Efficiently Fuelling a Quantum Engine with Incompatible Measurements

Sreenath K. Manikandan,1, 2, 3, ∗ Cyril Elouard,1, 4 Kater W. Murch,5 Alexia Auffeves,6 and Andrew N. Jordan7, 2, 1

1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA2Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627, USA

3Nordita, KTH Royal Institute of Technology and Stockholm University,Hannes Alfvens vag 12, SE-106 91 Stockholm, Sweden

4QUANTIC lab, INRIA Paris, 2 Rue Simone Iff, 75012 Paris, France5Department of Physics, Washington University, St. Louis, Missouri 63130

6Universite Grenoble Alpes, CNRS, Grenoble INP, Institut Neel, 38000 Grenoble, France7Institute for Quantum Studies, Chapman University, Orange, CA, 92866, USA

(Dated: July 29, 2021)

We propose a quantum harmonic oscillator measurement engine fueled by simultaneous quantummeasurements of the non-commuting position and momentum quadratures of the quantum oscillator.The engine extracts work by moving the harmonic trap suddenly, conditioned on the measurementoutcomes. We present two protocols for work extraction, respectively based on single-shot and time-continuous quantum measurements. In the single-shot limit, the oscillator is measured in a coherentstate basis; the measurement adds an average of one quantum of energy to the oscillator, whichis then extracted in the feedback step. In the time-continuous limit, continuous weak quantummeasurements of both position and momentum of the quantum oscillator result in a coherent state,whose coordinates diffuse in time. We relate the extractable work to the noise added by quadraturemeasurements, and present exact results for the work distribution at arbitrary finite time. Bothprotocols can achieve unit work conversion efficiency in principle.

Introduction.—Quantum thermodynamics is concernedwith how the exchange of heat and work can be under-stood and applied when quantum effects such as entan-glement and coherences are present [1–4]. The emergingfield of “quantum energetics” is applicable to stochas-tic energy exchanges when there are no thermal baths.As an example, quantum measurement powered engineshave been proposed with qubit systems, as well as con-tinuous variable systems [5–16]. This line of research isgreatly stimulated by successful demonstrations of quan-tum measurements and control in a variety of quantumplatforms including but not limited to superconductingcircuits [17–19], cavities [20–22], trapped ions [23–25],trapped nano-particles [26], single electron systems [27],and mechanical resonators [28, 29].

The prime focus in these models has been on the mea-surement of a single observable, either the spin alonga chosen axis in the case of finite dimensional sys-tems [8, 14, 16, 30], or a given quadrature with continuousvariable systems [5, 7, 31]. By making appropriate combi-nation of measurements and feedback operations [32–39],a quantum engine can be used to accelerate an electronto charge a capacitor, or to lift a tiny mass [5]. A quan-tum refrigerator based on measurements [13], measure-ment driven single temperature engines that require nofeedback [7, 12], interaction-free measurement engines [9],and quantum measurement engines driven by quantumentanglement [16] have also been conceptualized, extend-ing the scope of measurement based thermal-equivalentmachines.

In this Letter, we propose a quantum engine fueled bysimultaneous weak measurements of two non-commutingobservables: the position and momentum of a quantum

M M W

DD

T

(a) (b) (c)

M

FIG. 1. Cyclic operation of the quantum oscillator measure-ment engine fueled by incompatible measurements. (a) Thequantum oscillator is initially thermalized to a heat bath attemperature T. (b) A demon weakly measures both the po-sition and momentum of the quantum oscillator. The mea-surement results in a coherent state. (c) Work is extractedby displacing the trap conditioned on the measurement out-comes.

oscillator. Work is extracted by moving the bottom ofthe harmonic trap suddenly, conditioned on the mea-surement outcomes. We show that incompatible mea-surements have rewarding energetic consequences whencompared to similar protocols for a quantum engine fu-eled by measurement of a single quadrature [5, 31], andwhen compared to the classical limit which describes aheat engine fueled by measuring the position of a Brown-ian particle in a harmonic trap [40, 41]. With simultane-ous quantum measurements, the measurement strengthsfor each of the incompatible quadratures can be tunedsuch that the measurement results in a displaced groundstate (a coherent state [42, 43]) of the quantum oscilla-tor [44, 45]. This allows one to perfectly reset the engine’scycle by work extraction, in the same way as conceivedin the original version of the Szilard engine, where thedemon resets a gas of particles at no cost: it uses in-formation on the positions to compress the gas on the

arX

iv:2

107.

1323

4v1

[qu

ant-

ph]

28

Jul 2

021

2

left of a vacuum chamber, and then lets it expand backto equilibrium [46]. The engine is efficient because thework extraction step precisely brings it back to the equi-librium state. When simultaneous weak measurementsof position and momentum are performed on a thermalstate, it is also well known that they add an extra quan-tum of noise in to the oscillator [44, 47]. This addednoise from measurement allows us to extract work even atzero temperature, beating the previously known classicalbounds for a Brownian heat engine (see SupplementaryNote 1) [40, 41].The setup.—A quantum harmonic oscillator is described

by the Hamiltonian, H = p2

2m + mω2 x2

2 , where p and xobey the commutation relation, [x, p] = i~. When thetemperature is sufficiently small such that kBT � ~ω,thermal fluctuations are negligible and the quantum har-monic oscillator will be in its ground state |0〉. Thepremise of the measurement engine is that a suitableweak quantum measurement will excite the oscillator.Work can then be extracted by a feedback loop thatchanges the harmonic trap suddenly. The feedback ismost efficient if it resets the quantum engine to its initialquantum state at the end of each cycle, making the engineprepared for the next cycle to begin. For the quantumoscillator, a simultaneous weak quantum measurementof both position and momentum observables is uniquelysuited to this task because such a protocol realizes mea-surements in the coherent state basis [42–45].

We now proceed to describing two protocols for op-timal work extraction with incompatible quantum mea-surements of the quantum oscillator: a single-shot mea-surement protocol where measurements are described byprojection onto the coherent state basis [44], and thetime-continuous limit where continuous weak quantummeasurements of both position and momentum of thequantum oscillator results in a coherent state whose co-ordinates diffuse in time [44].Single-shot quantum measurements.—The protocol takesplace in three steps (see Fig. 1):

• Step (a): The quantum oscillator thermalizes withthe ambient inverse temperature, β = 1/kBT,

yielding a thermal state, ρ(0) = e−βH

Z , where Z =

tr{e−βH}.

