PHYSICAL REVIEW E 88, 042143 (2013)
Second law of information thermodynamics with entanglement transfer Hiroyasu Tajima Department of Physics, The University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan (Received 8 December 2012; revised manuscript received 1 July 2013; published 28 October 2013) We present an inequality which holds in the thermodynamical processes with measurement and feedback controls and uses only the Helmholtz free energy and the entanglement of formation: Wext −F − kB T EF . The quantity −EF , which is positive, expresses the amount of entanglement transfer from system S to probe P through the interaction Uˆ SP during the measurement. It is easier to achieve the upper bound in this inequality than in the Sagawa-Ueda inequality [Phys. Rev. Lett. 100, 080403 (2008)]. Our inequality has clear physical meaning: in the above thermodynamical processes, the work which we can extract from the thermodynamic system is greater than the upper bound in the conventional thermodynamics by the amount of the entanglement extracted by the measurement. DOI: 10.1103/PhysRevE.88.042143
PACS number(s): 05.30.−d, 03.67.−a, 05.70.Ln, 03.65.Ud
I. INTRODUCTION
The second law of thermodynamics appears to be violated in thermodynamic processes that include measurements and feedbacks. This well-known fact has been the center of attention and numerous studies have long been conducted on such processes [1–12]. The second law of information thermodynamics [6] derived by Sagawa and Ueda is a monumental landmark of such studies; in the case of an isothermal process, it is expressed as Wext −F + kB T IQC ,
(1)
where IQC is the QC-mutual information content [6]. This inequality gives an upper bound for the work extracted from a thermodynamic system when measurement and feedback are permitted on the system. † When the measurement is classical [[ρ,( ˆ Mˆ (k) Mˆ (k) )1/2 ] = 0, where {Mˆ (k) } is the measurement and ρˆ is the density matrix of the system with the baths], the QC-mutual information reduces to the classical mutual information content. Therefore, in the classical world, the work that we can extract from information thermodynamic processes is greater than the upper bound of the conventional thermodynamics by the amount of information which we obtain from the measurement. (In the present paper, we use the phrase “upper bound of the conventional thermodynamics” for the inequality Wext −F .) On the other hand, when the measurement is not classical, the physical meaning of the QC-mutual information is unclear. We will also show below that when we use finite systems for the heat baths, the upper bound of Eq. (1) is not necessarily achievable. In this paper, we present an information thermodynamic inequality by using only the Helmholtz free energy and the entanglement of formation: Wext −F − kB T EF ,
(2)
where the difference EF of the entanglement is taken between before and after the unitary interaction Uˆ SP between system S and probe P during the measurement. The quantity −EF is always non-negative and expresses the amount of entanglement transfer from S to P through Uˆ SP . Hence, the inequality (2) has clear physical meaning: the work that we can extract from information thermodynamic processes is greater than the upper bound of the conventional thermodynamics 1539-3755/2013/88(4)/042143(7)
by the amount of entanglement which we obtain from the measurement. In other words, from a thermodynamical point of view, we can interpret the entanglement transfer as the information transfer. In the above context, we introduce a new information content IE = −EF , which we refer to as the entanglement information. It has a clear physical meaning even when IQC does not. We also show that the condition for the achievement of the upper bound of inequality (2) is looser than that of inequality (1). II. SETUP OF WHOLE SYSTEM
As the setup, we consider a thermodynamic system S that is in contact with heat baths Bm for m = 1,2, . . . ,n which are at temperatures T1 , c, Tn , respectively. We refer to the whole set of heat baths {Bm } as B. Except when we perform measurement or feedback control, we express the Hamiltonian of the whole system as (t)) = Hˆ S (λ S (t)) + Hˆ (λ
n
SBm (t)) + Hˆ Bm ], [Hˆ SBm (λ
(3)
m=1
S (t)) is the Hamiltonian of the system S, Hˆ Bm where Hˆ S (λ SBm (t)) is the is the Hamiltonian of the bath Bm , and Hˆ SBm (λ interaction Hamiltonian between the system S and the heat bath Bm . The Hamiltonian is controlled through the external S (t) and λ SBm (t). We assume that there exists a parameters λ SBm ˆ We call the 0 ) = 0. value of λ (t) = λ0 such that Hˆ SBm (λ time evolution of the whole system with controlled values SBm (t) a thermodynamic operation. We further S (t) and λ of λ assume that we can realize a thermodynamic equilibrium state at temperature Tm by connecting S and Bm and waiting. Note that the equilibrium state may not be a canonical distribution. We define the energy U of a state ρˆ as tr[ρˆ Hˆ ] and define the Helmholtz free energy F for an equilibrium state at a temperature T as −kB T ln Z(β), where β ≡ (kB T )−1 and Z(β) ≡ tr[exp(−β Hˆ )]. III. INFORMATION THERMODYNAMIC PROCESS
Under the setup in Sec. 2, we consider the following thermodynamic processes from t = ti to t = tf (Fig. 1): At
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HIROYASU TAJIMA
PHYSICAL REVIEW E 88, 042143 (2013)
respectively. To put it simply, we can express IE as follows: IE ≡ −EFSB -R = EFSB -R before Uˆ SP − EFSB -R after Uˆ SP . (10) We also note that EFSB -R (|ψSBR ψSBR |) is equal to the amount of entanglement between SB and the rest of the whole system at t = t1 . Thus, we can interpret IE as the amount of entanglement between SB and R that is taken by the probe P during the interaction Uˆ SP . From t = t1 to t = t2 , we perform a projective measurement {Pˆ(k) = i |k,iP k,iP |} on the probe, where {|k,iP } are pure states of the probe. At t = t2 , we obtain a result k with probability pk , and then the state of SB becomes ρˆ2 = pk ρˆ2(k) = qi,k ρˆ2ik , (11) k
where FIG. 1. Schematic of the thermodynamic processes from ti to tf .
t = ti , we start the process with the following canonical initial state: exp −β Hˆ iS exp[−β1 Hˆ B1 ] exp[−βn Hˆ Bn ] ρˆi = ⊗ ⊗ · · · ⊗ , ZiS (β) Z B1 (β1 ) Z Bn (βn ) (4) S (ti )), ZiS = tr[exp[−β Hˆ iS ]], βm = where Hˆ iS = Hˆ S (λ −1 Bm (kB Tm ) , and Z = tr[exp[−βm Hˆ Bm ]] (m = 1, . . . ,n). From t = ti to t = t1 , we perform a thermodynamic operation † Uˆ init . At t = t1 , the state is therefore given by ρˆ1 = Uˆ init ρˆi Uˆ init . Adding a proper reference system R, we can find a pure state |ψSBR which satisfies trR [|ψSBR ψSBR |] = ρˆ1 .
