Non-thermal quantum channels as a thermodynamical resource

0.16 MB·English

by  Miguel Navascués

Non-thermal quantum channels as a thermodynamical resource
Miguel Navascu´es1 and Luis Pedro Garc´ıa-Pintos2
1Department of Physics, Bilkent University, Ankara 06800, Turkey
2School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K.
Quantumthermodynamicscanbeunderstoodasaresourcetheory,wherebythermalstatesarefree
andtheonlyallowedoperationsareunitarytransformationscommutingwiththetotalHamiltonian
of the system. Previous literature on the subject has just focused on transformations between
differentstateresources, overlookingthefact thatquantumoperationswhichdonotcommutewith
the total energy also constitute a potentially valuable resource. In this Letter, given a number of
non-thermalquantumchannels,westudytheproblemofhowtointegratetheminathermalengine
5 soas todistill amaximum amount of work. Wefindthat,in thelimit of asymptotically manyuses
1 of each channel, thedistillable work is an additivefunction of theconsidered channels, computable
0 for both finite dimensional quantum operations and bosonic channels. We apply our results to
2 bound the amount of distillable work due to the natural non-thermal processes postulated in the
Ghirardi-Rimini-Weber (GRW) collapse model. We find that, although GRW theory predicts the
n
possibility toextract work from thevacuumat nocost, thepower which acollapse engine could in
u
principle generate is extremely low.
J

5
1
Thefieldofquantumthermodynamicshasseenasurge tions. For instance, any unitary interaction of a system

ininterestinthepastyears,withincreasingattentionto- with an ancilla (or environment) generates a quantum
]
h wardstestingthevalidityoftherulesofclassicalthermo- channel, given by
p
dynamics in the quantum regime. A major topic within
-
t thermodynamicsisthatofextractingworkoutofagiven Ω(ρ)=TrA Vρ⊗σAV† , (1)
n
system and the optimal way to perform this. One way (cid:2) (cid:3)
a
u this has been approached in the quantum case was by whereσAisthestateoftheancilla,andV issomeunitary
q considering it from the perspective of a resource theory. operator In fact, it can be shown that any channel can
[ be generated via the above procedure [5].
The idea of a resourcetheory of thermodynamics is to

If, in the above expression, the ancilla is in a Gibbs
3 assume one has unlimited access to thermal baths (i.e.
stateoftemperatureT andtheunitaryV commuteswith
v Gibbs states of a fixed temperature T), the freedom to
7 apply any energy conserving unitary on system plus the the total Hamiltonian HT = HS ⊗ A + S ⊗ HA of
1 1
9 the target-ancilla system, the resulting map is called a
bath(anyunitaryV thatcommuteswiththetotalHamil-
5 thermal channel. Thesewillbethefreeoperationsinour
tonian), andthe possibility ofdiscarding partofthe sys-
2
theory, while any non-thermal map Ω will be considered
0 tem or bath (i.e. apply partial traces). These rules are
as a resource.
. imported from classical thermodynamics, where one as-
1
In this scenario, we define quantum work W as the
0 sumes access to infinite baths of constant temperature
process of exciting a two-level system with Hamiltonian
5 and any evolution where energy is conserved.
1 H =W|1ih1|fromitsgroundstate|0itotheexcitedstate
As a matter of fact, resource theories have been very
: |1i [6]. Different authors have explored how much work
v usefulindifferenttopicswithinquantuminformationthe-
i ory [1–4]. The idea is similar to above: considering one can extract from a non-thermal quantum state [6–
X
12]. When we regard the maximum average work as a
free accessto certainoperations and/orstates, any state
r figure of merit, a quantum generalizationof the classical
a and/oroperationthatisnotintheabovesetcaninprin-
free energy naturally emerges [6, 7, 12]:
ciple be used as a resource.
