Gibbs Free Energy Analysis of a Quantum Analog of the Classical Binary Symmetric Channel

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by  David K. Ford

Gibbs Free Energy Analysis of a Quantum Analog of the Classical Binary Symmetric
Channel
David Ford, Physics NPS∗
Department of Physics, Naval Post Graduate School, Monterey, California
(Dated: April 20, 2009)
TheGibbsfreeenergypropertiesofaquantumsend, receivecommunicationssystemarestudied.
ThecommunicationsmodelresemblestheclassicalIsingmodelofspinsonalatticeinthatthejoint
state of the quantum system is the product of sender and receiver states. However, the system
differsfromtheclassicalcaseinthatthesenderandreceiverspinstatesarequantumsuperposition
statescoupledbyaHamiltonianoperator. Abasicunderstandingofthesestatesisdirectlyrelevant
to communications theory and indirectly relevant to computation since the product states form a
9 basis (in function space) for entangled states. Highlights of the study include an exact method
0 for decimation for quantum spins. The main result is that the minimum Gibbs free energy of the
0 quantum system in the product state is higher (lower capacity) than a classical system with the
2 same parameter values. The result is both surprising and not. The channel characteristics of the
n quantum system in the product state are markedly inferior to those of the classical Ising system.
a Intuitively, it would seem that capacity should suffer as a result. Yet, one would expect entangled
J states, built from product states, to have better correlation properties.

9
1
Introduction To compute the Gibbs free energy of the system, an ex-

] actquantumdecimationoverthesenderandreceiverspin
h
sites is performed and coarse grained density operators
p Potentialapplicationsofinformationtheorytocompu-
- tation and communications include optimizations across are computed. In the case of the binary symmetric (i.e.
n coupledspinswithnoappliedfield)channelboththeclas-
complex processor/memory hierarchies, within complex
e
sical and quantum Gibbs free energy minima correspond
g biological networks, networks of quantum devices, etc.
. Often, non-stationarity or random spatial effects make to maximum single spin (the “sender’s”) entropy. How-
s
ever, it is seen that the Gibbs free energy minimum of
c analysis even more difficult. As such is the case, extend-
i thequantumsystemishigherthanthatoftheanalogous
s ingrelevantengineeringprinciples,suchastheconceptof
y channel capacity, to include well-established techniques classical system.
h
from statistical physics would be beneficial. Two points require some clarification. First, while
p
[ In a previous note [1] it was shown that the informa- by adopting this scheme the sender is in control of the
tion theoretic concept of channel capacity is a particular statistics of the sent messages, they are not in control of
1 case of the principle of minimum Gibbs free energy. An the content of any particular message. Left unchecked
v
1 analysis based on an extension of the second law of ther- this state of affairs does not directly lead to “commu-
1 modynamics is found in [2]. The system specific work nication” in the classical sense. However, through a
9 termintheGibbsformulationwasshowntobethe“work self-measurementoftheirchosensuperpositionstate,fol-
2 of communication”. That is, the work of separating the lowed by application, if necessary, of a unitary trans-
1. joint {send,receive} signal into its components {send} formation to the partially collapsed wave function, the
0 and {receive}. To make the necessary connections with sender may effectively communicate with the receiver.
9 thephysics,theanalysiswascarriedoutinthecontextof This procedure is discussed in the last section in more
0
the classical, static Ising model of magnetzation [3], [4]. detail. Secondly, while the states studied herein are not
:
v TheGibbsfreeenergypropertiesofaquantumanalogof entangled, they form a basis for the entangled states.
Xi that classical result are investigated below. A thorough understanding of the properties of the basis
statesmayleadtobettercomputationalmethodsandof-
r In communications theory, the sender of a message is
a fer alternative approaches for the solution of engineering
typicallyresponsibleforengineeringtheencodingscheme
problems.
which will efficiently and reliably transfer information to
thereceiver. Intermsofthevariationalproblem,asearch
over encoding schemes decides the statistics of the 1’s
and0’softhesentmessage. Tomirrorthis,thequantum
statesstudiedhereinareproductsofthesentandreceived Quantum mechanical background
superposition states. In this way, by choosing the sent
message superposition state, the sender has chosen the PerhapstheclosestanalogofaclassicalIsingspincon-
statisticsofthemeasured1’sand0’softhesentmessages. figuration (state) is the direct product of superposition
The fine grained Hamiltonian operator of the system spin states of spin 1 particles. The Pauli representation
2
correlates the the sent and received superposition states. of the hamiltonian for two coupled spin 1 particles in
2
2
zero applied field is given by In the chosen basis (1), the communication state admits
the representation
H=−J(σ ·σ +σ ·σ +σ ·σ ).