|ψP SBR = (Uˆ SP ⊗ 1ˆ B ⊗ 1ˆ R ) (|0P ⊗ |ψSBR ) ,
(6)
where 1ˆ B and 1ˆ R are the identity operators. At this point, we define the new quantity IE ; namely, the entanglement information, as follows: IE ≡ EFSB -R (|ψSBR ψSBR |) − EFSB -R (ρˆSBR ) (7) ), = S(ρˆ ) − E SB -R (ρˆ 1
F
SBR
where S(ρ) ˆ ≡ −tr [ρˆ ln ρ] ˆ , ρˆSBR ≡ trP [|ψP SBR ψP SBR |] , (8) SB -R (ρ) ˆ is the entanglement of formation [13] between and E F
SB and R: E SB -R (ρ) ˆ ≡ F
SB -R
min ρˆSBR = qj |φ j φ j |
qj E SB -R (|φ j ),
(9)
j
SBR
SBR
(12) (13) (14)
i ik and ρˆ2(k) being normalized. We can interpret the with |ψSBR above as performing a measurement {Mˆ (k) }, where Mˆ (k) = k,iP |Uˆ SP |0P , (15) i
on S from t = t1 to t = t2 . The QC-mutual information [6,8] is determined here. It is expressed as IQC ≡ S(ρˆ1 ) + H {pk } ˆ ˆ ˆ tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ], +
(16)
k † where Dˆ k ≡ Mˆ (k) Mˆ (k) . We emphasize the following two points: First, we can determine the unitary interaction Uˆ SP and the projective measurements Pˆ(k) for any measurement Mˆ (k) . Hence, if we can evaluate the QC-mutual information IQC , then we can also evaluate IE . Second, the timings at which IE and IQC are defined are different. The information IE is defined when only Uˆ SP is completed, whereas the information IQC is defined when the measurement {Mˆ (k) } is also completed. Thus, for two measurements with the same Uˆ SP , IE takes the same value but IQC may take different values. From t = t2 to t = t3 , we perform a feedback control depending on the measurement result k. To be precise, we perform a unitary transformations Uˆ (k) on SB. At t = t3 , the state of SB is given by † pk Uˆ (k) ρˆ2(k) Uˆ (k) . (17) ρˆ3 ≡ k
with E (|φ ) being the entanglement entropy [14] between SB and R for a pure state |φ j . Note that EFSB -R (|ψSBR ψSBR |) and EFSB -R (ρˆSBR ) indicate the amount of entanglement between SB and R at t = t1 and t = t1 , j
ik = (k,iP | ⊗ 1ˆ SBR )|ψP SBR , qi,k ψSBR ik ik ψSBR , ρˆ2ik = trR ψSBR
ik ik (k) , pk ρˆ = trP R ψ qi,k ψ
√
2
(5)
From t = t1 to t = t1 , we introduce a unitary interaction Uˆ SP between system S and probe P , which is initialized to a state |0P . At t = t1 , the state of the whole system is expressed as
i,k
From t = t3 to t = tf , we choose a thermodynamic operation Uˆ fin whose final state is assumed to be equilibrium and perform it. We also assume that by tf system S and heat bath Bm will have reached thermodynamic equilibrium at temperatures T
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SECOND LAW OF INFORMATION THERMODYNAMICS WITH . . .
and Tm , respectively. Note that we only assume that the final state is macroscopically in equilibrium; the final state may not be a canonical distribution given by exp −β Hˆ fS exp[−β1 Hˆ B1 ] can ρˆf ≡ ⊗ Z B1 (β1 ) ZfS (β ) exp[−βn Hˆ Bn ] , (18) ⊗ ··· ⊗ Z Bn (βn ) where β is the inverse temperature of the final state of the system. We hereafter call the above process the information thermodynamic process. IV. MAIN RESULTS
For the above information thermodynamic processes, we present five results. The first theorem is the new second law of information thermodynamics: Theorem 1. For any information thermodynamic process, the following inequality holds: U S − F S Qm U S − F S + − + kB IE , T T T m=1 m
PHYSICAL REVIEW E 88, 042143 (2013)
Condition 2. The thermodynamic operations Uˆ init and Uˆ fin satisfy the equation of the following inequality: U S − F S Qm U S − F S + − . T T T m=1 m n
(23)
0 is satisfied for t1 t t2 . SB (t) = λ Condition 3. λ Condition 2 dictates that we do not waste energy during the thermodynamic processes. Condition 3 implies that the system and baths do not interact during the measurement; if the system and baths interact during the measurement, the information obtained by the probe contains the information about the system as well as about the baths. Thus, we can interpret Theorem 4 as follows: We can completely use the information obtained by the probe with a proper interpretation {Pˆ(k) } and a proper feedback {Uˆ (k) } if we do not waste energy during the thermodynamic processes and if the information describes only the system. Theorem 5. Under conditions 1–3, there is a measurement {Mˆ (k) } for which we cannot achieve the equality of Eq. (22) with any Uˆ (k) .
n
S
where Qm ≡ tr[H (ρi − ρf )] and the quantities U , F , U , and F S are the energy and the Helmholtz free energy at ti and tf , respectively. When the the system undergoes an isothermal process in contact with a single heat bath B at temperature T , the inequality (19) reduces to ˆ Bm
S
Wext −F + kB T IE = −F − kB T EFSB−R .
S
(20) (21)
The second theorem shows that we can always achieve the upper bound of Eq. (19) when we use infinite systems for the heat baths. Theorem 2. When we use infinite systems as the heat baths, there is at least one set of projective measurement {Pˆ(k) } and feedback {Uˆ (k) } which achieve the upper bound of Eq. (19) for any interaction Uˆ SP . With the third, fourth, and fifth results, we will see that the condition for the achievement of the upper bound of inequality (19) is looser than that of the inequality U S − F S Qm U S − F S + − + kB IQC , T T T m=1 m
V. PROOFS OF MAIN RESULTS
(19)
n
(22)
Let us prove Theorems 1–5. Proof of Theorem 1. Theorem 1 is directly given by the following lemma: Lemma 1. For any measurement {Mˆ (k) }, the following inequality holds: IQC IE .
(24)
The inequalities (22) and (24) give Eq. (19). Lemma 1 and Theorems 4 and 5 seem to contradict each other. Although the upper bound of Eq. (19) is always achievable and although inequality (24) exists, there is a case in which the upper bound of Eq. (22) is not achievable. However, the contradiction is only spurious. Note that when IE is determined, we can take {Pˆ(k) } freely; in other words, we can choose the “best” interpretation of the information obtained by the probe. On the other hand, when IQC is determined, {Pˆ(k) } is also determined already, and thus our interpretation of the probe’s information is fixed uniquely. Let us prove Lemma 1. Because of the definitions (7) and (16), we prove EFSB -R (ρˆSBR ) ˆ ˆ ˆ −H {pk } − tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ]. k
when we use finite systems for the heat baths. First, let us present the third result: Theorem 3. If we can always achieve the upper bound of Eq. (22) with a proper feedback {Uˆ (k) }, we can always achieve the upper bound of Eq. (19) with a proper set of projective measurement {Pˆ(k) } and feedback {Uˆ (k) }. The fourth and fifth results show that the converse of Theorem 3 is not true. Theorem 4. When the following conditions are satisfied, we can always achieve the equality of Eq. (19) with proper choices of {Pˆ(k) } and {Uˆ (k) } for any Uˆ SP : Condition 1. The system S is a two-level system.
(25) We can express the above as follows: ˆ ˆ ˆ −H {pk } − tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ]
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k
=
k
pk S
ρˆ2(k)
pk S
qi,k
ρˆ2(ik)
pk i ik (ik) = qi,k S ρˆ2 qi,k E SB -R ψSBR
i,k
=
EFSB -R (ρˆSBR ),
k
i,k
(26)
HIROYASU TAJIMA
where ρˆSBR =
PHYSICAL REVIEW E 88, 042143 (2013)
we can transform Eq. (35) into EFSB -R (ρˆSBR ) = S(ρˆ3 ),
ik ik ψSBR = trP [|ψP SBR ψP SBR |] . qi,k ψSBR
i,k
(27) Proof of Theorems 2 and 3. First we prove Theorem 3. Let (k) } which satisfies us take an ensemble {q(k) ,|ψSBR E SB -R (ρˆ , (28) )≡ q E SB -R φ k SBR
F
ρˆSBR =
k
Let us take the projective measurement {Pˆ(k) } as {|kP kP |}. Then, pk reduces to qk , and thus IQC ≡ S(ρˆ1 ) + H {pk } tr[ Dˆ k ρˆSB Dˆ k ln Dˆ k ρˆSB Dˆ k ] + k
= S(ρˆ1 ) − = S(ρˆ1 ) −
k qk E SB -R φSBR = IE .