• Step (b): The quantum oscillator is weakly mea-sured in the coherent state basis {|α〉}, yielding re-sult α.

The Kraus operators describing the measurement are,K(α) = 1√

π|α〉〈α| [48, 49], and the normalized proba-

bility distribution corresponding to the readouts is,

PQ(r, n) =1

π〈α|e

−βH

Z|α〉 =

1

π(1 + n)e−r

2/(1+n), (1)

using the parametrization α = reiθ. We denote theaverage photon number in the initial thermal state by

n = (e~ωkBT − 1)−1, the Bose-Einstein occupation. The

probability density PQ(r, n) is also known as the HusimiQ distribution of the state ρ(0) [47, 50] which is obtainedwhen simultaneous weak measurements of position andmomentum are performed on a thermal state. The aver-age quanta in PQ(r, n) is n + 1, meaning that the mea-surement process adds one extra quantum to the oscilla-tor quantum state (see Supplementary Note 2).

We may allow the quantum harmonic oscillator to un-dergo free evolution, |α〉 → |αe−iωτ(α)〉 = |r〉 such thatthe coherent state is located along the positive x axisin the phase space, (x, p). Here τ(α) = θ/ω. This freeunitary evolution is essentially a rectifier which channelsan arbitrary displacement to a preferred direction, usingquantum feedback. The efficiency of this step will requirethat the time scale of thermalization, τ

T� 2π/ω.

• Step (c) (work extraction): We suddenly shift thequantum harmonic trap, such that the coherentstate |r〉 is the new quantum ground state. Inthe process, we extract the amount of work, W =~ωr2 > 0. The system thermalizes to the ambienttemperature T, completing the cycle.

In a quantum LC circuit implementation, the work ex-traction in step (c) would correspond to modifying theoffset voltage in the capacitor suddenly. Alternatively,one can also modify the branch flux/current in the cir-cuit, if the free evolution aligns the coherent state alongthe flux/current axis, or a combination of displacementsin both voltage and current by an appropriate choice,τ(α). An alternate Binary-valued feedback protocol forthe engine is presented in Supplementary Note 3.Extractable work.—The average amount of work ex-tractable from the quantum harmonic oscillator in steps(a)–(c) is,

〈W 〉 = ~ω∫ 2π

0

dθ

∫ ∞0

rdr r2PQ(r, n) = ~ω(1 + n).(2)

Note that the extra quantum added by the measurementprocess (see Supplementary Note 2) is also extracted per-fectly in the feedback step. This additional quantum ofenergy is a purely quantum effect, which exceeds the clas-sical bound on extractable work from the Brownian heatengine at zero temperature (see Supplementary Note 1).Addition of extra noise from simultaneous measurementof non-commuting observables is required by quantummechanics [44, 47, 51–55], and our measurement engineexploits the energetic consequence of this added noise bydemonstrating that it can be rectified to produce usefulwork.

The extracted work is also equal to the sum of theaverage energy given by the thermal bath (QT = ~ωn)and the average energy given by the measurement pro-cess (QM = ~ω). Hence energy is conserved and wehave 〈W 〉 = QT + QM ≡ Q. In addition, energy QM

3

-0.5 0.5 1.0

mω

ℏx(t)

-0.5

0.5

p (t)

ℏmω

(a)

1 2 3 4

W

ℏω

0.5

1.0

1.5

2.0

(W/ℏω)

(b)

0.5 1.0 1.5 2.0 2.5ωt

0.2

0.4

0.6

0.8

⟨W/ℏω⟩

(c)

-0.10 -0.05 0.05 0.10 0.15

mω

ℏx(t)

-0.15

-0.10

-0.05

0.05

0.10

p (t)

ℏmω

(d)

0.005 0.010 0.015

W

ℏω

100

200

300

400

500

(W/ℏω)

(e)

1 2 3 4 5ωt

0.1

0.2

0.3

0.4

0.5

Jτ

(f)

1/4

FIG. 2. (a) A single quantum trajectory resulting from continuous quantum measurement of both position and momentumobservables, for a duration ωt = 2.5. At time ωt = 2.5 the oscillator is reset (blue arrow). (b) The probability distributionof extractable work if the feedback is applied only once at ωt = 1, from simulation of 104 trajectories. The red curve is theexact prediction for the probability distribution of extractable work. (c) The average of extractable work when the feedback isperformed only once at a given ωt. (d) The average position and momentum of the harmonic oscillator after every single stepmeasurement, for a duration ωt = 5. Here the oscillator is reset to the origin by a feedback applied after each measurement.(e) The probability distribution of extractable work at ωt = 1 from simulation of 104 trajectories, for continuous feedbackapplied after every single step measurement. (f) The average power J delivered by the engine per measurement rate τ−1, as afunction of duration of the measurement. The engine reaches a steady state with unit efficiency in the large time limit whenthe quadrature variances of the oscillator (which are identical), ν(t) tends to 1/2.

is lost by the measuring apparatus such that the latterhas to be re-energized to be compatible with the Wigner-Araki-Yanase theorem [56–59]. The measurement enginein this case also has unit work conversion efficiency, i.e.,η = 〈W 〉/Q = 1. Above we have not accounted for theadditional cost to erase the memory of the Maxwell’s de-mon, which is discussed in Supplementary Note 4 [60].

Continuous weak quantum measurement protocol.—Wenow describe a time-continuous operation of our quan-tum oscillator measurement engine, where continuousweak quantum measurement of both position and mo-mentum of the quantum oscillator results in a coher-ent state whose coordinates diffuse in time. Work isextracted by moving the bottom of the harmonic trap;either time-continuously as measurement results are ac-cumulated, or at the end.