x x y y z z
(cid:18) (cid:19)
cosθ cosθ cosθ sinθ
s r s r . (5)
As is traditional, the basis of choice is the one that di- cosθ sinθ sinθ sinθ
r s r s
agonalizesthespinangularmomentumoperatorsS and
z
S2 of a single electron Note that the communications model lives in a subset
of all possible two spin states. It is not the most gen-
(cid:18) (cid:19) (cid:18) (cid:19)
1 0 eral possible representation. In particular, (5) is not an
=|↑(cid:105) =|↓(cid:105).
0 1 entangled state. However, entangled states may be well
approximated by sums of states of this type.
In the two electron direct product basis
The action of H on the communication state ψ is
c
{|↑↑(cid:105),|↑↓(cid:105),|↓↑(cid:105),|↓↓(cid:105)} (1)
H|ψ (cid:105) = −Jcosθ cosθ |↑↑(cid:105)−Jsinθ sinθ |↓↓(cid:105)
c s r s r
the hamiltonian takes the matrix form sin(θ +θ ) sin(θ −θ )
−J √s r |sym(cid:105)+3J √r s |anti(cid:105)
1 0 0 0 2 2
0 −1 2 0
H=−J .
0 2 −1 0 or
0 0 0 1
 3Jsin(θ −θ ) 
−Jcosθ cosθ 2 s r
Tdihaegohnaamlizilatbonleian and density matrix are simultaneously  s r −21Jsin(θs+θr) 
 .
H=(cid:88)E |i(cid:105)(cid:104)i| (2)  −32Jsin(θs−θr) −Jsinθ sinθ 
i −1Jsin(θ +θ ) s r
i 2 s r
The average energy of the system under H, a function of
(cid:88)e−βEi θs and θr, is given by
ρ= |i(cid:105)(cid:104)i| (3)
Q
i (cid:104)ψ |H|ψ (cid:105)=−Jcos(2(θ −θ )).
c c s r
whereQ=(cid:80) e−βEi. Inthechosenbasis(1),theenergy
i The density operator average, (cid:104)ψ |ρ|ψ (cid:105), also defines a
c c
eigenvalues and eigenstates, {E , |i(cid:105)}, are
i function of of θ and θ
s r
(cid:26) (cid:18) (cid:19)(cid:27) (cid:26) (cid:18) (cid:19)(cid:27)
1 0 0 0
−J, , −J, ,
0 0 0 1 P(θ ,θ )
s r = (6)
π2
(cid:40) (cid:32) (cid:33)(cid:41) (cid:40) (cid:32) (cid:33)(cid:41)
0 √1 0 √1 1 1+3eβ4J +(eβ4J −1)cos(2(θ −θ ))
−J, √1 02 , 3J, −√1 02 . π2 4(1+3eβ4J) s r .
2 2
In shorthand notation This function determines the probability density of the
communications state generated by the send, receive
H = −J|↑↑(cid:105)(cid:104)↑↑|−J|↓↓(cid:105)(cid:104)↓↓| state pairs {θ ,θ } ∈ [0,2π)2 under the influence of the
s r
−J|sym(cid:105)(cid:104)sym|+3J|anti(cid:105)(cid:104)anti|. hamiltonian H. The factor of π2 in the denominator is
thetotalstatisticalweightofψ underH. Thisprobabil-
c
Inthemodelitisassumedthatthesendertransmitsa ity density function is plotted in figure 1. The coupling
state of the form is apparent in the figure. The most likely states occur
when θ =θ . The least likely when θ and θ are phase
s r s r
ψsend =cosθs|↑(cid:105)+sinθs|↓(cid:105) (4) shifted by π.