(31)
Thus, for an arbitrary unitary Uˆ SP , there is a projective measurement {Pˆ(k) } that satisfies IE = IQC . Theorem 2 directly follows from Theorem 3. When we use infinite systems for the heat baths, we can always achieve the upper bound of (22) for any measurement {Mˆ (k) } [7]. Because of the above and Theorem 3, we can always achieve the upper bound of (19). Proof of Theorem 4. As in the derivation of Eq. (22) in Ref. [6], we can obtain the inequality (19) by transforming (32) S(ρˆi ) −tr ρˆf ln ρˆfcan + IE . Thus, we only have to prove that for any Uˆ SP , we can always take {Pˆ(k) } and {Uˆ (k) } that satisfy (33) S(ρˆi ) = −tr ρˆf ln ρˆfcan + IE . First, we prove that if ρˆ3 in Eq. (17) is a canonical distribution, we can transform Eq. (33) into (34) EFS -R (ρˆSR ) = S ρˆ3S , where ρˆSR ≡ trB [ρˆSBR ], ρˆ3S ≡ trB [ρˆ3 ], and EFS -R is the entanglement of formation between S and R. Thanks to Eq. (7) and S(ρˆi ) = S(ρˆ1 ), we can transform Eq. (33) into ). (35) 0 = −tr ρˆ ln ρˆ can − E SB -R (ρˆ f
f
F
F
SR1
BR2
(38) (39)
where we divide R into a two-level subsystem R1 and the rest R2 . Let us first prove Eq. (37). Owing to condition 2 and the fact that ρˆi is a canonical distribution, ρˆ1 is a canonical distribution as well. Because of condition 3, Hˆ (t1 ) = Hˆ S ⊗ Hˆ B is valid. Thus, because of condition 1, under the proper basis of R we can divide R into a two-level subsystem R1 and a subsystem R2 and express |ψSBR as follows: |ψSBR = ψSR1 ⊗ ψBR2 . (40) Owing to Eqs. (6) and (40), we can express |ψP SBR as follows: |ψP SBR = |ψP SR1 ⊗ |ψBR2 . Thus, we can express ρˆSBR as (41) ρˆSBR = ρˆSR1 ⊗ ψBR2 ψBR2 ,
pk S ρˆ2(k)
k EFSB -R (ρˆSBR )
SBR
EFS -R (ρˆSR ) = EFS -R1 ρˆSR1 , S(ρˆ3 ) = S ρˆ3S + E B -R2 ψBR2 ,
(29)
k
= S (ρˆ1 ) −
F
k
Then, we can take an orthonormal basis {|kP } which satisfies √ k |ψP SBR = . (30) qk |kP φSBR
k
where we use S(ρˆ3 ) = S(ρˆf ). Thus, we only have to transform Eq. (36) into Eq. (34). If the following three equations hold, (36) and (34) are equivalent:
) = E S -R1 ρˆ + E B -R2 ψ , (37) E SB -R (ρˆ
SBR
k k qk φSBR φSBR .
(36)
SBR
Note that a thermodynamic operation from a canonical distribution to an equilibrium state achieves the equality of Eq. (23) if and only if the final state is a canonical distribution, too [6]. Thus, because of condition 2, if ρˆ3 is a canonical distribution, ρˆf is the canonical distribution ρˆfcan in (18). Then
where ρˆSR1 ≡ trP [|ψP SR1 ψP SR1 |]. We have Eqs. (37) and (38) from Eq. (41). Next, we prove Eq. (39). Note that S has been isolated for t1 t t3 with proper {Uˆ (k) }. We can therefore express ρˆ3 as ρˆ3S ⊗ |ψBR ψBR | with E B -R2 (|ψBR2 ) = E B -R2 (|ψBR ). 2 2 2 Thus, we have Eq. (39). Now, we only have to find {Pˆ(k) } on P and {Uˆ (k) } on S such that ρˆ3 and ρˆ3S are canonical distributions and that Eq. (34) holds. We first prove that if ρˆ3S is a canonical distribution, ρˆ3 is also a canonical distribution. To prove this, we only have ψBR |] is a canonical distribution to note that trR2 [|ψBR 2 2 because B has been isolated for t1 t t3 and because trS [ρˆ1 ] is a canonical distribution. We second find {Pˆ(k) } on P and {Uˆ (k) } on S such that ρˆ3S is a canonical distribution and that Eq. (34) holds. Because both S and R1 are two-level systems, we can treat the state |ψP SR1 as a three-qubit pure state under a proper basis of P . In Appendix A, we prove the following with the approach used in Ref. [15]: we can perform a projective measurement {P˜(k) }k=0,1 on the probe P such that the results P˜(k) |ψP SR1 are local unitary (LU) equivalent for k = 0,1 and E S -R1 (P˜(k) |ψP SR1 ) = EFS -R1 (ρˆSR1 ) is valid (when we can transform a state into another state by local unitary transformation, we call the states are LU equivalent). Because the results P˜(k) |ψP SR1 are LU equivalent, there exists pre † {Vˆ(k) }k=0,1 on S, which satisfies ρˆ3 ≡ Vˆ(k) ρˆ2S(k) Vˆ(k) , where ρˆ2S(k) ≡ trR1 [P˜(k) |ψP SR1 ψP SR1 |P˜(k) ]. Owing to condition 1, if pre E S -R1 (P˜(k) |ψP SR1 ) = 0, we can make the state ρˆ3S = Vˆ ρˆ3 Vˆ † a canonical distribution with a unitary transformation Vˆ on S. Thus, {P˜(k) } and {Vˆ(k) } are the measurement and feedback that we want. Proof of Theorem 5. It is sufficient to prove the existence of a counterexample of the measurement {Mˆ (k) }. The equality of
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(22) is valid only if there exists a set of unitary transformations {Uˆ (k) } that satisfy k pk S(ρˆ2(k) ) = S(ρˆ3 ) [6]. We can transform S(ρˆ3 ) as follows: † (k) S(ρˆ3 ) = S pk Uˆ (k) ρˆ Uˆ (42) 2
(k)
k=0,1
=
k=0,1
† pk S ρˆ2(k) + pk D Uˆ (k) ρˆ2(k) Uˆ (k) ρˆ3 , k=0,1
ˆ ρˆ − ln ρˆ )]. Because D(ρ|| ˆ ρˆ ) = 0 where D(ρ|| ˆ ρˆ ) ≡ tr[ρ(ln if and only if ρˆ = ρˆ and because of Eq. (42), the equation (k) (k) k pk S(ρˆ2 ) = S(ρˆ3 ) is valid if and only if {ρˆ2 } are LU equivalent for k = 0,1; in other words, if and only if the measurement {Mˆ (k) } is a deterministic measurement. Because of this logic, if Theorem 5 were not valid, any measurement {Mˆ (k) } would be deterministic. This is clearly false, and thus Theorem 5 holds.