For the engine, we assume that the oscillator is initial-ized in a Gaussian state, so the first moments togetherwith the covariance matrix elements Vij = 〈1/2{qi −qi, qj − qj}〉, completely specify the quantum state [61].Here q1 = x, q2 = p, and qi = 〈qi〉, i = 1, 2. Simulta-neous weak measurements of both the position and mo-mentum observables of the quantum harmonic oscillatorgiven the continuous readouts r1(t) for position measure-ment and r2(t) for momentum measurement results in thefollowing stochastic quantum evolution of the quantum

harmonic oscillator state (in dimensionless units wherex→

√mω/~x, p→ p/

√~mω, t→ ωt) [45]:

dq1

dt= q2 +

q3

2τ1(r1 − q1) +

q4

2τ2(r2 − q2),

dq2

dt= −q1 +

q4

2τ1(r1 − q1) +

q5

2τ2(r2 − q2),

dq3

dt= 2q4 −

q23

2τ1− q2

4

2τ2+

1

2τ2,

dq4

dt= q5 − q3 −

q3q4

2τ1− q4q5

2τ2,

dq5

dt= −2q4 −

q24

2τ1− q2

5

2τ2+

1

2τ1. (3)

The covariances are labeled by, q3 = 2(〈x2〉−〈x〉2), q4 =〈xp+px〉−2〈x〉〈p〉, and q5 = 2(〈p2〉−〈p〉2). The readoutsare stochastic variables, ri = qi +

√τiζi, where ζi are a

Gaussian white noise (due to quantum fluctuations) satis-fying 〈ζi(t)ζi(0)〉 = δ(t) and τi are the characteristic mea-surement times [45, 62–65]. Work can be extracted in theform of an instantaneous linear feedback Hamiltonian,Hfb = f1(t)x+f2(t)p, where f2(t) = − q3

2τ1[r1(t)− q1(t)]−

q42τ2

[r2(t)−q2(t)] and f1(t) = q42τ1

[r1(t)−q1(t)]+ q52τ2

[r2(t)−q2(t)](see Supplementary Note 5). Here qi, i = 1, 2are the predicted evolution of qi in the absence of mea-

4

surements, given by, q1(t) = q1(0) cos t + q2(0) sin t andq2(t) = −q1(0) sin t+ q2(0) cos t.

We restrict to the case when τ1 = τ2 = τ which pro-duces coherent states of the quantum oscillator as mea-surement outcomes and maximizes the engine’s efficiency(see Supplementary Note 7). For the covariance matrixinitialized in its normal form, V = diag{ν, ν} with a cor-responding mean number of thermal photons n = ν−1/2,the quantum measurement induced evolution is such thatit preserves the normal form of the covariance matrix [45].The work along an individual trajectory obeys (in theStratonovich form),

d(W/~ω)

dt= q1q1 + q2q2 = q1

[q3

2√τζ1(t) +

q4

2√τζ2(t)

]+ q2

[q4

2√τζ1(t) +

q5

2√τζ2(t)

]. (4)

The probability distribution of work extracted at arbi-trary finite time is given by,

P(W/~ω, t) =τ

σ(t)exp

[− τW

σ(t)~ω

], (5)

so the work extracted on an average is given by 〈W/~ω〉 =σ(t)/τ . The parameter σ(t) equals ν2(t)dt if work isextracted after every measurement of duration dt. Ifthe controller decides to apply feedback only after a du-ration t, the work distribution corresponds to the caseσ(t) =

∫ t0dt′ν2(t′) in Eq. (5). The nonzero average work

results from rectifying the quantum noise in the measure-ment process, and is nonzero even at zero temperature(see Supplementary Note 6). The average power J of thequantum measurement engine is given by,

J(t) = d〈W/~ω〉/dt = ν2(t)/τ, (6)

which serves as a useful quantity to infer the relation be-tween rate of information acquisition (the measurementrate) and work extraction for the continuous measure-ment engine; work is extracted at a faster rate as themeasurement rate τ−1 is increased. Further, a larger ν(t)corresponding to a higher thermal quanta in the initialstate also allows higher power.Steady state of the engine.—The dynamical equationswhich describe the evolution of covariance matrix el-ements are deterministic, and they achieve the steadystate value, lim

t�τν(t) = 1/2. In this limit—which is also

the steady state of the quantum measurement engine—the quantum measurement dynamics describes coherentstate diffusion. The measurement engine also achievesunit efficiency in its steady state; since the measurementmerely displaces the ground state (where ν(t) = 1/2), thefeedback resets the engine perfectly, closing the engine’scycle.Results.—The characteristics of the continuous quantummeasurement engine are shown in Fig. 2. We first con-sider a situation where work is extracted by applying a

feedback after continuously measuring the oscillator fora finite duration t. Figure 2(a) displays a simulation ofa single such trajectory, which undergoes two types ofdynamics; while the variance decreases to its minimumuncertainty deterministically, the average displacementdiffuses as energy is added through the measurement.After a time t the oscillator is reset to extract work.Figure 2(b) displays the probability distribution of ex-tracted work from the feedback step. The average workextracted [Fig. 2(c)] increases with the duration of themeasurement. Subsequently, we consider the situationwhere work is extracted after each step of the measure-ment. Figure 2(d) displays the average position and mo-mentum of the oscillator after each measurement, whichare then reset to the origin by the feedback. Figure 2(e)displays the probability distribution of extracted work.The average power J delivered by the quantum engine inunit of the measurement rate τ−1 is shown in Fig. 2(f),which decreases and reaches its steady state value as thequantum oscillator is purified by measurements.

Conclusions.—We have characterized a quantum enginefueled by simultaneous quantum measurements of bothposition and momentum observables of a simple har-monic oscillator. We discussed two protocols for oper-ation of the engine, respectively powered by single-shotand continuous quantum measurements. In both cases,the measurement produces a displaced ground state ofthe quantum oscillator, and work is extracted by shift-ing the bottom of the harmonic trap suddenly. Whencompared to their classical counterpart, the quantum en-gines yield non-zero work output even at zero temper-ature, demonstrating the energetic consequence of thequantum of noise added when both quadratures are mea-sured simultaneously; at zero temperature the nonzerowork output results from the feedback utilizing the quan-tum of energy inserted by the phase preserving measure-ment [44, 47, 51–55]. We also derived exact analytical ex-pressions for the probability distributions of extractablework in both transient and steady state of the quantumengine. The availability of exact probability distribu-tions of extractable work may further aid research to-wards thermodynamically characterizing quantum mea-surement engines, and derive quantum fluctuation theo-rems for quantum engines and refrigerators fueled solelyby the quantum measurement process [13, 14, 30, 66, 67].

Note added.—Towards the completion of this work, webecame aware of a closely related preprint [31] inves-tigating work extraction from thermal resources usingphase sensitive measurements, as opposed to the phasepreserving measurements discussed in the present Letter.