This work complements previous researchin quantum
F(ρ)=U(ρ)−K TS(ρ). (2)
B
thermodynamics by accommodating the possibility of
consideringnon-thermalmaps,orchannels,asaresource. Here ρ is the state of the system from which we wish
Physicaloperations are representedby quantum chan- to extract work; U(ρ)=Tr[ρH], its average energy; and
nels, i.e., completely positive trace preserving maps Ω : S(ρ) = −Tr[ρlogρ], its von Neumann entropy. K is
B
B(H) → B(H′) acting on a state space B(H). For sim- Boltzmann’s constant. The maximum amount of work
plicity, we will assume that the input and output spaces one can extract (on average) from the state ρ can then
(and, as we will see later, Hamiltonians) of each channel be shown to be F(ρ)−F(τ ), where τ represents the
th th
are the same, although the results can be easily general- thermal state at temperature T.
ized. In this Letter we want to address the following re-
Unitaryevolutionisaparticularinstanceofaquantum lated problem: suppose we want to build a thermal en-
channel,determinedbytheevolutionoperator. However, gine,whereweareallowedtointegrateanumberofnon-
quantum channels allow to express more general evolu- thermal gates {Ω }N , each of which is assumed to act
i i=1
2
on a system with Hamiltonian H . More specifically, our Note, though, that, unless the catalysts are diago-
i
machine can make free use of any amount of thermal nal in the energy basis, an extra amount of coherence,
states andoperations,andwe caninvokeone use ofeach sub-linear in n, may be needed to rebuild them (see
ofthechannels{Ω }N ,inanyorderwewantatanystep. Appendix E of [7]). More specifically, for each energy
i i=1
We are also allowed to use catalysts, i.e., we can use any transition E → E in the Hamiltonian H , the proto-
s t i
numberofnon-thermalstates,aslongaswereturnthem col proposed in [7] requires a system with Hamiltonian
intheend. Undertheseconditions,whatisthemaximum Hs,t = O(m)(E −E )k|kihk|instate 1 m |ki,with
i k=0 s t √m k=0
amount of work that our device can extract?
msublinPearinthenumbernofusesofeachPchannel. Like
There are two ways to approach this problem:
thecatalyststates,attheendoftheprotocolsuch‘coher-
ent states’ will be approximately rebuilt with vanishing
1. We can restrict to thermodynamical processes
error.
whichdistillworkdeterministically,i.e.,alwaysthe
Inordertoprovetheaboveresult,andsomelaterones,
sameamount. Thecorrespondingdeterministic ex-
the next lemma will be invoked extensively:
tractable work can then be shown to behave very
badly: not only is it not additive, but it can be Lemma 1. Let σ(N) be an N-partite quantumstate, and
super-activated. That is, there exist channels Ω let{Ω }N beacollectionofN single-sitequantumchan-
i i=1
such that no work can be distilled from a single nels. Defining Ω ≡ N Ω , we have that
1...N i=1 i
use, but two uses of the channel can be combined
N
toproduceanon-zeroamountofdeterministicwork
(see the Supplemental Information). N
S(σ )−S(Ω (σ ))≥S(σ(N))−S(Ω (σ(N))). (5)
i i i 1...N
2. Alternatively, we can consider thermodynamical Xi=1
processes which generate a given amount of work
Theproofisastraightforwardapplicationofthecontrac-
with high probability. Here the figure of merit
tivity of the relative entropy [14].
would be the maximum amount of work that can
An almostimmediate consequence of Lemma 1 is that
be distilled asymptotically (on average) when we
W(Ω,H),asdefinedbyeq.(3),hastheremarkableprop-
have access to n uses of each channel.
ertyofbeingadditive. Thatis,ifthe bipartitesystem12
We will follow the second approach: in the next pages isdescribedbytheHamiltonianH12 =H1⊗ 2+ 1⊗H2,
1 1
we will show that the asymptotically extractable work is andthechannelsΩ1 andΩ2 actontherespectiveHilbert
upper bounded by Ni=1W(Ωi,Hi), where spaces H1,H2, then, W(Ω1 ⊗Ω2,H12) = W(Ω1,H1)+
W(Ω ,H ).
P 2 2
Indeed, let Ω ≡ Ω ⊗Ω act on the bipartite state
12 1 2
W(Ω,H)≡max∆F(ρ,Ω), (3) ρ . By choosing ρ = ρ ⊗ρ in (3) we trivially have
12 12 1 2
ρ
that W(Ω ⊗Ω ,H )≥W(Ω ,H )+W(Ω ,H ), since
1 2 12 1 1 2 2
with ∆F(ρ,Ω) denoting the free energy difference be- maximizing over states in 12 is more general than maxi-
tween the states Ω(ρ) and ρ, i.e. mizing over 1 and 2 independently. Let us then focus on
the opposite inequality. By Lemma 1, we have
∆F(ρ,Ω)≡Tr[(Ω(ρ)−ρ)H]−K T[S(Ω(ρ))−S(ρ)].
