The noise every amplifier must add

The signal from a single superconducting qubit, read out by bouncing a microwave pulse off its resonator, carries a few photons. The signal from a distant galaxy’s hydrogen at 21 centimetres is a whisper below the noise of the sky. A pulse of light at the far end of a hundred kilometres of optical fibre has lost all but a percent of a percent of its power. Each is too weak to be used directly and must be amplified, and amplifiers add noise. Most of that noise comes from the amplifier’s warmth — the jiggling of its electrons, the half a kT in a piece of wire that every resistor carries — and can be reduced by cooling it.

Not all of it. In 1962 Hermann Haus and James Mullen showed that an amplifier which magnifies a signal without regard to its phase must add a minimum of noise that no cooling removes, and in 1982 Carlton Caves stated the limit for every such amplifier: referred to its input, at least half a quantum. The limit is not a property of any material. It is a consequence of the commutation rule that also makes position and momentum uncertain, and the reason is short enough to follow in full.

A wave’s two halves #

Any narrow-band signal — a radio wave, a microwave, a beam of light — can be written as two oscillations a quarter of a cycle apart: an in-phase part and an out-of-phase part, called its quadratures. Its amplitude and phase are the length and angle of the arrow those two make. In quantum mechanics the two quadratures are like the position and momentum of an oscillator, which is exactly what each mode of the field is: the questions that can be asked together found that two quantities whose operators do not commute cannot both be sharp, and the quadratures do not commute. Their uncertainties have a product no smaller than a fixed amount, and in the field’s lowest state — the vacuum — both are at that minimum, equal, and nonzero.

A coherent state, the quantum description of a classical wave from an ideal laser or oscillator, has the vacuum’s uncertainty carried along with its amplitude: drawn in the plane of the two quadratures, it is a small round disc, one standard deviation across, displaced from the origin by the wave’s amplitude. That disc is the half quantum of vacuum fluctuation the motion that cannot be stopped found in every oscillator’s ground state, and it is the noise floor of any measurement of the wave.

Why the copy cannot be exact #

An amplifier of power gain $G$ should multiply both quadratures by $\sqrt G$, so that the disc’s centre moves out by that factor. The disc itself, if the amplifier added nothing, would also grow by $\sqrt G$ — larger than the vacuum, but in proportion to the signal, so the signal-to-noise ratio would be untouched. That is the dashed circle in the figure, and quantum mechanics forbids it.

The reason is in the mode operators. The input field’s annihilation operator $a$ obeys $[a, a^\dagger] = 1$, which is the commutation rule from which the quadratures’ uncertainty follows. An amplified output $b = \sqrt G\,a$ would have $[b, b^\dagger] = G$, which is not a field mode at all. The output must also obey $[b, b^\dagger] = 1$, and the only way for a linear amplifier to arrange it is to mix in a second mode — some internal degree of freedom of the amplifier — with the opposite sign:

$$b = \sqrt G\,a + \sqrt{G - 1}\,c^\dagger,$$

whose commutator is $G - (G - 1) = 1$. The second mode $c$ is at best in its own vacuum, and its vacuum fluctuations, multiplied by $\sqrt{G - 1}$, come out with the amplified signal. They are the added noise. Referred back to the input, by dividing by the gain, they are $\tfrac12(1 - 1/G)$ of a quantum: nothing at unit gain, approaching half a quantum at high gain.

Half a quantum added to the half quantum of vacuum the signal already carried doubles the noise. A coherent signal therefore leaves the best possible phase-insensitive amplifier with half the signal-to-noise ratio it entered with — three decibels worse — and no design, no material and no cooling changes that. The amplifier in the figure, at a gain of 4, has already added 0.375 of a quantum; its output disc is wider than the forbidden noiseless one by a third.

The amplifier that does not add #

The proof assumed the amplifier treats both quadratures alike. Drop that assumption and the limit goes. An amplifier can stretch the in-phase quadrature by $\sqrt G$ while shrinking the out-of-phase one by $1/\sqrt G$: the output is $b = \sqrt G\,X_1 + iX_2/\sqrt G$ in quadrature form, its commutator is unchanged, no second mode is needed, and nothing is added. The disc becomes an ellipse — the red one in the figure — whose long axis carries the amplified signal with the vacuum’s noise scaled in proportion, and whose short axis has had its noise squeezed below the vacuum’s.