Acknowledgements.—This work was supported by theJohn Templeton Foundation grant no. 61835. The workof SKM was supported in part by the Wallenberg Ini-tiative on Networks and Quantum Information (WINQ).We acknowledge fruitful discussions with Tathagata Kar-makar, Philippe Lewalle, Masahito Ueda, Benjamin

5

Huard, Remy Dassonneville, and Lea Bresque.

6

M M W

T

(a) (b) (c)

M

D

FIG. 3. Cyclic operation of the classical Brownian heat engine [40, 41]. (a) The Brownian particle in a harmonic trap is initiallythermalized to a heat bath at temperature T. (b) The demon measures position of the Brownian particle. (c) Work is extractedby displacing the trap conditioned on the measurement outcome.

SUPPLEMENTARY MATERIALS

Supplementary Note 1: The Brownian heat engine

The classical limit of the engine is described by a Brownian particle in a harmonic trap, whose position x is

distributed according to the equilibrium distribution P(x) =√

k2πkBT exp

(− kx2

2kBT

), where k is the spring constant,

kB is Boltzmann’s constant and T is the temperature of the heat bath (see Fig. 3) [40, 41]. A Maxwell’s demonextracts work from the heat bath by measuring the particle’s position x, followed by shifting the bottom of theharmonic trap suddenly to the new position of the particle. If the measurement performed is error free with idealfeedback, it is possible to convert all the available information to extractable work, W = kx2/2 [40]. The demon inthis case extracts work on an average, 〈W 〉 = kBT/2 per cycle, which also saturates the achievable upper-bound forextractable work by moving the trap in the classical Brownian engine limit [40].

Supplementary Note 2: Energy contributed by measurement

The unconditional state of the quantum oscillator after a coherent state basis measurement (implemented by asimultaneous weak measurement of both position and momentum quadratures, also known as a phase-preservingmeasurement) can be written as,

ρA =

∫d2αQ(α)|α〉〈α|, (7)

where Q(α) = 1π(1+n)e

−|α|2/(1+n), is the Husimi Q distribution [50] of a thermal state (denoted as PQ in Eq. (1) of

the main text) with average number of thermal quanta n. We can now compute the average number of quanta in theunconditional post-measurement state ρ, given by,

〈N〉A = tr{ρAN} = tr[ρAa†a] =

∫d2αQ(α)|α|2 = n+ 1. (8)

We find that the measurement, on an average, adds one quantum of energy to the oscillator, which is then perfectlyextracted in the feedback step. The additional quantum in Q(α) stems from the fact that the coherent state basis isovercomplete; even measurement of the vacuum state will in general yield a coherent state with finite α.

Note that the addition of single quantum is a generic property of coherent state basis measurements which can beimplemented by a simultaneous weak measurement of both position and momentum of the quantum particle [44]. Tosee this, we can write an arbitrary initial quantum state ρ in the P representation as using the coherent state basis|β〉,

ρ =

∫d2βP(β)|β〉〈β|. (9)

The P representation can be derived from the initial density matrix ρ in different equivalent ways, for instance seeRef. [68], where it is defined as,

P(β) =e|β|

2

π2

∫〈−γ|ρ|γ〉 exp(|γ|2 + βγ∗ − β∗γ)d2γ, (10)

7

where |γ〉 are again coherent states. The different equivalent P representations of ρ are constrained to obey theoptical equivalence theorem for the density matrix ρ for the expectation value of any normally ordered operator:〈(a†)nam〉 = tr[ρ(a†)nam] =

∫d2βP(β)(β∗)nβm. As an example, the average number of quanta in the initial density

matrix ρ maybe computed using the P function as, 〈N〉 = 〈a†a〉 = tr[ρa†a] =∫d2βP(β)|β|2 = n. The P function for

a coherent state |α〉〈α| is the delta function, P(β) = δ(β−α) and a P representation for a thermal state with averagenumber of thermal quanta n is, P(β) = 1

πn exp(−|β|2/n).On the other hand, performing a measurement in the coherent state basis produces coherent states |α〉 with

probability Q(α) = 1π 〈α|ρ|α〉, which is known as the Q representation of the density matrix ρ. The unconditional

post measurement state ρA can be written as the following ensemble where the probabilities are given by the Qfunction [50, 69],

ρA =

∫d2αQ(α)|α〉〈α|. (11)

So a method to probe the Q function of a quantum state experimentally is to perform measurements in the coherentstate basis [47]. We now proceed to compute the average quanta in the post measurement state. In order to do that,we can relate the Q function (which appear as the probability density describing the post measurement state) to the

P function via the transformation rule [69]: Q(α) = 1π

∫d2βP(β)e−|α−β|

2

, and write,

ρA =

∫d2α

(1

π

∫d2βP(β)e−|α−β|

2

)|α〉〈α|. (12)

The average number of quanta in the unconditional post measurement state is, 〈N〉A = tr[ρAa†a]. This can be

evaluated by changing the order of integrals as,

〈N〉A = tr[ρAa†a] =

1

π

∫d2βP(β)

∫d2αe−|α−β|

2

|α|2 =

∫d2βP(β)(1 + |β|2) = 1 + n, (13)

where n is the average quanta in the state prior to measurement. Clearly, this extra quantum comes from themeasurement process, and the addition of a quantum of noise is in the same spirit as how it is typically describedas added to the variance of quadratures after the measurement, as discussed in Refs. [44, 47]. Here we look at thechange in the mean quanta instead, which is the observable relevant to the engine’s energetics.

Supplementary Note 3: Binary feedback

The engine’s implementation with single shot measurements and a binary-valued feedback is as follows:

• Step (a)–(b) are identical to that discussed in the main text.

• Step (c) (work extraction): When the amplitude of the measured coherent state is greater than r0/2, work isextracted by shifting the trap to r0, and the system is let to thermalize. Otherwise, no action is performed. Thework extracted is zero if r < r0/2 and the work extracted is ~ω(2rr0 − r2

0) for r ≥ r0/2.

The average work extracted in this protocol is,

〈W ′〉 = ~ω∫ 2π

0

dθ

∫ ∞r0/2

rdrP(r, n)(2rr0 − r20)

= ~ωr0

√π(1 + n) erfc

r0

2√

1 + n. (14)

Here erfc(u) = 2π−1/2∫∞ue−y

2

dy. The efficiency is less than that of the continuous feedback scheme, η′ = 〈W ′〉Q < 1,

and has a maximum, η′max ≈ 0.85 (Fig. 4). The classical analogue of this protocol is presented in Ref. [41].