B
(4) S(ρ )−S(Ω (ρ ))≥S(ρ )−S(Ω (ρ )). (6)
i i i 12 12 12
The quantity W(Ω,H) will be called the distillable work iX=1,2
of channel Ω. From the inequality ∆F(τ ,Ω) ≥ 0, it
th
follows that W(Ω,H)≥0 for any Ω. Substituting into (3) gives
The bound N W(Ω ,H ) can be achieved asymptot-
i=1 i i
∆F(ρ ,Ω )≥∆F(ρ ,Ω ) ∀ ρ , (7)
ically via aPsimple protocol where we prepare suitable i i 12 12 12
initial states σ (the catalysts) maximizing eq. (3) for iX=1,2
cat
eachchannel,andthenleteachchannelactoveritscorre-
It follows that W(Ω ,H )≥W(Ω ,H ).
spondingmaximizer. Theresultofthisprotocolwillbea i=1,2 i i 12 12
state with free energy F(σ )+ N W(Ω ,H ). Given We are nowPready to prove that W(Ω,H) quantifies
cat i=1 i i the maximum (average)amount of work one can extract
access to n uses of each channel,Pwe can thus prepare n
from channel Ω.
copies of the latter state, whose free energy can be con-
vertedtoworkviathermaloperationsusing the protocol Proposition 1. Let {Ω }N be a set of quantum chan-
i i=1
depicted in [7]. Following [7], part of this work (roughly nels, defined over different quantum systems with Hamil-
nF(σcat)) can then be used to regenerate the catalysts tonians {Hi}Ni=1. Suppose that we integrate n uses of all
uptoasmallerror[13]. Theaverageworkextractedwith such channels in a thermal engine T that produces a net
n
this procedure (namely, the total work divided by n) is amountofworkW withprobability 1−ǫ . Letusfurther
n n
thus given by N W(Ω ,H ). assumethattheprobability offailurevanishes inthelimit
i=1 i i
P
3
of large n, i.e., lim ǫ =0. Under these conditions, This result allows to quantify the work extraction ca-
n n
the average asymptot→ic∞work W¯ ≡limsupWn satisfies pabilities of different channels. One can check, for in-
n
n stance, that no work can be distilled from a dephasing
→∞
channel. Meanwhile, for a two-level system with Hamil-
N
W¯ ≤ W(Ω ,H ). (8) tonian H = E|1ih1|, E > 0, the channel that takes any
i i
Xi=1 state to the excited state |1i provides the highest distill-
able work.
As indicated above, this bound is achievable with the
use of catalysts and a sublinear amount of quantum co- Gaussian channels
herence.
If our target system is infinite dimensional, in princi-
Proof. In any protocol for work extraction, the initial pletheremayexistquantumstatespossessinganinfinite
state of the system will be given by the catalysts σcat, amount of energy. If we regard such states as unphys-
a number of thermal states τth, and the work system ical, we should replace the maximization in eq. (3) by
in state |0iw. The initial state of the system is hence an optimization over all states of finite energy. The re-
ρ ≡σ ⊗τ ⊗|0ih0| ,withfreeenergyF(σ )+F(τ ). sulting quantity will hence bound the maximum amount
0 cat th w cat th
Suppose that now we apply a sequence of energy- ofwork generatedin physically conceivable quantumen-
conserving unitaries. At time t, the state of the overall gines, where the overallstate of the system alwayshas a
systemis ρ , and we apply the channelΩ overpartof finite amount of energy.