2
Similarly the receiver will obtain
The quantum channel
ψ =cosθ |↑(cid:105)+sinθ |↓(cid:105)
rcv r r
Thejointcommunicationstateψ generatedbythesend, In practice, the communication of messages occurs via
c
receive process is a set of codewords. A choice of codeword set fixes the
statistics (the sender’s marginal distribution) of the sent
ψ =ψ ⊗ψ . signal. The received statistics may be different due to
c send rcv
3
TABLE I: Decimated Density Operator (sender)
label weight projection amplitudes operator
(u) e−β(2h−J) (cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) |↑(cid:105)(cid:104)↑|
r r
(d) e−β(−2h−J) (cid:104)ψ |↓(cid:105)(cid:104)↓|ψ (cid:105) |↓(cid:105)(cid:104)↓|
r r
(s1) eβJ 1(cid:104)ψ |↓(cid:105)(cid:104)↓|ψ (cid:105) |↑(cid:105)(cid:104)↑|
2 r r
(s2) eβJ 1(cid:104)ψ |↓(cid:105)(cid:104)↑|ψ (cid:105) |↑(cid:105)(cid:104)↓|
2 r r
(s3) eβJ 1(cid:104)ψ |↑(cid:105)(cid:104)↓|ψ (cid:105) |↓(cid:105)(cid:104)↑|
2 r r
(s4) eβJ 1(cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) |↓(cid:105)(cid:104)↓|
2 r r
FIG. 1: (cid:104)ψc|ρ|ψc(cid:105) for the hamiltonian −Jσ ·σ with zero (a1) e−β3J 1(cid:104)ψ |↓(cid:105)(cid:104)↓|ψ (cid:105) |↑(cid:105)(cid:104)↑|
applied field. 2 r r
(a2) e−β3J −1(cid:104)ψ |↓(cid:105)(cid:104)↑|ψ (cid:105) |↑(cid:105)(cid:104)↓|
2 r r
(a3) e−β3J −1(cid:104)ψ |↑(cid:105)(cid:104)↓|ψ (cid:105) |↓(cid:105)(cid:104)↑|
2 r r
(a4) e−β3J 1(cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) |↓(cid:105)(cid:104)↓|
2 r r
transmission errors. In equation (4), the sender’s statis-
tics are governed by ψ . That is, by the angle θ .
send s
For each θ , the hamiltonian of the system determines
s
the joint statistics. In other words, for each θ , the sys-
s ρ = (7)
tem hamiltonian induces a distribution on the received ↑↑
messages statistics. That is, it induces a distribution on 1 (cid:90) 2π dθr e−β(2h−J)(cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) (cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105)
θ . Q π r r s s
r 0
For example, at fixed θ , the probability of sending an
s ρ = (8)
↑↓
↑ and receiving an ↑ varies with the receiver’s θ . Over
r 1 (cid:90) 2π dθ (cid:16) (cid:17)
a sufficiently long observation period, an average prob- r e−β(−J)+e−β3J (cid:104)ψ |↓(cid:105)(cid:104)↓|ψ (cid:105)(cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105).
ability characterizing the θr variation emerges so that Q 0 π 2 r r s s
it makes sense to speak of an averaged sender’s density
operator. It is assumed that this empirical averaging is Observe that in the calculations above, the integra-
equal to the thermal average over θr. tionoverθr isuniformlyweightedovertheregion[0,2π].
However, as has been mentioned, the choice of codeword
The eigenstates described so far, for example in equa-
setfixesthestatistics(thesender’smarginaldistribution)
tions (2) and (3), correspond to operators which act on
of the sent signal. To mirror this, the θ integration is
the electron pair. These joint eigenstates may be de- s
performedwithadeltafunctionsothatonlythesender’s
composed into pieces (herein referred to as eigenpieces)
chosen value, “θ ” contributes to the sum over states.
according to their action on, say, the sender’s spin state. ∗
Theneteffectisthattherearefeweravailablestatesand
Such a decomposition facilitates thermal averaging over
Q is interpreted as
θ . Table I gathers together the information necessary
r
to analyze the thermal averages of the receiver and con-
structthesender’sdensityoperatorinthepresenceofan Q = (9)
applied field. (cid:90) 2πdθ δ(θ −θ ) (cid:90) 2π dθr (cid:88)e−βEi (cid:104)ψ |i(cid:105)(cid:104)i|ψ (cid:105).