Supplementary Note 4: Cost of memory erasure and the efficiency of the engine

Here we first discuss Landauer erasure in the Binary feedback protocol discussed above. The memory in thisexample behaves essentially like a logical bit, with states L or R, indicating if the particle is to the left of r0/2 or tothe right of r0/2. The minimum energy cost to erase the memory can be accounted by the Landauer’s bound as [70],

W ′res = kBTDH(p0). (15)

8

Here H(p0) = −p0 log p0 − (1 − p0) log (1− p0), where we have defined p0 as the concatenated probability of findingthe particle to the right of r0/2:

p0 =

∫dθ

∫ ∞r0/2

rdrP(r, n) = e−r20

4(n+1) . (16)

The Shannon entropy is indeed bounded from above by log 2 as expected for a classical bit memory. The efficiencyη′T of the thermodynamic cycle can now be computed,

η′T =〈W ′〉 −W ′res

Q=〈W ′〉 − kBTDH(p0)

Q. (17)

We now look at continuous measurement examples where at each step both the information stored, and the feedbackare continuous-valued. The associated probability density is P(r1, r2), where r1, r2 are the readout variables. As anextension of the above example, we consider a countable discretization for the probability distribution of the readoutsP(r1i, r2j) where the sampling is done according to bins around points r1i, r2j having bin area δ2 such that,

P(r1i, r2j)δ2 =

∫ (i+1)δ

iδ

∫ (j+1)δ

jδ

dr1dr2P(r1, r2), (18)

by the mean value theorem [71]. Here δ2 can be related to the resolution of the detector. The entropy of this discreteprobability distribution is given by,

H(P) = −∑i,j

δ2P(r1i, r2j) log [P(r1i, r2j)δ2] = −

∑i,j

δ2P(r1i, r2j) log [P(r1i, r2j)]−∑i,j

δ2P(r1i, r2j) log δ2. (19)

Assuming P(r1, r2) logP(r1, r2) is Riemann integrable, we note that the first term−∑i,j δ

2P(r1i, r2j) log P(r1i, r2j) → −∫dr1dr2P(r1, r2) logP(r1, r2) = H(P) when δ → 0. We also note that∑

i,j δ2P(r1i, r2j) = 1. Therefore for a discretization into squares of area δ2, the associated Shannon entropy scales

approximately as, H(P) = H(P)− log δ2 [71]. The efficiency of the thermodynamic cycle at each step becomes,

ηT =〈W 〉 −Wres

Q=〈W 〉 − kBTDH(P)

Q, (20)

where 〈W 〉 is the average work extracted at each step.

Supplementary Note 5: Continuous feedback

The work extraction protocol is essentially a linear feedback stabilization protocol for the quantum oscillator. Theeffect of a linear Hamiltonian Hfb = f1x + f2p on the quantum mechanical averages of x and p when measurementsare done continuously is,

dq1

dt= q2 +

q3

2τ1(r1 − q1) +

q4

2τ2(r2 − q2) + f2,

dq2

dt= −q1 +

q4

2τ1(r1 − q1) +

q5

2τ2(r2 − q2)− f1, (21)

where q1 = 〈x〉 and q2 = 〈p〉. Therefore we can choose the amplitude of the linear feedback Hamiltonian, f2(t) =− q3

2τ1[r1(t)− q1(t)]− q4

2τ2[r2(t)− q2(t)] and f1(t) = q4

2τ1[r1(t)− q1(t)] + q5

2τ2[r2(t)− q2(t)] such that the unitary evolution

resulting from feedback effectively cancels the measurement back-action on the quantum state. Here qi, i = 1, 2 isthe predicted evolution of qi in the absence of measurements, given by, q1(t) = q1(0) cos t + q2(0) sin t and q2(t) =−q1(0) sin t+ q2(0) cos t, in units where t→ ωt.

Supplementary Note 6: Work extracted on an average in the continuous measurement protocol

As discussed in the main text, the work extracted along an individual quantum trajectory in the continuousmeasurement protocol satisfies the following stochastic equation,

d(W/~ω)

dt= q1q1 + q2q2 = q1

[q3

2√τζ1(t) +

q4

2√τζ2(t)

]+ q2

[q4

2√τζ1(t) +

q5

2√τζ2(t)

]. (22)

9

We further assume initial conditions q1(0) = q2(0) = 0 and q3(0) = q5(0), q4(0) = 0. As a result, the extractable

work at arbitrary time t is the integral, W (t)/~ω =∫ t

0dt′[q1(t′) q3(t′)

2√τζ1(t′) + q2(t′) q5(t′)

2√τζ2(t′)

]. The above integral is

cast in the Stratonovich form and can be discretized as,

∫ t

0

dt′[q1(t′)

q3(t′)

2√τζ1(t′) + q2(t′)

q5(t′)

2√τζ2(t′)

]=

dt′

2√τ

N−1∑i=1

qi1 + qi+11

2qi3ζ

i1 +

dt′

2√τ

N−1∑i=1

qi2 + qi+12

2qi5ζ

i2. (23)

Here q3(t′), q5(t′) are the quadrature variances which evolve deterministically according to Eq. (3) of the main text.

We can now use the relations, qi+11 = qi1 + dt′

(qi2 +

qi32√τζi1

), and qi+1

2 = qi2 + dt′(− qi1 +

qi52√τζi2

)(together with the

observation that the stochastic averages of terms which are linear in ζ vanish, and ζ2i dt′ = 1 by Ito’s rule) in order to

compute the stochastic average of extractable work. We have,

〈W/~ω〉 =

⟨dt′

2√τ

N−1∑i=1

2qi1 + dt′(qi2 +

qi32√τζi1

)2

qi3ζi1 +

dt′

2√τ

N−1∑i=1

2qi2 + dt′(− qi1 +

qi52√τζi2

)2

qi5ζi2

⟩,

=dt′

8τ

N−1∑i=1

(qi3)2 + (qi5)2 ≈ 1

τ

∫ t

0

dt′ν(t′)2. (24)

Above we have defined ν(t′) = q3(t′)/2 = q5(t′)/2, when q3(0) = q5(0), and q4(0) = 0. It follows that work isextracted at a rate, d〈W/~ω〉/dt′ = ν2(t′)/τ. The above calculation also shows that the extracted work originatesfrom the noise due to quantum fluctuations in the simultaneous quantum measurement process, with each quadraturemeasurement channel contributing work at a rate ν2(t′)/(2τ).