t s(t)
the whole system, possibly followed by some other ther- In infinite dimensional systems Gaussian quantum
mal operation. Let us analyze how the free energy of ρ channelshaveaspecialrelevance: theyareeasytoimple-
t
can increase in the above step. Calling H the Hamilto- ment in the lab, and are extensively used to model par-
T
nianofthewholesystem,fromthedefinitionofW(Ω,H) ticle interactions with a macroscopicenvironment. They
and the additivity of the distillable work we have that: are defined as channels which, when composed with the
identity map, transform Gaussian states into Gaussian
∆F(Ωs(t)⊗ ,ρt)≤W(Ωs(t)⊗ ,HT)=W(Ωs(t),Hs(t)). states, the latter being those states with a Gaussian
1 1
(9) Wignerfunction[15]. Anm-modeGaussianstateiscom-
Now, any intermediate energy-conserving unitary in- pletely defined via its displacement vector d =hR i and
i i
between the use of any two of the channels {Ω }N will covariance matrix γ = h{R −d ,R −d } i, where
i i=1 ij i i j j +
1 1
keepthefreeenergyoftheoverallsystemconstant. Call- (R ,R ,...,R ) ≡ (Q ,P ,...,Q ,P ) are the optical
1 2 2m 1 1 m m
ing ρ¯the state of the system at the end of the protocol, quadratures. The action of a Gaussian channel is fully
we hence have that specified by its action over the displacement vector and
covariance matrix, given by:
N
F(ρ¯)≤n W(Ω ,H )+F(σ )+F(τ ). (10)
i i cat th d→Xd+z,γ →XγXT +Y, (11)
Xi=1
From the subadditivity of the von Neumann entropy, it whereY +iσ−iXTσX ≥0. Hereσ denotesthesymplec-
follows that F(ρ¯) ≥ F(ρ¯cat) + F(ρ¯th) + F(ρ¯w), where tic form σ = ⊕m 0 1 . If the Hamiltonian of the
σ¯ ,ρ¯ ,ρ¯ are, respectively, the reduced density matri- i=1(cid:18)−1 0 (cid:19)
cat th w
ces of the catalyst, thermal and work systems. systemunderstudyhappenstobeaquadraticfunctionof
theopticalquadratures,i.e.,H =R~TGR~+~h·R~,forsome
At the end of the protocol, the catalyst must be re-
realsymmetricmatrixGandrealvector~h,thentheaver-
generated, i.e., σ¯ = σ . Also, F(ρ¯ ) ≥ F(τ ). It
cat cat th th age energy of a state with displacement vector d~and co-
followsthatthefreeenergyoftheworksystemisbounded
by n Ni=1W(Ωi,Hi). variancematrixγ isgivenbyE = 21Tr[Gγ]+d~TGd~+~h·d~.
Stateswithfinite energyhencecorrespondtostateswith
ThPis system is expected to end up in state |1i with
finite first and second moments.
probability 1 − ǫ , i.e., ρ¯ = (1 − ǫ )|1ih1| + ǫ σ¯. It
n w n n When the quadratic Hamiltonian has no zero energy
follows that F(ρ¯ ) ≥ (1 − ǫ )W − K Th(ǫ ), with
w n n B n modes (that is, when G > 0), Proposition 1 allows to
h(p) = −pln(p)−(1−p)ln(1−p). In the asymptotic
easily classify generic Gaussian channels according to
limit,withn→∞,ǫ →0,theaverageasymptoticwork
n their capacity to generate an infinite amount of work.
limsupWn is hence bounded by N W(Ω ,H ).
n i=1 i i Indeed, for XTGX − G 6≤ 0, the channel’s distillable
n
→No∞te that this bound also hoPlds if the catalysts are work is unbounded: this can be seen by inputting a se-
recoveredup to an error, as long as F(σ )−F(σ¯ )≤ quence of Gaussian states with constant covariance ma-
cat cat
o(n). trix but increasing displacement vector parallel to any
positive eigenvector of XTGX −G. Conversely, as we
4
show in the Supplemental Information, for channels sat- with ρ the solution of eq. (13) for the initial state ρ .