This is a phase-sensitive amplifier, and it is the same device that the noise pushed below the floor found making squeezed light: a parametric amplifier, pumped at twice the signal frequency, which amplifies motion in one phase and suppresses it in the other, as the swing that is pumped, not pushed does for a playground swing. It is noiseless only for a signal in the quadrature it amplifies. A signal of unknown phase gains nothing, and on average loses as much in the squeezed quadrature as it gains in the stretched one. The theorem’s real content is the trade: amplify everything and pay half a quantum, or amplify one half of the wave and destroy the other.

Half a quantum as a temperature #

Engineers describe an amplifier’s noise by a noise temperature: the temperature of a resistor at its input that would add as much noise. Half a quantum per mode is, in that language, $\hbar\omega/2k$ — a temperature proportional to the frequency.

At radio frequencies the limit is tiny. At the hydrogen line it is 34 millikelvin, a hundred times below the noise of the cold sky itself, and the receivers of radio telescopes, cooled to a few kelvin and adding several kelvin of their own, are nowhere near it — their noise is technology, not quantum mechanics. At 5 gigahertz it is 0.12 kelvin, comparable to the temperatures of the dilution refrigerators in which superconducting qubits are kept; the transistor amplifiers that read qubits out add some ten to twenty quanta, and the Josephson parametric amplifiers placed in front of them, built from the superconducting circuits whose states they measure, come within a factor of two of the limit.

At optical frequencies everything is reversed. At the 1550-nanometre wavelength of telecommunications fibre, half a quantum is 4,655 kelvin. A fibre amplifier — a length of erbium-doped glass pumped by a laser — sits at room temperature, and its thermal noise at that frequency is negligible; its noise is entirely quantum, and the best of them add almost exactly half a quantum. Their famous three-decibel minimum noise figure, the halving of the signal-to-noise ratio a fibre engineer budgets for at every amplifier, is Caves’s theorem at work. Light arrives in lumps found that a faint light beam’s shot noise is the counting noise of its photons; an optical amplifier adds a second helping of the same size, from the spontaneous emission of its excited atoms, which is the second mode’s vacuum doing what the algebra said it would.

The first stage decides #

A receiver is a chain of amplifiers, and the noise of a chain is dominated by its first stage: the noise each later stage adds is divided, referred to the input, by the gain of everything before it — Friis’s rule for cascades.

So the quantum limit is reached, if at all, by putting a quantum-limited amplifier first and giving it enough gain to make everything behind it irrelevant. Behind a transistor amplifier adding twenty quanta, a quantum-limited first stage needs 16 decibels of gain before the chain’s noise is down to one quantum — twice the limit — and more to get closer. The progress in reading out qubits quickly enough for error correction, over the last fifteen years, has been largely the progress in building first-stage amplifiers with enough gain and bandwidth at the quantum limit.

The first amplifiers to approach the limit were masers. In the early 1960s the most sensitive radio receivers in the world put a ruby maser, cooled in liquid helium, directly behind the antenna, because its noise — a few kelvin — was ten times lower than anything else then available. It was a maser receiver of this kind, on a horn antenna in New Jersey, with which Arno Penzias and Robert Wilson found in 1964 an extra three kelvin of noise that no part of their equipment could account for, and that turned out to be the cosmic microwave background. Their noise budget, accounting for the antenna, the waveguide, the maser and the atmosphere to a fraction of a kelvin each, was possible only because the maser’s own contribution was small and known. The maser was nowhere near half a quantum at 4 gigahertz — about 0.1 kelvin — but it was the first amplifier for which the question could be asked.

What the half quantum costs in time #

The cost of amplifier noise is easiest to feel as time. To tell a qubit’s two states apart, a readout pulse is sent through its resonator and comes back with its phase shifted one way or the other; the two possible outputs are two discs in the plane of the quadratures, a small distance apart, and the measurement succeeds when the distance between their centres is several times the width of the noise. The distance grows with the number of photons collected, which grows with the time the pulse is integrated. The noise is the vacuum’s half quantum plus whatever the amplifiers add.

If the amplifier chain adds twenty quanta, the noise is forty-one times the vacuum’s half quantum, and reaching a given separation takes forty-one times as many photons as a perfect measurement — forty-one times as long at a given photon rate, or forty-one times as many photons in the resonator, which disturbs the qubit. If a quantum-limited parametric amplifier goes first with enough gain, the noise is twice the vacuum’s, and the measurement is about twenty times faster. That difference, between a measurement that takes a few microseconds and one that takes a few hundred nanoseconds, is the difference between reading a qubit faster than it decays, which error correction requires, and not.