Ito interpretation

The extracted work is given by, W/~ω = (q21 + q2

2)/2 = f(q1, q2). Using Ito’s lemma, we have,

df =∂f

∂t′dt′ +

∂f

∂q1dq1 +

∂f

∂q2dq2 +

1

2

∂2f

∂q21

dq21 +

1

2

∂2f

∂q22

dq22 +

∂2f

∂q1q2dq1dq2 + ... (25)

Also note that ∂f∂t′ = 0, ∂f

∂q1= q1, ∂f

∂q2= q2, ∂

2f∂q2

1= ∂2f

∂q22

= 1 and ∂2f∂q1q2

= 0. We can also use the coordinate equations

in the Ito form, given by:

dq1 = q2dt′ +

q3

2√τdW1 +

q4

2√τdW2, and dq2 = −q1dt

′ +q4

2√τdW1 +

q5

2√τdW2. (26)

Here Wi, i = 1, 2., are the Wiener increments. Substituting in Eq. (25), we get the following differential,

df = q1

(q2dt

′ +q3

2√τdW1 +

q4

2√τdW2

)+ q2

(− q1dt

′ +q4

2√τdW1 +

q5

2√τdW2

)+

1

2

[(q2dt

′ +q3

2√τdW1 +

q4

2√τdW2

)2

+

(− q1dt

′ +q4

2√τdW1 +

q5

2√τdW2

)2]+ ... (27)

We are interested in the case when q3(0) = q5(0), and q4(0) = 0 [leading to q3(t′) = q5(t′) = 2ν(t′), and q4(t′) = 0].Further we use dW2

i = dt′ and keep terms of order dt′ or less to obtain,

df = q1q3

2√τdW1 + q2

q5

2√τdW2 +

1

2

(q23

4τdt′ +

q25

4τdt′)

=ν(t′)2

τdt′ +

q1q3

2√τdW1 +

q2q5

2√τdW2. (28)

The drift terms vanish upon stochastic average, demonstrating that the average power J delivered by the engine is

J(t′) = ν(t′)2

τ . the unit of energy, ~ω, is implicit in the energy (power) definition, J.

10

FIG. 4. The efficiency of the engine in the binary feedback η′ = 〈W ′〉/Q, as a function of the average number of thermalphotons in the initial state n and r0. The dashed black line indicates the contour line of maximum η′ : η′max(n, r0) ≈ 0.85.

Supplementary Note 7: Efficiency of the Engine with continuous feedback

We define the stochastic energy of the quantum oscillator as Q/~ω = H/~ω − 1/2 = (〈p2〉 + 〈x2〉)/2 = (var(x) +var(p) + 〈x〉2 + 〈p〉2)/2− 1/2 = (q2

1 + q22)/2 + (q3 + q5)/4− 1/2 = W/~ω + (q3 + q5)/4− 1/2. We have identified the

energy stored in the displacement as extractable work, W . The subtracted half corresponds to the zero-point energyof the quantum oscillator. When τ1 = τ2 = τ , q3(t) = q5(t) = 2ν(t), and q4(t) = 0. In this case we obtain,

Q/~ω = W/~ω + 4ν(t)/4− 1/2t�τ−−−→W/~ω, (29)

since in the steady state (t� τ) we have limt�τ

ν(t) = 1/2. We thus notice that in the continuous feedback case, when

work is extracted after each measurement, the extracted work W → Q in the long time limit, demonstrating that thework conversion efficiency of the engine η = W/Q → 1. Here the energy extracted as work differs from the averageHamiltonian change for the quantum oscillator in the transient regime, which includes energy stored in the variancesof x and p that evolve according to Eq. (3). Moving the bottom of the trap—which extracts part of the averageHamiltonian change as work—preserves the variances. Extracted work becomes equal to the average Hamiltonianchange in the steady state, when the variances reach their fixed point values. The efficiency of the engine in threecases, τ2 = τ1, τ2 = 0.9τ1, and τ2 = 1.2τ1 are shown in Fig. 5. We note that unit efficiency is reached in the steadystate when τ2 = τ1.

∗ sreenath.k.manikandan@su.se[1] S. Vinjanampathy and J. Anders, Quantum thermodynamics, Contemporary Physics 57, 545 (2016), publisher: Taylor &

Francis eprint: https://doi.org/10.1080/00107514.2016.1201896.[2] J. Goold, M. Huber, A. Riera, L. d. Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics-a topical

review, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016), publisher: IOP Publishing.[3] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, eds., Thermodynamics in the Quantum Regime: Funda-

mental Aspects and New Directions, 1st ed. (Springer, 2019).[4] A. N. Jordan, C. Elouard, and A. Auffeves, Quantum measurement engines and their relevance for quantum interpretations,

Quantum Studies: Mathematics and Foundations 7, 203 (2020).[5] C. Elouard and A. N. Jordan, Efficient Quantum Measurement Engines, Physical Review Letters 120, 260601 (2018),

publisher: American Physical Society.[6] K. Jacobs, Second law of thermodynamics and quantum feedback control: Maxwell’s demon with weak measurements,

Physical Review A 80, 012322 (2009), publisher: American Physical Society.

11

2 4 6 8 10 12ωt

0.2

0.4

0.6

0.8

1.0

η

τ2=τ1

τ2=0.9τ1

τ2=1.2τ1

FIG. 5. The efficiency of the engine in three cases, τ2 = τ1, τ2 = 0.9τ1, and τ2 = 1.2τ1, averaged over 104 quantum trajectories.We note that unit efficiency is reached in the steady state when τ2 = τ1.

[7] X. Ding, J. Yi, Y. W. Kim, and P. Talkner, Measurement-driven single temperature engine, Physical Review E 98, 042122(2018), publisher: American Physical Society.

[8] A. Solfanelli, L. Buffoni, A. Cuccoli, and M. Campisi, Maximal energy extraction via quantum measurement, Journal ofStatistical Mechanics: Theory and Experiment 2019, 094003 (2019), publisher: IOP Publishing.