t 0
isfying XTGX−G<0 only a finite amountof workcan We suppose that the particle under consideration is
be distilled. subject to a harmonic potential, i.e., H = mω2X2 +
2
For such channels there is still the dilemma of how 1 P2, and that, despite the GRW dynamics, the bath’s
2m
much work can be extracted. The next Proposition temperature T is constant. A physical justification for
greatlysimplifiesthisproblembyshowingthat,forGaus- this last assumption is that the temperature-increasing
sianchannelsΩ,themaximizationin(3)canberestricted GRW dynamics is countered by radiation from the bath
to Gaussian states: into outer space. Hence, as a function of time, the tem-
perature will converge to a stationary value T = T
Proposition 2. Consider a continuous variable quan- eq
above the temperature of the cosmic microwave back-
tum system of m modes, with quadratic Hamiltonian H,
ground (CMB) [21]. Finally, we suppose that our set of
let Ω be a Gaussian channel mapping m modes to m
resource operations remains the same: that is, in spite
modes, and denote by G the set of all m-mode Gaussian
of the modified Schr¨odinger equation (13), we can still
states. Then,
switchonandoffany unitary interactionthat commutes
W(Ω,H)=max∆F(ρ,Ω). (12) withthetotalenergyofthesystem. NoticethattheGRW
ρ∈G dynamics can be modelled by an open system approach,
The proposition can be proven by combining Lemma 1 wheretheparticleisinteractingwithsomeunknownsys-
with the ‘gaussification’ protocol described in [16], see tem such that the resulting evolution is given by (13).
the Supplemental Information. From this viewpoint, we are simply assuming that we
Since W(Ω,H) just involves an optimization over a still have the capacity to interact with the system in the
finite set of parameters subject to positive semidefinite usual way.
constraints,(inprinciple)itcanbe computedexactlyfor In these conditions, we wish to find the maximum
any Gaussian channel Ω. amount of work that a collapse engine could extract if
it had access to the evolution equation (13) for a finite
One application: collapse engines amountoftime t. FromProposition(1), this amountsto
computing lim t W(Ω ,H).
Inordertoaddressthemeasurementproblem[17],and, δt→0 δt δt
First,noticethatwecan(reversibly)evolvethesystem
independently, the decoherence effects that a quantum
with the Hamiltonians H or −H, since this corresponds
theory of gravitycould impose on the wave-function[18,
to a thermal operation. This implies that we can ignore
19],differentauthorshaveproposedthatclosed quantum
the first term in the right hand side of (13), and what
systemsshouldevolveaccordingtotheLindbladequation
remains is a Gaussian channel given by
0 0
d i Λ
dtρt =−~[H,ρt]− 4[X,[X,ρt]], (13) d→d,γ →γ+(cid:18)0 ~2Λ2δt (cid:19). (15)
where X is the position operator for the particle consid- It follows that the energy of any input state will in-
ered. crease by ∆U ≡ ~2Λδt. From Proposition 2, we can
4m
The effect of the non-unitary term is a suppression
estimate the entropy increase by just considering Gaus-
of coherences in the position basis, effectively destroying sianstates. Now,theentropyofa1-modeGaussianstate
quantum superpositions. The value of the constant Λ,
isanincreasingfunctionofthe determinantofits covari-
which can be interpreted as the rate at which this local-
ance matrix[15], which, by the aboveequation,canonly
ization process occurs, depends on the particular theory increase with time. Hence, W(Ω ,H)≤∆U.
δt
invoked to justify eq. (13). In the Ghirardi-Rhimini-
On the other hand, suppose we input a squeezedstate
Weber(GRW)theory[17],thelocalizationprocessispos- with γ = diag(1/r,~2r). Then the determinants of the
tulated to solve the measurement problem in quantum covariance matrices of initial and final states will be ~2
mechanics. Toachievethisgoalandavoidcontradictions and ~2+ ~2Λδt, respectively. The entropy change of the
2r
with past experimental results, Λ must be roughly be- state can thus be made as small as desired by increasing
tween 10 2s 1m 2 and 106s 1m 2, according to latest
− − − − − the value ofr, andso the aboveboundcanbe saturated,
estimations [20].
leading to W(Ω ,H) = ∆U. Consequently, the maxi-
δt
Note that the above evolution is non-thermal. Hence, mum poweratwhich a collapse engine couldin principle
it could be used in principle to extract work from noth-
operate is given by
ingness by means of a suitable thermal engine. We will
call such a hypothetical device a collapse engine. dW ~2Λ
= . (16)
Toconnectthistoourprevioussetting,noticethatthe dt 4m
evolution equation (13) defines a quantum channel
Using the upper range estimation Λ ∼ 106s 1m 2, we
− −
Ω (ρ )=ρ , (14) have that a collapse engine powered by a single electron
δt t t+δt
5
wouldproduce dW ∼10 32watt. Assumingtotalcontrol At the end of the protocol, the work system will have
dt −
over the electrons of a macroscopic sample, one would evolved to |1i.