The row corresponding to eigenpiece (u) of Table I in- 0 s s ∗ 0 π i c c
dicates how the receiver averaged statistical weight of
sending an ↑ and receiving an ↑ (as a function of θ ) is
s The probability of receiving an ↑ conditional upon an
obtained. There is a projection of ψ onto the appro-
send ↑ being sent is computed as
priateeigenpieceandanintegrationovertheψ ampli-
rcv
tudes. Thecalculationispresentedinequation(7). Sim-
ρ
ilarly, the receiver averaged statistical weight of sending P = ↑↑
↑|↑ ρ +ρ
an↑andreceivinga↓(asafunctionofθ )isobtainedby ↑↑ ↑↓
s
integrating over ψrcv amplitudes in eigenpieces (s1) and e−β(2h−J)cos2(θ∗)
(a1). That calculation is presented in equation (8). It = Q
e−β(2h−J)cos2(θ∗) + e−βJcos2(θ∗)cosh(2βJ)
should be noted that the mixed terms, eigenpieces (s2), Q Q
(s3), (a2) and (a3) make no contribution to the receiver
averaged operator. Similarly, the probability of receiving an ↓ conditional
4
upon an ↑ being sent is computed as Recall from equation (6) that the density operator aver-
age, (cid:104)ψ |ρ|ψ (cid:105), defines P, a function of θ and θ with
ρ c c s r
P = ↑↓
↓|↑ ρ +ρ
↑↑ ↑↓ (cid:90) 2π dθ (cid:18)(cid:90) 2π dθ (cid:19)
= e−βJcos2(θQ∗)cosh(2βJ) 0 πs 0 πrP(θs,θr) =1. (14)
e−β(2h−J)cos2(θ∗) + e−βJcos2(θ∗)cosh(2βJ)
Q Q Theresultofinnerintegrationdefinesacoarseneddensity
P˜(θ ), the senders marginal. One may also use the inte-
s
In this way the elements of the quantum channel
grationoverθ todefinethequantumdecimatedhamilto-
r
(cid:18) (cid:19) nian operator. The details necessary for its construction
P P
↑|↑ ↓|↑ (10) are gathered together in Table II. Recall that, to mirror
P P
↑|↓ ↓|↓ the effect of the sender being in control of the sent mes-
are determined from the thermal averages. sage statistics, the integral over θs is performed with a
delta function. See equation (9).
At zero applied field, P and P are equal to each
↑|↑ ↓|↓
other and (10) becomes a symmetric channel. Due to
the symmetry of the hamiltonian and the fact that the
Decimation of the quantum system
rows of the channel must sum to one, a single entry (say
P ) is enough to determine the matrix. Under these
↑|↑
conditions (10) becomes
TABLE II: Decimated Density Operator
 
α 1−α label weight projection amplitudes operator
  (11)
(u) e−β(2h−J) (cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) |↑(cid:105)(cid:104)↑|
1−α α r r
(s1) eβJ 1(cid:104)ψ |↓(cid:105)(cid:104)↓|ψ (cid:105) |↑(cid:105)(cid:104)↑|
2 r r
with (a1) e−β3J 21(cid:104)ψr|↓(cid:105)(cid:104)↓|ψr(cid:105) |↑(cid:105)(cid:104)↑|
2e4Jβ (d) e−β(−2h−J) (cid:104)ψr|↓(cid:105)(cid:104)↓|ψr(cid:105) |↓(cid:105)(cid:104)↓|
α= .
1+3e4Jβ (s4) eβJ 21(cid:104)ψr|↑(cid:105)(cid:104)↑|ψr(cid:105) |↓(cid:105)(cid:104)↓|
(a4) e−β3J 1(cid:104)ψ |↑(cid:105)(cid:104)↑|ψ (cid:105) |↓(cid:105)(cid:104)↓|
2 r r
Thechannelpropertiesareindependentofψ ,i.e. the
send
angle θ . This is the reminiscent of the classical case
s Themixedtermeigenpieceprojections,(s2),(a2),(s3),
where the“noise” isindependent ofthe properties ofthe
(a3),inTableIaveragetozero. Theremainingtermsare
information source. However, it differs from the classical
collectedtogetheraccordingtosingleelectroneigenstates
case in that arbitrarily increasing the coupling strength,
in Table II.
J,doesnotarbitrarilyimprovethechannelprobabilities.
Theformofthereducedhamiltonianoperatorisgiven
by
Construction of the coarse grained operators H˜ =H˜ |↑(cid:105)(cid:104)↑|+H˜ |↓(cid:105)(cid:104)↓|.