[9] C. Elouard, M. Waegell, B. Huard, and A. N. Jordan, An Interaction-Free Quantum Measurement-Driven Engine, Foun-dations of Physics 50, 1294 (2020).

[10] A. Das and S. Ghosh, Measurement Based Quantum Heat Engine with Coupled Working Medium, Entropy 21, 1131(2019), number: 11 Publisher: Multidisciplinary Digital Publishing Institute.

[11] T. Debarba, G. Manzano, Y. Guryanova, M. Huber, and N. Friis, Work estimation and work fluctuations in the presenceof non-ideal measurements, New Journal of Physics 21, 113002 (2019), publisher: IOP Publishing.

[12] J. Yi, P. Talkner, and Y. W. Kim, Single-temperature quantum engine without feedback control, Physical Review E 96,022108 (2017), publisher: American Physical Society.

[13] L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, and M. Campisi, Quantum Measurement Cooling, Physical ReviewLetters 122, 070603 (2019), publisher: American Physical Society.

[14] C. Elouard, D. Herrera-Marti, B. Huard, and A. Auffeves, Extracting Work from Quantum Measurement in Maxwell’sDemon Engines, Physical Review Letters 118, 260603 (2017), publisher: American Physical Society.

[15] S. Seah, S. Nimmrichter, and V. Scarani, Maxwell’s Lesser Demon: A Quantum Engine Driven by Pointer Measurements,Physical Review Letters 124, 100603 (2020), publisher: American Physical Society.

[16] L. Bresque, P. A. Camati, S. Rogers, K. Murch, A. N. Jordan, and A. Auffeves, Two-Qubit Engine Fueled by Entanglementand Local Measurements, Physical Review Letters 126, 120605 (2021), publisher: American Physical Society.

[17] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, Stabilizing Rabioscillations in a superconducting qubit using quantum feedback, Nature 490, 77 (2012), number: 7418 Publisher: NaturePublishing Group.

[18] P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson, P. Morfin, M. Mirrahimi, M. H. Devoret, F. Mallet, and B. Huard,Persistent Control of a Superconducting Qubit by Stroboscopic Measurement Feedback, Physical Review X 3, 021008(2013), publisher: American Physical Society.

[19] D. Riste, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Y. M. Blanter, K. W. Lehnert, R. N. Schouten, andL. DiCarlo, Deterministic entanglement of superconducting qubits by parity measurement and feedback, Nature 502, 350(2013), number: 7471 Publisher: Nature Publishing Group.

[20] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune,J.-M. Raimond, and S. Haroche, Real-time quantum feedback prepares and stabilizes photon number states, Nature 477,73 (2011), number: 7362 Publisher: Nature Publishing Group.

[21] X. Zhou, I. Dotsenko, B. Peaudecerf, T. Rybarczyk, C. Sayrin, S. Gleyzes, J. M. Raimond, M. Brune, and S. Haroche,Field Locked to a Fock State by Quantum Feedback with Single Photon Corrections, Physical Review Letters 108, 243602(2012), publisher: American Physical Society.

[22] D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya, and S. Habib, Quantum Feedback Control of Atomic Motion in anOptical Cavity, Physical Review Letters 92, 223004 (2004), publisher: American Physical Society.

[23] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Reviews of Modern Physics75, 281 (2003), publisher: American Physical Society.

[24] P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt, ExperimentalRepetitive Quantum Error Correction, Science 332, 1059 (2011), publisher: American Association for the Advancementof Science Section: Report.

12

[25] P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, FeedbackCooling of a Single Trapped Ion, Physical Review Letters 96, 043003 (2006), publisher: American Physical Society.

[26] J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, Subkelvin Parametric Feedback Cooling of a Laser-Trapped Nanopar-ticle, Physical Review Letters 109, 103603 (2012), publisher: American Physical Society.

[27] B. D’Urso, B. Odom, and G. Gabrielse, Feedback Cooling of a One-Electron Oscillator, Physical Review Letters 90, 043001(2003), publisher: American Physical Society.

[28] M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, Measurement-based quantum control of mechanical motion,Nature 563, 53 (2018), number: 7729 Publisher: Nature Publishing Group.

[29] V. Sudhir, D. Wilson, R. Schilling, H. Schutz, S. Fedorov, A. Ghadimi, A. Nunnenkamp, and T. Kippenberg, Appearanceand Disappearance of Quantum Correlations in Measurement-Based Feedback Control of a Mechanical Oscillator, PhysicalReview X 7, 011001 (2017), publisher: American Physical Society.

[30] C. Elouard, D. A. Herrera-MartA, M. Clusel, and A. Auffeves, The role of quantum measurement in stochastic thermody-namics, npj Quantum Information 3, 1 (2017), number: 1 Publisher: Nature Publishing Group.

[31] T. Opatrny, A. Misra, and G. Kurizki, Work Generation from Thermal Noise by Quantum Phase-Sensitive Observation,arXiv:2106.09653 [quant-ph] (2021), arXiv: 2106.09653.

[32] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura,Information-to-work conversion by Maxwell’s demon in a superconducting circuit quantum electrodynamical system, NatureCommunications 9, 1291 (2018), number: 1 Publisher: Nature Publishing Group.

[33] H. S. Leff and A. F. Rex, Maxwell’s Demon: Entropy, Information, Computing (Princeton University Press, 2014) google-Books-ID: GiIABAAAQBAJ.

[34] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffeves, R. Azouit, P. Rouchon, andB. Huard, Observing a quantum Maxwell demon at work, Proceedings of the National Academy of Sciences 114, 7561(2017), publisher: National Academy of Sciences Section: Physical Sciences.

[35] M. Naghiloo, J. Alonso, A. Romito, E. Lutz, and K. Murch, Information Gain and Loss for a Quantum Maxwell’s Demon,Physical Review Letters 121, 030604 (2018), publisher: American Physical Society.

[36] P. A. Camati, J. P. Peterson, T. B. Batalhao, K. Micadei, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra,Experimental Rectification of Entropy Production by Maxwell’s Demon in a Quantum System, Physical Review Letters117, 240502 (2016), publisher: American Physical Society.

[37] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, Experimental realization of a Szilard engine with a single electron,Proceedings of the National Academy of Sciences 111, 13786 (2014), publisher: National Academy of Sciences Section:Physical Sciences.