Where the noise comes from #

The algebra says a second mode must be mixed in; a physical amplifier says what that mode is. In a maser or a laser amplifier the signal is amplified by stimulated emission from excited atoms, and the same atoms emit spontaneously, into the same mode, at random phases. Stimulated emission is the gain; spontaneous emission is the added noise; and the ratio between them is fixed by the fact that both come from the same atoms coupling to the same field. If every atom is excited — complete inversion — the spontaneous emission adds exactly half a quantum referred to the input, the minimum. If some atoms are in the lower state, they absorb as well as emit, more pumping is needed for the same gain, and the noise rises: the spontaneous-emission factor that fibre-amplifier engineers quote is the ratio of the noise to its quantum minimum, and it is one only for complete inversion.

In a parametric amplifier the second mode is the idler — the partner frequency into which the pump splits each of its photons along with a signal photon. The idler starts in its vacuum, and its fluctuations, mixed into the signal by the same process that amplifies it, are the added noise. Only when signal and idler are the same mode — the degenerate, phase-sensitive case — is there no separate vacuum to mix in, which is why that case can be noiseless. Every phase-insensitive amplifier, whatever it is built from, has some such second mode, and the theorem says only that its vacuum cannot be avoided.

Why an amplifier is not a copying machine #

The theorem has a second derivation, and it explains why the limit is as large as it is. The state that cannot be copied found that no device can take an unknown quantum state and produce two exact copies of it. A noiseless phase-insensitive amplifier would be such a device: amplify a coherent state by a gain of 2 and split the result in two at a beam splitter, and each half would be the original state.

With the half quantum included, the copies are noisy, and the noise is exactly what the no-cloning theorem requires. Two copies made this way have a fidelity of two-thirds with the original — and two-thirds is the proven best that any device whatever can achieve for copying unknown coherent states. Many copies approach a fidelity of one half, which is what is obtained by measuring both quadratures of the original as well as possible and preparing fresh coherent states from the result: amplifying to infinite gain is a measurement. The quantum-limited amplifier sits exactly on the boundary between what physics allows and what it forbids, and the half quantum is the width of the boundary.

The same boundary appears in measurement. Measuring both quadratures of a signal at once — as a heterodyne receiver does — must add the same half quantum, because such a measurement can be built from an amplifier and a classical readout; measuring one quadrature — homodyne detection — need not, which is why the most sensitive measurements of a wave, such as the gravitational-wave interferometers that inject squeezed light, measure one quadrature and arrange for the signal to be in it.

What the figures leave out #

The figures describe linear amplifiers acting on a single mode, with Gaussian noise, in the limit of narrow bandwidth. A real amplifier has a bandwidth over which its gain varies and a noise that varies with it; the theorem applies frequency by frequency. Amplifiers that are not linear — a photon counter, a threshold detector — are outside it, which is how single-photon detectors register light without adding half a quantum, at the price of discarding its phase. The temperatures in the limit figure use the convention that counts noise power as $kT$ per unit bandwidth; with the Planck form of thermal noise the numbers differ at frequencies where $\hbar\omega$ approaches $kT$, and engineers quote quantum-limited amplifiers in both conventions. The domain is linear amplification of a weak signal in a single mode, by an amplifier that treats both quadratures alike or, in the phase-sensitive case, treats them oppositely.

Still open: amplifying at the limit without the limit’s price #

Two directions push against the theorem’s edges. One avoids it by not amplifying phase-insensitively: measuring a qubit’s state needs only one quadrature of the readout signal, and phase-sensitive amplifiers can read it with less than the half quantum, while non-reciprocal amplifiers that pass signals one way protect the qubit from the amplifier’s own noise travelling backwards. The other probabilistically beats it: “noiseless linear amplifiers” that succeed only some of the time can amplify a coherent state without adding noise in the runs where they succeed, because a probabilistic process escapes the linearity the proof assumed; whether such devices can be useful for communication, given how rarely they succeed at high gain, is an active question. And building quantum-limited amplifiers with the bandwidth and power handling to read out thousands of qubits at once, without filling a dilution refrigerator with heat, is an engineering problem on which the scaling of superconducting quantum computers currently depends.

The limit itself is exact. An amplifier that multiplies both quadratures of a wave by $\sqrt G$ must mix in a second mode to keep $[b, b^\dagger] = 1$, and that mode’s vacuum adds at least $\tfrac12(1 - 1/G)$ of a quantum referred to the input — 0.45 at a gain of 10, approaching ½ — halving a coherent signal’s signal-to-noise ratio; as a temperature that is $\hbar\omega/2k$, 0.12 K at 5 GHz and 4,655 K at 1550 nm; and it is exactly the noise that makes amplify-and-split copies no better than the two-thirds fidelity quantum mechanics allows. Magnifying a wave without regard to its phase is a measurement in disguise, and a measurement of two incompatible things cannot be free.