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where CSR1 (ρˆSR1 ) is the concurrence of ρˆSR1 and h(x) ≡ −x ln x − (1 − x) ln(1 − x). Thus, we only have to find a projective measurement {P˜(k) }k=0,1 such that P˜(k) |ψP SR1 for k = 0,1 are LU equivalent to each other and CSR1 (P˜(k) |ψP SR1 ) = CSR1 (ρˆSR1 ). Before giving the projective measurement {P˜(k) }, we first present preparations. First we express |ψP SR1 in the form of the generalized Schmidt decomposition [17]: ψP SR = λ0 |000 + λ1 eiϕ |100 + λ2 |101 1
+ λ3 |110 + λ4 |111
ACKNOWLEDGMENTS
This work was supported by the Grants-in-Aid for Japan Society for Promotion of Science (JSPS) Fellows (Grant No. 24E8116). The author thanks Professor Naomichi Hatano for useful discussions. APPENIDX A: THE PROOF OF THE EXISTENCE OF { P˜(k) }
In the present Appendix, we prove the following theorem: Theorem 6. For an arbitrary three-qubit pure state |ψP SR1 , there exists a projective measurement {Pˆ(k) }k=0,1 such that the results Pˆ(k) |ψP SR1 are LU equivalent for k = 0,1 and
(A1) E S -R1 Pˆ(k) ψP SR1 = EFS -R1 ρˆSR1 is valid. Proof. Because ρˆSR1 is a two-qubit mixed state, we can express EFS -R1 (ρˆSR1 ) in the form of the concurrence [16]: ⎛
⎞ 2 1 + 1 − CSR ρ ˆ SR
1 1 ⎠, EFS -R1 ρˆSR1 = h ⎝ (A2) 2
(A3)
and introduce the following eight parameters [15]:
VI. CONCLUSION
To conclude, we obtain an alternate information thermodynamic inequality. In this inequality, the information gain is the entanglement gain; the new information content IE represents the amount of the entanglement between the system and the reference system which the probe takes from the system. The new information content depends only on the premeasurement state of the system and the unitary interaction between the probe and the system, and thus when IE is determined, we can take {Pˆ(k) } freely. The QC-mutual information IQC does not have this freedom. Theorems 4 and 5 follow from this difference of the freedom between IE and IQC . Thus, in the above context, we can state that the substance of information is the entanglement. The information gain is already completed when the unitary interaction is over and the projective measurement is only the interpretation of the information.
(0 ϕ π )
KP S ≡ CP2 S + τP SR1 ,
(A4)
KP R1 ≡ CP2 R1 + τP SR1 ,
(A5)
2 KSR1 ≡ CSR + τP SR1 , 1
(A6)
J5 ≡ 4λ20 |λ1 λ4 eiϕ − λ2 λ3 |2 + λ22 λ23 − λ21 λ24 ,
(A7)
K5 ≡ J5 + τP SR1 ,
(A8)
J ≡ K52 − KP S KP R1 KSR1 ,
(A9)
λ2 λ3 − λ1 λ4 eiϕ e−i ϕ˜5 ≡ , |λ2 λ3 − λ1 λ4 eiϕ |
+ J τ P SR 5
, Qe ≡ sgn sin ϕ λ20 − 2 1 2 CSR1 + τP SR1
(A10) (A11)
where τP SR1 is the tangle of |ψP SR1 and sgn[x] is the sign function, x/|x| (x = 0) sgn[x] = (A12) 0 (x = 0). When Qe = 0, there are two possible decompositions which satisfy Eq. (A3). We then choose the decomposition with a greater coefficient λ0 . Now we have completed the preparation. In the basis of Eq. (A3), the projective measurement {P˜(k) }k=0,1 is given as follows: a ke−iθ 1 − a −ke−iθ ˜ ˜ , P(1) = , P(0) = keiθ b −keiθ 1 − b (A13) where the measurement parameters a, b, k, and θ are defined as follows: √ 2 ˙ J CSR K5 τP SR1 ± 1 1 a= − , (A14) 2 2K 2 2 SR1 K5 − KP S KP R1 CSR1 b = 1 − a, k = a(1 − a),
(A15)
θ = −ϕ˜5 ,
(A17)
(A16)
˙ means −Qe and when Qe = 0 and when Qe = 0 the mark ± ˙ means −. By using Eqs. (A14)–(A17) and the mark ± Lemma 1 of Ref. [15] and after straightforward algebra, we
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can confirm that the measurement {P˜(k) } is the measurement that we sought. APPENIDX B: A TOY MODEL
In the present appendix, we give an example which satisfies the equality of (2). Let us consider for the system S a qubit with the Hamiltonian Hˆ S = −λx σˆ x − λy σˆ y − λz σˆ z ,
(B1)
where σˆ x , σˆ y , and σˆ z are the Pauli matrices and λx , λy , and λz are the control parameters. We can interpret the control parameters as a magnetic field on the qubit S. Let the control parameters (λx ,λy ,λz ) initially set to (0,0,1) and the qubit initially set in the canonical distribution of temperature T . We add the reference system R and make SR a pure state. We also prepare the probe P , which is initialized to a state |0P . We perform the following unitary interaction Uˆ SP between S and P : −2β 1 − e 1 + e−2β |0P 0S 0P 0S | − |1P 0S 0P 0S | Uˆ SP = 2 2 1 + e−2β |0P 0S 1P 0S | + 2 1 − e−2β |1P 0S 1P 0S | + 2 (B2) + |1P 1S 0P 1S | + |0P 1S 1P 1S | . Then, the concurrence and the tangle of SP R change as shown in Fig. 2 during the interaction USP . Thus, because of Eqs. (10)
and (A2), the entanglement information is IE = −EF = EFSB -R before Uˆ SP − EFSB -R after Uˆ SP = EFS -R before Uˆ SP − EFS -R after Uˆ SP ⎛ ⎞ 2(1−e−2β ) 1 + 1 − (e β +e−β )2 eβ ⎠, =h β − h⎝ e + e−β 2
(B3) (B4) (B5)
where h(x) ≡ −x ln x − (1 − x) ln(1 − x). After the unitary interaction Uˆ SP , we perform the following projective measurement {Pˆ(k) } on P : 1 1 1 1 1 −1 ˆ ˆ , P(1) = . (B6) P(0) = 2 1 1 2 −1 1 Then, we perform the following feedback control: (1) When the result of the measurement {Pˆ(k) } is k = 0, we first set (λx ,λy ,λz ) to (1,0,0) from t = 0 to t = π . Second, we quasistatically vary (λx ,λy ,λz ) from (0,0,1) √ to (0,0,β −1 Arctanh[ tanhβ]). The work which is extracted from √S during√ this feedback is W (0) = −tanh[β] + β −1 Arctanh[ tanhβ] tanhβ. (2) When the result of the measurement {Pˆ(k) } is k = 1, we quasistatically vary (λx ,λy ,λz ) from (0,0,1) √ to (0,0,β −1 Arctanh[ tanhβ]). The work which is extracted from √S during√ this feedback is W (1) = −tanh[β] + β −1 Arctanh[ tanhβ] tanhβ. At the end of the feedback control, the system S is in the canonical distribution √ of temperature T with the Hamiltonian −β −1 Arctanh[ tanhβ]σˆ z . Thus, the difference of the Helmholtz free energy is 1 ln[2coshβ] β 1 + ln{2cosh[Arctanh( tanhβ)]}. β
−F = −
(B7)
Because of W (0) = W (1) , the average work Wext , on the other hand, is 1 Wext = −tanh[β] + Arctanh[ tanhβ] tanhβ. (B8) β With a straightforward algebra, we have Wext − (−F ) 1 = −tanhβ + ln [2coshβ] β 1 − (−Arctanh[ tanhβ] tanhβ β + ln{2cosh[Arctanh( tanhβ)]}) ⎛ ⎡ β 1 + 1− 1 e ⎝ = ⎣h β − h β e + e−β 2
2(1−e−2β ) (eβ +e−β )2
⎞⎤ ⎠⎦ . (B9)
FIG. 2. The entanglement transfer from S to P during the interaction USP . The straight line denotes the concurrence and the circle denotes the tangle. Note that the sum of the concurrence and the tangle is conserved.