needakilotonofHydrogentopowera40wattlightbulb. If, rather than implementing step 3, we add the state
|ψi in the preparation stage, then it is trivial to find an
Conclusion energy-conservingunitary U˜ thatat the laststep would
2
produce exactly the same amount of work. That is, we
In this Letter we addressed the problem of determin- would have extracted work from the resource state |ψi.
ing how much work can be extracted from operational In [6], however, it is shown that no work can be ex-
-as opposed to state- resources. We proved that the so-
tracted from a single copy of |ψi, even with the use of
lution to this problem in the asymptotic limit is given
catalysts. This implies that the deterministic distillable
by a single-letter formula that quantifies the amount of
work of a single use of Ω is zero.
distillableworkthatachannelcan,inprinciple,generate
Suppose, now, that we have access to two uses of Ω.
whensupplementedwiththermaloperationsandcatalyst
Thenwecanpreparetwocopiesof|ψi,fromwhichanon-
states. Moreover,we found how this quantity can be de-
zero amount of work can be deterministically extracted
termined for bosonic channels, and computed it exactly
via thermal operations [6].
for the case of the GRW dynamics, hence determining
the maximum power which a hypothetical collapse en-
gine could provide for free. Gaussian channels with finite distillable work
Note that we have only studied operational resources
regarding their capacity to generate work. An interest- Let Ω be a Gaussian channel whose action on the dis-
ing topic for future research is to extend our results and placement vector d~ and covariance matrix γ of the m-
drawamapoftheinter-conversionrelationsbetweendif-
mode input state is given by
ferentoperationalresources. Inthecaseofstateresources
thereisauniquemonotonicquantity,thefreeenergy,de-
terminingtheoptimalratesforstatetransformations[7]. d→Xd+z,γ →XγXT +Y. (17)
Inthisworkwehaveidentifiedanoperationalmonotone,
the distillable work, but we suspect that there may be Suppose that Ω acts on a system with Hamiltonian H =
many others. R~TGR~ +~h·R~, where R~ is the vector of optical quadra-
tures. Under the assumption that G˜ ≡G−XTGX >0,
Acknowledgements we wonder if the difference between the free energies of
We thank RalphSilva andNoahLindenfor useful dis- the input and output states is bounded, i.e., whether
cussions. ∆F(ρ,Ω)<K, for some K <∞.
Call E the energy of the input state; and γ,d~, its co-
0
variance matrix and displacement vector. Then we have
The deterministic distillable work can be that
superactivated
1
ConsiderthechannelΩthattakesanystateofatarget E0 = Tr[Gγ]+(d~−d~0)TG(d~−d~0)−E¯, (18)
2
two-level system with Hamiltonian H = E |1ih1| to the
1
state |ψi ∝ |0i+e βE1/2|1i . That is, Ω(ρ) = |ψihψ| where d~ ≡ −G 1~h/2, and E¯ ≡ d~TGd~ . Hence, defining
− 0 − 0 0
for all ρ. (cid:16) (cid:17) µmin > 0 (µmax > 0) to be the minimum (maximum)
eigenvalue of G we have that
Now, any protocol that pretends to extract work out
via a single use of Ω can be divided in three steps:
E +E¯ 1 E +E¯
1. The system is prepared in a state which comprises 0 ≤ Tr[γ]+kd~−d~0k2 ≤ 0 , (19)
µ (cid:18)2 (cid:19) µ
max min
catalysts, thermal states and the work system (in
state |0i). That is, ρ =σ ⊗τ ⊗|0ih0| . and consequently
0 cat th w
2. Weapplyanenergy-conservingunitaryU overthe
1
whole system. Tr[γ]≤O(E ),kd~k≤O( E ),
0 0
1 p
3. We apply Ω, hence replacing a subsystem’s state Tr[γ]+kd~−d~0k2 ≥O(E0). (20)
2
by |Ψi.