[38] J. Koski, A. Kutvonen, I. Khaymovich, T. Ala-Nissila, and J. Pekola, On-Chip Maxwell’s Demon as an Information-PoweredRefrigerator, Physical Review Letters 115, 260602 (2015), publisher: American Physical Society.

[39] J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley,R. Uzdin, and E. Poem, Experimental Demonstration of Quantum Effects in the Operation of Microscopic Heat Engines,Physical Review Letters 122, 110601 (2019), publisher: American Physical Society.

[40] Y. Ashida, K. Funo, Y. Murashita, and M. Ueda, General achievable bound of extractable work under feedback control,Physical Review E 90, 052125 (2014), publisher: American Physical Society.

[41] G. Paneru, D. Y. Lee, T. Tlusty, and H. K. Pak, Lossless Brownian Information Engine, Physical Review Letters 120,020601 (2018), publisher: American Physical Society.

[42] E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams,Physical Review Letters 10, 277 (1963), publisher: American Physical Society.

[43] R. J. Glauber, Coherent and Incoherent States of the Radiation Field, Physical Review 131, 2766 (1963), publisher:American Physical Society.

[44] E. Arthurs and J. L. Kelly, On the Simultaneous Measurement of a Pair of Conjugate Observables, Bell System TechnicalJournal 44, 725 (1965), eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.1538-7305.1965.tb01684.x.

[45] T. Karmakar, P. Lewalle, and A. N. Jordan, Stochastic Path Integral Analysis of the Continuously Monitored QuantumHarmonic Oscillator, arXiv:2103.06111 [quant-ph] (2021), arXiv: 2103.06111.

[46] L. Szilard, uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen,Zeitschrift fur Physik 53, 840 (1929).

[47] C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, Quantum limits on phase-preserving linear amplifiers, Physical ReviewA 86, 063802 (2012), publisher: American Physical Society.

[48] K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory (Springer Berlin Heidelberg, 1983)google-Books-ID: fRBBAQAAIAAJ.

[49] M. A. Ochoa, W. Belzig, and A. Nitzan, Simultaneous weak measurement of non-commuting observables: a generalizedArthurs-Kelly protocol, Scientific Reports 8, 15781 (2018), number: 1 Publisher: Nature Publishing Group.

[50] K. Husimi, Some Formal Properties of the Density Matrix, Proceedings of the Physico-Mathematical Society of Japan. 3rdSeries 22, 264 (1940).

[51] R. Loudon and P. L. Knight, Squeezed Light, Journal of Modern Optics 34, 709 (1987), publisher: Taylor & Franciseprint: https://doi.org/10.1080/09500348714550721.

[52] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement,and amplification, Reviews of Modern Physics 82, 1155 (2010), publisher: American Physical Society.

13

[53] C. M. Caves, Quantum limits on noise in linear amplifiers, Physical Review D 26, 1817 (1982), publisher: AmericanPhysical Society.

[54] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M.Girvin, and M. H. Devoret, Phase-preserving amplification near the quantum limit with a Josephson ring modulator,Nature 465, 64 (2010), number: 7294 Publisher: Nature Publishing Group.

[55] H. A. Haus and J. A. Mullen, Quantum Noise in Linear Amplifiers, Physical Review 128, 2407 (1962), publisher: AmericanPhysical Society.

[56] E. P. Wigner, Die Messung quantenmechanischer Operatoren, in Philosophical Reflections and Syntheses, The CollectedWorks of Eugene Paul Wigner, edited by J. Mehra (Springer, Berlin, Heidelberg, 1995) pp. 147–154.

[57] H. Araki and M. M. Yanase, Measurement of Quantum Mechanical Operators, Physical Review 120, 622 (1960), publisher:American Physical Society.

[58] M. M. Yanase, Optimal Measuring Apparatus, Physical Review 123, 666 (1961), publisher: American Physical Society.[59] M. Ahmadi, D. Jennings, and T. Rudolph, The Wigner-Araki-Yanase theorem and the quantum resource theory of asym-

metry, New Journal of Physics 15, 013057 (2013), publisher: IOP Publishing.[60] R. Landauer, Irreversibility and Heat Generation in the Computing Process, IBM Journal of Research and Development

5, 183 (1961), conference Name: IBM Journal of Research and Development.[61] R. Simon, N. Mukunda, and B. Dutta, Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and

normal forms, Physical Review A 49, 1567 (1994), publisher: American Physical Society.[62] A. Chantasri, J. Dressel, and A. N. Jordan, Action principle for continuous quantum measurement, Physical Review A 88,

042110 (2013), publisher: American Physical Society.[63] A. Chantasri and A. N. Jordan, Stochastic path-integral formalism for continuous quantum measurement, Physical Review

A 92, 032125 (2015), publisher: American Physical Society.[64] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan, K. W. Murch, and I. Siddiqi, Mapping the optimal route be-

tween two quantum states, Nature 511, 570 (2014), bandiera abtest: a Cg type: Nature Research Journals Number:7511 Primary atype: Research Publisher: Nature Publishing Group Subject term: Atomic and molecular interactionswith photons;Quantum mechanics;Quantum metrology;Qubits Subject term id: atomic-and-molecular-interactions-with-photons;quantum-mechanics;quantum-metrology;qubits.

[65] A. N. Korotkov, Continuous quantum measurement of a double dot, Physical Review B 60, 5737 (1999), publisher:American Physical Society.

[66] S. K. Manikandan, C. Elouard, and A. N. Jordan, Fluctuation theorems for continuous quantum measurements andabsolute irreversibility, Physical Review A 99, 022117 (2019), publisher: American Physical Society.

[67] M. Jayaseelan, S. K. Manikandan, A. N. Jordan, and N. P. Bigelow, Quantum measurement arrow of time and fluctuationrelations for measuring spin of ultracold atoms, Nature Communications 12, 1847 (2021), number: 1 Publisher: NaturePublishing Group.

[68] C. L. Mehta, Diagonal Coherent-State Representation of Quantum Operators, Physical Review Letters 18, 752 (1967),publisher: American Physical Society.

[69] W. P. Schleich, Quantum Optics in Phase Space (John Wiley & Sons, 2011).[70] K. Maruyama, F. Nori, and V. Vedral, Colloquium: The physics of Maxwell’s demon and information, Reviews of Modern

Physics 81, 1 (2009), publisher: American Physical Society.[71] T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012) google-Books-ID:

VWq5GG6ycxMC.