The right-hand side of Eq. (B9) is equal to −kB T EF . Thus, the equality of (2) holds in the present information thermodynamic process; the extracted work can exceed the upper bound of the conventional thermodynamics just by the amount of the entanglement extracted by the measurement.
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SECOND LAW OF INFORMATION THERMODYNAMICS WITH . . .
The above achievement of the upper limit of Eq. (2) is given by the best interpretation of the information; if we perform a projective measurement other than {Pˆ(k) }, for example, the measurement {Pˆ(k) }, 1 0 Pˆ(0) = , 0 0
0 0 Pˆ(1) = , 0 1
(B10)
we cannot achieve the upper limit of Eq. (2) with any feedback. For the present S, we can choose the best projective
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
L. Szilard, Z. Phys. 53, 840 (1929). R. Landauer, IBM J. Res. Dev. 5, 183 (1961). C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982). B. Piechocinska, Phys. Rev. A 61, 062314 (2000). M. A. Nielsen, C. M. Caves, B. Schumacher, and H. Barnum, Proc. R. Soc. London, Ser. A 454, 277 (1998). T. Sagawa and M. Ueda, Phys. Rev. Lett. 100, 080403 (2008). K. Jacobs, Phys. Rev. A 80, 012322 (2009). T. Sagawa and M. Ueda, Phys. Rev. Lett. 102, 250602 (2009). F. J. Cao and M. Feito, Phys. Rev. E 79, 041118 (2009). S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. 6, 988 (2010).
PHYSICAL REVIEW E 88, 042143 (2013)
measurement with which we can achieve the upper limit of Eq. (2) for arbitrary USP , by using Eqs. (A13)–(A17). } is an example of We finally see that the above {Pˆ(k) unachievableness of the Sagawa-Ueda inequality (1). Note that the general measurement which consists of Uˆ SP and {Pˆ(k) } is not deterministic. Namely, the state which corresponds to the results k = 0 is not LU equivalent to that of k = 1. Thus, because of the proof of Theorem 5, we cannot achieve the equality of (1) by using any finite heat bath; we cannot achieve the equality of (1) with any feedback controls.
[11] J. M. Horowitz and J. M. R. Parrondo, New J. Phys. 13, 123019 (2011). [12] S. De Liberato and M. Ueda, Phys. Rev. E 84, 051122 (2011). [13] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [14] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). [15] H. Tajima, Ann. Phys. (NY) 329, 1 (2013). [16] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [17] A. Ac´ın, A. Andrianov, L. Costa, E. Jan´e, J. I. Latorre, and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000).
042143-7
Second law of information thermodynamics with entanglement transfer Hiroyasu Tajima Department of Physics, The University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan (Received 8 December 2012; revised manuscript received 1 July 2013; published 28 October 2013) We present an inequality which holds in the thermodynamical processes with measurement and feedback controls and uses only the Helmholtz free energy and the entanglement of formation: Wext −F − kB T EF . The quantity −EF , which is positive, expresses the amount of entanglement transfer from system S to probe P through the interaction Uˆ SP during the measurement. It is easier to achieve the upper bound in this inequality than in the Sagawa-Ueda inequality [Phys. Rev. Lett. 100, 080403 (2008)]. Our inequality has clear physical meaning: in the above thermodynamical processes, the work which we can extract from the thermodynamic system is greater than the upper bound in the conventional thermodynamics by the amount of the entanglement extracted by the measurement. DOI: 10.1103/PhysRevE.88.042143
PACS number(s): 05.30.−d, 03.67.−a, 05.70.Ln, 03.65.Ud
I. INTRODUCTION
The second law of thermodynamics appears to be violated in thermodynamic processes that include measurements and feedbacks. This well-known fact has been the center of attention and numerous studies have long been conducted on such processes [1–12]. The second law of information thermodynamics [6] derived by Sagawa and Ueda is a monumental landmark of such studies; in the case of an isothermal process, it is expressed as Wext −F + kB T IQC ,
(1)
where IQC is the QC-mutual information content [6]. This inequality gives an upper bound for the work extracted from a thermodynamic system when measurement and feedback are permitted on the system. † When the measurement is classical [[ρ,( ˆ Mˆ (k) Mˆ (k) )1/2 ] = 0, where {Mˆ (k) } is the measurement and ρˆ is the density matrix of the system with the baths], the QC-mutual information reduces to the classical mutual information content. Therefore, in the classical world, the work that we can extract from information thermodynamic processes is greater than the upper bound of the conventional thermodynamics by the amount of information which we obtain from the measurement. (In the present paper, we use the phrase “upper bound of the conventional thermodynamics” for the inequality Wext −F .) On the other hand, when the measurement is not classical, the physical meaning of the QC-mutual information is unclear. We will also show below that when we use finite systems for the heat baths, the upper bound of Eq. (1) is not necessarily achievable. In this paper, we present an information thermodynamic inequality by using only the Helmholtz free energy and the entanglement of formation: Wext −F − kB T EF ,
(2)
where the difference EF of the entanglement is taken between before and after the unitary interaction Uˆ SP between system S and probe P during the measurement. The quantity −EF is always non-negative and expresses the amount of entanglement transfer from S to P through Uˆ SP . Hence, the inequality (2) has clear physical meaning: the work that we can extract from information thermodynamic processes is greater than the upper bound of the conventional thermodynamics 1539-3755/2013/88(4)/042143(7)
by the amount of entanglement which we obtain from the measurement. In other words, from a thermodynamical point of view, we can interpret the entanglement transfer as the information transfer. In the above context, we introduce a new information content IE = −EF , which we refer to as the entanglement information. It has a clear physical meaning even when IQC does not. We also show that the condition for the achievement of the upper bound of inequality (2) is looser than that of inequality (1). II. SETUP OF WHOLE SYSTEM
As the setup, we consider a thermodynamic system S that is in contact with heat baths Bm for m = 1,2, . . . ,n which are at temperatures T1 , c, Tn , respectively. We refer to the whole set of heat baths {Bm } as B. Except when we perform measurement or feedback control, we express the Hamiltonian of the whole system as (t)) = Hˆ S (λ S (t)) + Hˆ (λ
n
SBm (t)) + Hˆ Bm ], [Hˆ SBm (λ
(3)
m=1
S (t)) is the Hamiltonian of the system S, Hˆ Bm where Hˆ S (λ SBm (t)) is the is the Hamiltonian of the bath Bm , and Hˆ SBm (λ interaction Hamiltonian between the system S and the heat bath Bm . The Hamiltonian is controlled through the external S (t) and λ SBm (t). We assume that there exists a parameters λ SBm ˆ We call the 0 ) = 0. value of λ (t) = λ0 such that Hˆ SBm (λ time evolution of the whole system with controlled values SBm (t) a thermodynamic operation. We further S (t) and λ of λ assume that we can realize a thermodynamic equilibrium state at temperature Tm by connecting S and Bm and waiting. Note that the equilibrium state may not be a canonical distribution. We define the energy U of a state ρˆ as tr[ρˆ Hˆ ] and define the Helmholtz free energy F for an equilibrium state at a temperature T as −kB T ln Z(β), where β ≡ (kB T )−1 and Z(β) ≡ tr[exp(−β Hˆ )]. III. INFORMATION THERMODYNAMIC PROCESS
Under the setup in Sec. 2, we consider the following thermodynamic processes from t = ti to t = tf (Fig. 1): At
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©2013 American Physical Society
HIROYASU TAJIMA
PHYSICAL REVIEW E 88, 042143 (2013)
respectively. To put it simply, we can express IE as follows: IE ≡ −EFSB -R = EFSB -R before Uˆ SP − EFSB -R after Uˆ SP . (10) We also note that EFSB -R (|ψSBR ψSBR |) is equal to the amount of entanglement between SB and the rest of the whole system at t = t1 . Thus, we can interpret IE as the amount of entanglement between SB and R that is taken by the probe P during the interaction Uˆ SP . From t = t1 to t = t2 , we perform a projective measurement {Pˆ(k) = i |k,iP k,iP |} on the probe, where {|k,iP } are pure states of the probe. At t = t2 , we obtain a result k with probability pk , and then the state of SB becomes ρˆ2 = pk ρˆ2(k) = qi,k ρˆ2ik , (11) k
where FIG. 1. Schematic of the thermodynamic processes from ti to tf .
