossible feat for deterministic linear amplifiers due to the constraints of quantum mechanics.

In the conditional, experimentally accessible realization, the process is engineered via a sequence of photon addition ($a^\dagger$) and subtraction ($a$) operating as $$\hat{G} = (g-2)a^\dagger a + a a^\dagger = (g-1)\hat{n} + 1$$ [Eq. (2), (Zavatta et al., 2010)]. For $g=2$, this simplifies to $\hat{G} = a a^\dagger$.

Key quantitative metrics include:

• Effective gain:

$$g_\text{eff} = \frac{1}{\alpha} \frac{\langle\alpha | \hat{G} a \hat{G} |\alpha\rangle}{\langle\alpha|\hat{G}^2|\alpha\rangle},$$

with $g_\text{eff} \approx g$ for weak $|\alpha|$.

• Fidelity with the target state:

$$F = \frac{|\langle g\alpha|\hat{G}|\alpha\rangle|^2}{\langle\alpha|\hat{G}^2|\alpha\rangle}.$$

For small $|\alpha|$, fidelities exceed $90\%$, with effective gains close to the nominal $g$.

2. Experimental Realization via Photon Addition and Subtraction

The reference implementation employs ultrafast pulsed parametric down-conversion and high-transmissivity beam splitters:

• Light preparation: Short-pulse Ti:sapphire laser at $786\,\text{nm}$ seeds the experiment.
• Photon addition: Achieved by stimulated parametric down-conversion in a nonlinear $\beta$-barium-borate crystal. Detection by D$_a$ in the idler channel projects the signal onto $a^{\dagger} |\alpha\rangle$.
• Photon subtraction: Followed by transmission through a beam splitter with $5\%$ reflectivity. Detection of a photon in D$_s$ signals a subtraction event. Coincidence detection (D$_a$ and D$_s$) heralds successful application of $a a^{\dagger}$.
• Quantum state reconstruction: Time-domain balanced homodyne detection is phase-stabilized and synchronized with coincidence events. The full quantum state (density matrix) is reconstructed using iterative maximum-likelihood quantum tomography.

Calibration is performed by measuring amplitude quadrature expectation values and reconstructing the density matrices of input, amplified, and vacuum states, ensuring gain and fidelity determinations are not affected by detection efficiency since losses cancel in the ratios.

3. Figures of Merit and Noise Properties

Effective gain and fidelity remain near-ideal for input amplitudes $|\alpha| \leq 0.65$. The equivalent input noise $N_\text{eq}$ (as defined in units of vacuum noise) is negative; for $g=2$, $N_\text{eq} < -0.48$, demonstrating that the process introduces less noise than the minimum quantum-limited noise of a deterministic amplifier.

A crucial property is that the combined action of photon addition and subtraction is fundamentally non-Gaussian and non-deterministic, but, by conditioning on successful events, the operation avoids noise amplification that would degrade phase information, enabling discrimination of closely spaced quantum states and recovery of information from lossy channels.

4. Significance for Quantum Communication and Metrology

The high-fidelity, low-noise properties of probabilistic noiseless linear amplifiers directly address critical challenges in quantum communications:

• State discrimination: NLAs increase the separation between partially overlapping coherent states, facilitating more reliable readout of quantum information encoded in amplitude or phase.
• Loss recovery: In quantum key distribution and metrology, NLAs can effectively pre-amplify weak signals at the receiver, compensating for transmission loss and thereby enhancing secure range and metrological precision (e.g., variance of phase estimators is reduced by $g_\text{eff}^{-2}$).
• Resource limits: NLAs enable performance surpassing the deterministic quantum noise limit, allowing negative added noise in the postselected subensemble—at the cost of a bounded, generally small, heralding probability (Zavatta et al., 2010).

5. Operational Constraints: Trade-Offs and Heralding Probability

The probabilistic character of NLAs is a direct consequence of the no-cloning theorem and the nonunitary nature of the operation. The success probability (probability of heralded amplification) decreases with increasing gain $g$ and input amplitude $|\alpha|$.

For the photon addition/subtraction protocol, heralding relies on simultaneous single-photon detection events, intrinsically limiting the maximum rate of successful amplification. In practical systems, optimizing the input amplitude and gain is essential to balance effective amplification, fidelity, and throughput. Extremely high gains or larger input amplitudes degrade both fidelity and success probability.

6. Broader Implications and Advances

The design and high-fidelity operation of probabilistic noiseless linear amplifiers, as introduced in (Zavatta et al., 2010), have inspired extensive theoretical and experimental developments across quantum information science. The photon addition/subtraction approach has been integrated into advanced protocols for quantum repeaters, error-correction networks, and entanglement distillation. More advanced architectures, including measurement-based NLAs and cluster-state amplification networks, extend the applicability to broader classes of quantum states and are compatible with both optical and microwave photonic platforms (Chrzanowski et al., 2014, Khalifa et al., 2023).

The noise properties and heralded operation principle established in this work underpin the role of NLAs as indispensable tools for overcoming transmission losses, extending quantum communication distances, and enhancing metrological sensitivity beyond deterministic quantum amplification limits.

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Key Properties Summary Table

Figure of Merit	Description	Values for $g=2$ ($|\alpha|\leq0.65$)
Effective gain ($g_\text{eff}$)	Ratio of amplified to input amplitude	$\approx$ 2
Fidelity ($F$)	Overlap with ideal amplified state	$>90\%$
Equivalent input noise ($N_\text{eq}$)	Added noise relative to vacuum noise	$< -0.48$
Heralding probability	Fraction of events yielding successful amplification	Decreases with larger $|\alpha|$, $g$

The scheme establishes the paradigm wherein probabilistic, conditional, and non-Gaussian quantum operations facilitate amplification that is otherwise forbidden in deterministic, linear, phase-insensitive amplifiers by quantum theory, with direct implications for quantum measurement, communication, and information processing (Zavatta et al., 2010).