↑ ↓
ThedetailsarereadilyobtainedfromTableIIbyinspec-
The hamiltonian and density operators
tion
The Pauli representation of the hamiltonian for two H˜ = −1 log(cid:90) 2π dθr e−β(2h−J) |(cid:104)ψ |↑(cid:105)|2
coupled spin 1 particles in an applied field h is given by ↑ β π r
2 0
1 1
H=−J(σx·σx+σy·σy+σz·σz)+h(σz·σo+σo·σz). (12) +eβJ2 |(cid:104)ψr|↓(cid:105)|2 +e−β3J2 |(cid:104)ψr|↓(cid:105)|2
(cid:18) (cid:19)
In the energy basis the hamiltonian takes the form = −1 log e−β(2h−J)+ 1eβJ + 1e−β3J
β 2 2
H = (2h−J)|↑↑(cid:105)(cid:104)↑↑|+(−2h−J)|↓↓(cid:105)(cid:104)↓↓|
−J|sym(cid:105)(cid:104)sym|+3J|anti(cid:105)(cid:104)anti| 1 (cid:90) 2π dθ
H˜ = − log r e−β(−2h−J) |(cid:104)ψ |↓(cid:105)|2
↓ β π r
and the density operator 0
1 1
+eβJ |(cid:104)ψ |↑(cid:105)|2 +e−β3J |(cid:104)ψ |↑(cid:105)|2
1 2 r 2 r
ρ = (exp−β(2h−J)|↑↑(cid:105)(cid:104)↑↑|+exp−β(−2h−J)|↓↓(cid:105)(cid:104)↓↓|
Q (cid:18) (cid:19)
1 1 1
= − log e−β(−2h−J)+ eβJ + e−β3J
+exp−β(−J)|sym(cid:105)(cid:104)sym|+exp−β3J|anti(cid:105)(cid:104)anti|). (13) β 2 2
5
The form of the reduced density operator is then measurestheamountoffreeenergyinthesystemthatis
not involved in the “work of communication”, i.e. sep-
e−βH˜↑ e−βH˜↓ arating the joint state into its products [1]. The energy
ρ˜= |↑(cid:105)(cid:104)↑|+ |↓(cid:105)(cid:104)↓| (15)
Q˜ Q˜ eigenpiececontributionstotheworkcalculationaregath-
ered together in Table III
with (recall equation (9))
(cid:90) 2π
Q˜ = dθsδ(θs−θ∗) TABLE III: accumulated eigenpiece contributions
0
(cid:16) (cid:17)
× e−βH˜↑ |(cid:104)ψs|↑(cid:105)|2 +e−βH˜↓ |(cid:104)ψs|↓(cid:105)|2 label H˜ Hˆ H ρ
= (cid:18)e−β(2h−J)+ 1eβJ + 1e−β3J(cid:19)cos2θ (u) H˜↑cos2θ∗ Hˆ↑cos2θ∗ (2h−J)cos2θ∗ e−β(2h−J)
2 2 ∗
(cid:18) (cid:19) (s1) H˜↑cos2θ∗ Hˆ↓cos2θ∗ −Jcos2θ∗ eβJ
1 1 2 2 2
+ e−β(−2h−J)+ eβJ + e−β3J sin2θ
2 2 ∗ (s4) H˜↓sin2θ∗ Hˆ↑sin2θ∗ −Jsin2θ∗ eβJ
1 2 2 2
= e−β(2h−J)cos2θ + eβJ
∗ 2 (a1) H˜↑cos2θ∗ Hˆ↓cos2θ∗ 3Jcos2θ∗ e−β3J
1 2 2 2
+ e−β3J +e−β(−2h−J)sin2θ
2 ∗ (a4) H˜↓sin2θ∗ Hˆ↑sin2θ∗ 3Jsin2θ∗ e−β3J
= Q. 2 2 2
Asimilardecimationiscarriedoutoverthesender’sspin (d) H˜↓sin2θ∗ Hˆ↓sin2θ∗ (−2h−J)sin2θ∗ eβ(2h+J)
site. A delta function ensures that all the probability
mass is assigned to the sender’s choice of θ =θ . Table
s ∗ Infigure2,thesystem’sGibbsfreeenergyisshownfor
II eigenpieces (u), (s4) and (a4) contribute
a binary symmetric channel, governed by hamiltonian
1 (cid:90) 2π (12), at parameter values β = 1, J = 1 and h = 0. It is
Hˆ↑ = −β log dθsδ(θs−θ∗) interesting to note that the Gibbs free energy minimum
0 for the quantum product state is higher than that of the
×(e−β(2h−J) |(cid:104)ψ |↑(cid:105)|2
s analogous classical Ising system [1].