t = ti , we start the process with the following canonical initial state: exp −β Hˆ iS exp[−β1 Hˆ B1 ] exp[−βn Hˆ Bn ] ρˆi = ⊗ ⊗ · · · ⊗ , ZiS (β) Z B1 (β1 ) Z Bn (βn ) (4) S (ti )), ZiS = tr[exp[−β Hˆ iS ]], βm = where Hˆ iS = Hˆ S (λ −1 Bm (kB Tm ) , and Z = tr[exp[−βm Hˆ Bm ]] (m = 1, . . . ,n). From t = ti to t = t1 , we perform a thermodynamic operation † Uˆ init . At t = t1 , the state is therefore given by ρˆ1 = Uˆ init ρˆi Uˆ init . Adding a proper reference system R, we can find a pure state |ψSBR which satisfies trR [|ψSBR ψSBR |] = ρˆ1 .
|ψP SBR = (Uˆ SP ⊗ 1ˆ B ⊗ 1ˆ R ) (|0P ⊗ |ψSBR ) ,
(6)
where 1ˆ B and 1ˆ R are the identity operators. At this point, we define the new quantity IE ; namely, the entanglement information, as follows: IE ≡ EFSB -R (|ψSBR ψSBR |) − EFSB -R (ρˆSBR ) (7) ), = S(ρˆ ) − E SB -R (ρˆ 1
F
SBR
where S(ρ) ˆ ≡ −tr [ρˆ ln ρ] ˆ , ρˆSBR ≡ trP [|ψP SBR ψP SBR |] , (8) SB -R (ρ) ˆ is the entanglement of formation [13] between and E F
SB and R: E SB -R (ρ) ˆ ≡ F
SB -R
min ρˆSBR = qj |φ j φ j |
qj E SB -R (|φ j ),
(9)
j
SBR
SBR
(12) (13) (14)
i ik and ρˆ2(k) being normalized. We can interpret the with |ψSBR above as performing a measurement {Mˆ (k) }, where Mˆ (k) = k,iP |Uˆ SP |0P , (15) i
on S from t = t1 to t = t2 . The QC-mutual information [6,8] is determined here. It is expressed as IQC ≡ S(ρˆ1 ) + H {pk } ˆ ˆ ˆ tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ], +
(16)
k † where Dˆ k ≡ Mˆ (k) Mˆ (k) . We emphasize the following two points: First, we can determine the unitary interaction Uˆ SP and the projective measurements Pˆ(k) for any measurement Mˆ (k) . Hence, if we can evaluate the QC-mutual information IQC , then we can also evaluate IE . Second, the timings at which IE and IQC are defined are different. The information IE is defined when only Uˆ SP is completed, whereas the information IQC is defined when the measurement {Mˆ (k) } is also completed. Thus, for two measurements with the same Uˆ SP , IE takes the same value but IQC may take different values. From t = t2 to t = t3 , we perform a feedback control depending on the measurement result k. To be precise, we perform a unitary transformations Uˆ (k) on SB. At t = t3 , the state of SB is given by † pk Uˆ (k) ρˆ2(k) Uˆ (k) . (17) ρˆ3 ≡ k
with E (|φ ) being the entanglement entropy [14] between SB and R for a pure state |φ j . Note that EFSB -R (|ψSBR ψSBR |) and EFSB -R (ρˆSBR ) indicate the amount of entanglement between SB and R at t = t1 and t = t1 , j
ik = (k,iP | ⊗ 1ˆ SBR )|ψP SBR , qi,k ψSBR ik ik ψSBR , ρˆ2ik = trR ψSBR
ik ik (k) , pk ρˆ = trP R ψ qi,k ψ
√
2
(5)
From t = t1 to t = t1 , we introduce a unitary interaction Uˆ SP between system S and probe P , which is initialized to a state |0P . At t = t1 , the state of the whole system is expressed as
i,k
From t = t3 to t = tf , we choose a thermodynamic operation Uˆ fin whose final state is assumed to be equilibrium and perform it. We also assume that by tf system S and heat bath Bm will have reached thermodynamic equilibrium at temperatures T
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SECOND LAW OF INFORMATION THERMODYNAMICS WITH . . .
and Tm , respectively. Note that we only assume that the final state is macroscopically in equilibrium; the final state may not be a canonical distribution given by exp −β Hˆ fS exp[−β1 Hˆ B1 ] can ρˆf ≡ ⊗ Z B1 (β1 ) ZfS (β ) exp[−βn Hˆ Bn ] , (18) ⊗ ··· ⊗ Z Bn (βn ) where β is the inverse temperature of the final state of the system. We hereafter call the above process the information thermodynamic process. IV. MAIN RESULTS
For the above information thermodynamic processes, we present five results. The first theorem is the new second law of information thermodynamics: Theorem 1. For any information thermodynamic process, the following inequality holds: U S − F S Qm U S − F S + − + kB IE , T T T m=1 m
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Condition 2. The thermodynamic operations Uˆ init and Uˆ fin satisfy the equation of the following inequality: U S − F S Qm U S − F S + − . T T T m=1 m n
(23)
0 is satisfied for t1 t t2 . SB (t) = λ Condition 3. λ Condition 2 dictates that we do not waste energy during the thermodynamic processes. Condition 3 implies that the system and baths do not interact during the measurement; if the system and baths interact during the measurement, the information obtained by the probe contains the information about the system as well as about the baths. Thus, we can interpret Theorem 4 as follows: We can completely use the information obtained by the probe with a proper interpretation {Pˆ(k) } and a proper feedback {Uˆ (k) } if we do not waste energy during the thermodynamic processes and if the information describes only the system. Theorem 5. Under conditions 1–3, there is a measurement {Mˆ (k) } for which we cannot achieve the equality of Eq. (22) with any Uˆ (k) .
n
S
where Qm ≡ tr[H (ρi − ρf )] and the quantities U , F , U , and F S are the energy and the Helmholtz free energy at ti and tf , respectively. When the the system undergoes an isothermal process in contact with a single heat bath B at temperature T , the inequality (19) reduces to ˆ Bm
S
Wext −F + kB T IE = −F − kB T EFSB−R .
S
(20) (21)
The second theorem shows that we can always achieve the upper bound of Eq. (19) when we use infinite systems for the heat baths. Theorem 2. When we use infinite systems as the heat baths, there is at least one set of projective measurement {Pˆ(k) } and feedback {Uˆ (k) } which achieve the upper bound of Eq. (19) for any interaction Uˆ SP . With the third, fourth, and fifth results, we will see that the condition for the achievement of the upper bound of inequality (19) is looser than that of the inequality U S − F S Qm U S − F S + − + kB IQC , T T T m=1 m
V. PROOFS OF MAIN RESULTS
(19)
n
(22)
Let us prove Theorems 1–5. Proof of Theorem 1. Theorem 1 is directly given by the following lemma: Lemma 1. For any measurement {Mˆ (k) }, the following inequality holds: IQC IE .