We can now bound the energy difference between the
4. We apply a second energy-conserving unitary U input andoutput states. First, note that ∆E ≡E −E
2 f 0
over the whole system. can be written as:
6
Proof. Let the action of Ω over the displacement vec-
tor and covariance matrix be given by eq. (17). Since
1
∆E =− Tr γG˜ −(d~−d~0)TG˜(d~−d~0)+O(d~). (21) von Neumann entropies remain the same after a unitary
2
h i
transformation,withoutlossofgeneralitywewillassume
Defining λ >0 to be the smallest eigenvalue of G˜, we that ρ’s displacement vector is null. Similarly, we will
min
thus arrive at take z = 0 in the channel description (17) of Ω. Now,
let U be the n-system ‘Gaussification’ transformation
n
described in [16]. Calling ρ˜(n) =U ρ nU , we have that
1 n ⊗ n†
∆E ≤−λ Tr[γ]+kd~−d~ )k2 +O(d~)
min 0
(cid:18)2 (cid:19)
n
≤−O(E )+O( E )=−O(E ). (22)
0 0 0 L ≡ S(ρ˜)−S(Ω(ρ˜))
n i i
Let us now bound the entroppy of the input state: by Xi=1
the subadditivity of the von Neumann entropy, S(ρ) is ≥S(ρ˜(n))−S(Ω⊗n(ρ˜(n))) (27)
boundedfromaboveby m S(ρ ),whereρ denotesthe =S(ρ n)−S(Ω n(ρ n))
i=1 i i ⊗ ⊗ ⊗
reduceddensity matrix oPf eachmode i. S(ρi), inturn, is =n{S(ρ)−S(Ω(ρ))}, (28)
bounded by the von Neumman entropy of the Gaussian
state with the same first and second moments as ρ , i.e., where the first inequality is due to Lemma 1 inthe main
i
a Gaussian state with covariance matrix γ . Note that text, and the equality follows from the fact that, for all
i
m Tr[γ ] = Tr[γ] ≤ O(E ), where the last inequality states σ,
i=1 i 0
iPs due to eq. (20). In particular, Tr[γi] ≤ O(E0) for
i=1,...,m.
The entropy of a 1-mode Gaussian state with covariance UnΩ⊗n(σ)Un† =Ω⊗n(UnσUn†). (29)
matrix γ˜ is given by
This identity follows from three observations: 1) any
Gaussian channel with z = 0 is the result of applying
(N +1)log(N +1)−Nlog(N)=O(log(N)), (23) a symplectic unitary V over the target system and an
S
ancillary Gaussian state ω with zero displacement vec-
where N = det(γ˜)/~ ≤ Tr[γ˜]/2~ (this last inequality
tor; 2) n copies of ω are invariant with respect to a
reflectsthefapctthatthegeometricmeanofγ˜’stwoeigen- Gaussification operation UnA; 3) [VS⊗n,Un′] = 0, where
values is smaller than their arithmetic mean). It follows U = U ⊗UA represents the Gaussification of n copies
n′ n n
that
of the system target-ancilla.
Since the vonNeumann entropyis continuous in trace
normwith respectto collections ofstates with finite sec-
S(ρ)≤O(log(E )). (24)
0
ond moments, by [16], we end up with
Putting all together, we have that
n
L 1
n
lim = lim S(ρ˜)−S(Ω(ρ˜))
i i
∆F(ρ,Ω)≤∆E+KBTS(ρ) n→∞ n n→∞nXi=1
≤−O(E0)+O(log(E0)). (25) =S(ρG)−S(Ω(ρG)). (30)
The last expression cannot thus take arbitrarily large The lemma hence follows from (28) and (30).
values, and so the distillable work of channel Ω is
bounded.
Now,letρbeanarbitrarystatewithfiniteenergy(and
thus finite first and second moments), and let ρ be the
G
Proof of Proposition 2 Gaussian state with the same first and second moments.
Then, ρ and ρ have the same averageenergy, andsince
G
Proposition 2 follows straightforwardly from the fol- Ω is a Gaussian channel the same is true for Ω(ρ) and
lowing Lemma: Ω(ρG). However, by the previous Lemma, the entropic
change is bigger for ρ , and so ∆F(ρ ,Ω) ≥ ∆F(ρ,Ω),
G G
Lemma 2. Let ρ be an arbitrary state with finite first
proving Proposition 2.
and second moments, and let ρ be the unique Gaussian
G
state with the same first and second moments. Then, for
any Gaussian channel Ω, we have that
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