1 1
+eβJ |(cid:104)ψ |↓(cid:105)|2 +e−β3J |(cid:104)ψ |↓(cid:105)|2)
2 s 2 s
1
= − log(e−β(2h−J)cos2θ
β ∗
1 1
+eβJ sin2θ +e−β3J sin2θ ).
2 ∗ 2 ∗
Eigenpieces (d), (s3) and (a3) yield
1 (cid:90) 2π
Hˆ = − log dθ δ(θ −θ )
↓ β s s ∗
0
×(e−β(2h−J) |(cid:104)ψ |↓(cid:105)|2
s
1 1
+eβJ |(cid:104)ψ |↑(cid:105)|2 +e−β3J |(cid:104)ψ |↑(cid:105)|2)
2 s 2 s
1
= − log(e−β(−2h−J)sin2θ
β ∗
1 1
+eβJ cos2θ +e−β3J cos2θ ).
2 ∗ 2 ∗
The Gibbs free energy of the binary symmetric
channel FIG. 2: Upper frame: −βTr{ρ(H˜+Hˆ−H)}, the work of
separation and −log(Q), β times the Helmholtz free energy
Theinversetemperaturetimesthesystem’sGibbsfree ofthesystem. Lowerframe: theirsum,theGibbsfreeenergy
as a function of θ ∈[0,2π).
energy s
(cid:18) (cid:19)
1
β −Tr{ρ(H˜+Hˆ−H)}− log(Q) (16)
β
6
A comment on applications [3] H. Nishimori, “Statistical Physics of Spin Glasses and
Information Processing”, Oxford Science, 2001
[4] T. Tanaka, “Statistical mechanics of CDMA multiuser
Figure 1 demonstrates that for every choice of sender
demodulation”, Euro- phys. Lett., vol. 54, pp. 540546,
superposition state (the angle θ ), there are many possi-
s 2001
blereceiverstates(theangleθ )accessibletothesystem.
r
Yet, the performance of the quantum spin system under
study has better Gibbs free energy properties than the
correspondingclassicalsystem[1]. Inotherwordsthereis
atradeoff,thequantumanalogcarriesmoreinformation
per bit but the sender does not have control over which
bit is sent. One way to benefit from the better commu-
nication properties of the quantum system and exercise
a degree of control over the sent message is to collapse
thewavefunctiononthesender’ssideofthechanneland
(dependingupontheresultsofthatmeasurement)apply
a unitary transformation to the system. The operators
|↑(cid:105) |↑(cid:105)
1 =
|↓(cid:105) |↓(cid:105)
corresponding to a phase shift θ →θ and
|↑(cid:105) |↓(cid:105)
X =
|↓(cid:105) |↑(cid:105)
corresponding to a phase shift θ → θ+ π are sufficient
2
to meet requirements of the present discussion.
Suppose that the sender intends to transmit the spe-
cific message ↑, ↓, ↑. The sender prepares three systems
in states
(|ψ (cid:105)⊗|ψ (cid:105))
send rcv 1
(|ψ (cid:105)⊗|ψ (cid:105))
send rcv 2
(|ψ (cid:105)⊗|ψ (cid:105)) .
send rcv 3
Uponmeasurementthesendermayfindthepartiallycol-
lapsed states
(|↑(cid:105)⊗|ψ (cid:105))
rcv 1
(|↑(cid:105)⊗|ψ (cid:105))
rcv 2
(|↓(cid:105)⊗|ψ (cid:105)) .
rcv 3
There are “errors” in messages 2 and 3 in the sense that
the sent message did not collapse to the intended state.
To remedy this, the sender applies the identity transfor-
mation 1⊗1 to the state (|↑(cid:105)⊗|ψ (cid:105)) and the trans-
rcv 1
formation 1⊗X to states 2 and 3.
∗ [email protected]
[1] D. Ford, “A note on the equivalence of Gibbs
free energy and information theoretic capacity”,
http://arxiv.org/abs/0809.3540, 2008
[2] O. Shental, I. Kanter, “Shannon Meets Carnot:
Mutual Information Via Thermodynamics”,
http://arxiv.org/pdf/0806.3133v1, 2008