(24)
The inequalities (22) and (24) give Eq. (19). Lemma 1 and Theorems 4 and 5 seem to contradict each other. Although the upper bound of Eq. (19) is always achievable and although inequality (24) exists, there is a case in which the upper bound of Eq. (22) is not achievable. However, the contradiction is only spurious. Note that when IE is determined, we can take {Pˆ(k) } freely; in other words, we can choose the “best” interpretation of the information obtained by the probe. On the other hand, when IQC is determined, {Pˆ(k) } is also determined already, and thus our interpretation of the probe’s information is fixed uniquely. Let us prove Lemma 1. Because of the definitions (7) and (16), we prove EFSB -R (ρˆSBR ) ˆ ˆ ˆ −H {pk } − tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ]. k
when we use finite systems for the heat baths. First, let us present the third result: Theorem 3. If we can always achieve the upper bound of Eq. (22) with a proper feedback {Uˆ (k) }, we can always achieve the upper bound of Eq. (19) with a proper set of projective measurement {Pˆ(k) } and feedback {Uˆ (k) }. The fourth and fifth results show that the converse of Theorem 3 is not true. Theorem 4. When the following conditions are satisfied, we can always achieve the equality of Eq. (19) with proper choices of {Pˆ(k) } and {Uˆ (k) } for any Uˆ SP : Condition 1. The system S is a two-level system.
(25) We can express the above as follows: ˆ ˆ ˆ −H {pk } − tr[ Dk ρˆSB Dk ln Dk ρˆSB Dˆ k ]
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k
=
k
pk S
ρˆ2(k)
pk S
qi,k
ρˆ2(ik)
pk i ik (ik) = qi,k S ρˆ2 qi,k E SB -R ψSBR
i,k
=
EFSB -R (ρˆSBR ),
k
i,k
(26)
HIROYASU TAJIMA
where ρˆSBR =
PHYSICAL REVIEW E 88, 042143 (2013)
we can transform Eq. (35) into EFSB -R (ρˆSBR ) = S(ρˆ3 ),
ik ik ψSBR = trP [|ψP SBR ψP SBR |] . qi,k ψSBR
i,k
(27) Proof of Theorems 2 and 3. First we prove Theorem 3. Let (k) } which satisfies us take an ensemble {q(k) ,|ψSBR E SB -R (ρˆ , (28) )≡ q E SB -R φ k SBR
F
ρˆSBR =
k
Let us take the projective measurement {Pˆ(k) } as {|kP kP |}. Then, pk reduces to qk , and thus IQC ≡ S(ρˆ1 ) + H {pk } tr[ Dˆ k ρˆSB Dˆ k ln Dˆ k ρˆSB Dˆ k ] + k
= S(ρˆ1 ) − = S(ρˆ1 ) −
k qk E SB -R φSBR = IE .
(31)
Thus, for an arbitrary unitary Uˆ SP , there is a projective measurement {Pˆ(k) } that satisfies IE = IQC . Theorem 2 directly follows from Theorem 3. When we use infinite systems for the heat baths, we can always achieve the upper bound of (22) for any measurement {Mˆ (k) } [7]. Because of the above and Theorem 3, we can always achieve the upper bound of (19). Proof of Theorem 4. As in the derivation of Eq. (22) in Ref. [6], we can obtain the inequality (19) by transforming (32) S(ρˆi ) −tr ρˆf ln ρˆfcan + IE . Thus, we only have to prove that for any Uˆ SP , we can always take {Pˆ(k) } and {Uˆ (k) } that satisfy (33) S(ρˆi ) = −tr ρˆf ln ρˆfcan + IE . First, we prove that if ρˆ3 in Eq. (17) is a canonical distribution, we can transform Eq. (33) into (34) EFS -R (ρˆSR ) = S ρˆ3S , where ρˆSR ≡ trB [ρˆSBR ], ρˆ3S ≡ trB [ρˆ3 ], and EFS -R is the entanglement of formation between S and R. Thanks to Eq. (7) and S(ρˆi ) = S(ρˆ1 ), we can transform Eq. (33) into ). (35) 0 = −tr ρˆ ln ρˆ can − E SB -R (ρˆ f
f
F
F
SR1
BR2
(38) (39)
where we divide R into a two-level subsystem R1 and the rest R2 . Let us first prove Eq. (37). Owing to condition 2 and the fact that ρˆi is a canonical distribution, ρˆ1 is a canonical distribution as well. Because of condition 3, Hˆ (t1 ) = Hˆ S ⊗ Hˆ B is valid. Thus, because of condition 1, under the proper basis of R we can divide R into a two-level subsystem R1 and a subsystem R2 and express |ψSBR as follows: |ψSBR = ψSR1 ⊗ ψBR2 . (40) Owing to Eqs. (6) and (40), we can express |ψP SBR as follows: |ψP SBR = |ψP SR1 ⊗ |ψBR2 . Thus, we can express ρˆSBR as (41) ρˆSBR = ρˆSR1 ⊗ ψBR2 ψBR2 ,
pk S ρˆ2(k)
k EFSB -R (ρˆSBR )
SBR
EFS -R (ρˆSR ) = EFS -R1 ρˆSR1 , S(ρˆ3 ) = S ρˆ3S + E B -R2 ψBR2 ,
(29)
k
= S (ρˆ1 ) −
F
k
Then, we can take an orthonormal basis {|kP } which satisfies √ k |ψP SBR = . (30) qk |kP φSBR
k
where we use S(ρˆ3 ) = S(ρˆf ). Thus, we only have to transform Eq. (36) into Eq. (34). If the following three equations hold, (36) and (34) are equivalent:
) = E S -R1 ρˆ + E B -R2 ψ , (37) E SB -R (ρˆ
SBR
k k qk φSBR φSBR .
(36)
SBR
Note that a thermodynamic operation from a canonical distribution to an equilibrium state achieves the equality of Eq. (23) if and only if the final state is a canonical distribution, too [6]. Thus, because of condition 2, if ρˆ3 is a canonical distribution, ρˆf is the canonical distribution ρˆfcan in (18). Then
where ρˆSR1 ≡ trP [|ψP SR1 ψP SR1 |]. We have Eqs. (37) and (38) from Eq. (41). Next, we prove Eq. (39). Note that S has been isolated for t1 t t3 with proper {Uˆ (k) }. We can therefore express ρˆ3 as ρˆ3S ⊗ |ψBR ψBR | with E B -R2 (|ψBR2 ) = E B -R2 (|ψBR ). 2 2 2 Thus, we have Eq. (39). Now, we only have to find {Pˆ(k) } on P and {Uˆ (k) } on S such that ρˆ3 and ρˆ3S are canonical distributions and that Eq. (34) holds. We first prove that if ρˆ3S is a canonical distribution, ρˆ3 is also a canonical distribution. To prove this, we only have ψBR |] is a canonical distribution to note that trR2 [|ψBR 2 2 because B has been isolated for t1 t t3 and because trS [ρˆ1 ] is a canonical distribution. We second find {Pˆ(k) } on P and {Uˆ (k) } on S such that ρˆ3S is a canonical distribution and that Eq. (34) holds. Because both S and R1 are two-level systems, we can treat the state |ψP SR1 as a three-qubit pure state under a proper basis of P . In Appendix A, we prove the following with the approach used in Ref. [15]: we can perform a projective measurement {P˜(k) }k=0,1 on the probe P such that the results P˜(k) |ψP SR1 are local unitary (LU) equivalent for k = 0,1 and E S -R1 (P˜(k) |ψP SR1 ) = EFS -R1 (ρˆSR1 ) is valid (when we can transform a state into another state by local unitary transformation, we call the states are LU equivalent). Because the results P˜(k) |ψP SR1 are LU equivalent, there exists pre † {Vˆ(k) }k=0,1 on S, which satisfies ρˆ3 ≡ Vˆ(k) ρˆ2S(k) Vˆ(k) , where ρˆ2S(k) ≡ trR1 [P˜(k) |ψP SR1 ψP SR1 |P˜(k) ]. Owing to condition 1, if pre E S -R1 (P˜(k) |ψP SR1 ) = 0, we can make the state ρˆ3S = Vˆ ρˆ3 Vˆ † a canonical distribution with a unitary transformation Vˆ on S. Thus, {P˜(k) } and {Vˆ(k) } are the measurement and feedback that we want. Proof of Theorem 5. It is sufficient to prove the existence of a counterexample of the measurement {Mˆ (k) }. The equality of
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(22) is valid only if there exists a set of unitary transformations {Uˆ (k) } that satisfy k pk S(ρˆ2(k) ) = S(ρˆ3 ) [6]. We can transform S(ρˆ3 ) as follows: † (k) S(ρˆ3 ) = S pk Uˆ (k) ρˆ Uˆ (42) 2
(k)
k=0,1
=
k=0,1
† pk S ρˆ2(k) + pk D Uˆ (k) ρˆ2(k) Uˆ (k) ρˆ3 , k=0,1
ˆ ρˆ − ln ρˆ )]. Because D(ρ|| ˆ ρˆ ) = 0 where D(ρ|| ˆ ρˆ ) ≡ tr[ρ(ln if and only if ρˆ = ρˆ and because of Eq. (42), the equation (k) (k) k pk S(ρˆ2 ) = S(ρˆ3 ) is valid if and only if {ρˆ2 } are LU equivalent for k = 0,1; in other words, if and only if the measurement {Mˆ (k) } is a deterministic measurement. Because of this logic, if Theorem 5 were not valid, any measurement {Mˆ (k) } would be deterministic. This is clearly false, and thus Theorem 5 holds.
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where CSR1 (ρˆSR1 ) is the concurrence of ρˆSR1 and h(x) ≡ −x ln x − (1 − x) ln(1 − x). Thus, we only have to find a projective measurement {P˜(k) }k=0,1 such that P˜(k) |ψP SR1 for k = 0,1 are LU equivalent to each other and CSR1 (P˜(k) |ψP SR1 ) = CSR1 (ρˆSR1 ). Before giving the projective measurement {P˜(k) }, we first present preparations. First we express |ψP SR1 in the form of the generalized Schmidt decomposition [17]: ψP SR = λ0 |000 + λ1 eiϕ |100 + λ2 |101 1
+ λ3 |110 + λ4 |111
ACKNOWLEDGMENTS
This work was supported by the Grants-in-Aid for Japan Society for Promotion of Science (JSPS) Fellows (Grant No. 24E8116). The author thanks Professor Naomichi Hatano for useful discussions. APPENIDX A: THE PROOF OF THE EXISTENCE OF { P˜(k) }
In the present Appendix, we prove the following theorem: Theorem 6. For an arbitrary three-qubit pure state |ψP SR1 , there exists a projective measurement {Pˆ(k) }k=0,1 such that the results Pˆ(k) |ψP SR1 are LU equivalent for k = 0,1 and
(A1) E S -R1 Pˆ(k) ψP SR1 = EFS -R1 ρˆSR1 is valid. Proof. Because ρˆSR1 is a two-qubit mixed state, we can express EFS -R1 (ρˆSR1 ) in the form of the concurrence [16]: ⎛
⎞ 2 1 + 1 − CSR ρ ˆ SR
1 1 ⎠, EFS -R1 ρˆSR1 = h ⎝ (A2) 2
(A3)
and introduce the following eight parameters [15]:
VI. CONCLUSION
To conclude, we obtain an alternate information thermodynamic inequality. In this inequality, the information gain is the entanglement gain; the new information content IE represents the amount of the entanglement between the system and the reference system which the probe takes from the system. The new information content depends only on the premeasurement state of the system and the unitary interaction between the probe and the system, and thus when IE is determined, we can take {Pˆ(k) } freely. The QC-mutual information IQC does not have this freedom. Theorems 4 and 5 follow from this difference of the freedom between IE and IQC . Thus, in the above context, we can state that the substance of information is the entanglement. The information gain is already completed when the unitary interaction is over and the projective measurement is only the interpretation of the information.
(0 ϕ π )
KP S ≡ CP2 S + τP SR1 ,
(A4)
KP R1 ≡ CP2 R1 + τP SR1 ,
(A5)
2 KSR1 ≡ CSR + τP SR1 , 1
(A6)
J5 ≡ 4λ20 |λ1 λ4 eiϕ − λ2 λ3 |2 + λ22 λ23 − λ21 λ24 ,
(A7)
K5 ≡ J5 + τP SR1 ,
(A8)
J ≡ K52 − KP S KP R1 KSR1 ,
(A9)
λ2 λ3 − λ1 λ4 eiϕ e−i ϕ˜5 ≡ , |λ2 λ3 − λ1 λ4 eiϕ |
+ J τ P SR 5
, Qe ≡ sgn sin ϕ λ20 − 2 1 2 CSR1 + τP SR1
(A10) (A11)
where τP SR1 is the tangle of |ψP SR1 and sgn[x] is the sign function, x/|x| (x = 0) sgn[x] = (A12) 0 (x = 0). When Qe = 0, there are two possible decompositions which satisfy Eq. (A3). We then choose the decomposition with a greater coefficient λ0 . Now we have completed the preparation. In the basis of Eq. (A3), the projective measurement {P˜(k) }k=0,1 is given as follows: a ke−iθ 1 − a −ke−iθ ˜ ˜ , P(1) = , P(0) = keiθ b −keiθ 1 − b (A13) where the measurement parameters a, b, k, and θ are defined as follows: √ 2 ˙ J CSR K5 τP SR1 ± 1 1 a= − , (A14) 2 2K 2 2 SR1 K5 − KP S KP R1 CSR1 b = 1 − a, k = a(1 − a),
(A15)
θ = −ϕ˜5 ,
(A17)
(A16)
˙ means −Qe and when Qe = 0 and when Qe = 0 the mark ± ˙ means −. By using Eqs. (A14)–(A17) and the mark ± Lemma 1 of Ref. [15] and after straightforward algebra, we
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can confirm that the measurement {P˜(k) } is the measurement that we sought. APPENIDX B: A TOY MODEL
In the present appendix, we give an example which satisfies the equality of (2). Let us consider for the system S a qubit with the Hamiltonian Hˆ S = −λx σˆ x − λy σˆ y − λz σˆ z ,
(B1)
where σˆ x , σˆ y , and σˆ z are the Pauli matrices and λx , λy , and λz are the control parameters. We can interpret the control parameters as a magnetic field on the qubit S. Let the control parameters (λx ,λy ,λz ) initially set to (0,0,1) and the qubit initially set in the canonical distribution of temperature T . We add the reference system R and make SR a pure state. We also prepare the probe P , which is initialized to a state |0P . We perform the following unitary interaction Uˆ SP between S and P : −2β 1 − e 1 + e−2β |0P 0S 0P 0S | − |1P 0S 0P 0S | Uˆ SP = 2 2 1 + e−2β |0P 0S 1P 0S | + 2 1 − e−2β |1P 0S 1P 0S | + 2 (B2) + |1P 1S 0P 1S | + |0P 1S 1P 1S | . Then, the concurrence and the tangle of SP R change as shown in Fig. 2 during the interaction USP . Thus, because of Eqs. (10)
and (A2), the entanglement information is IE = −EF = EFSB -R before Uˆ SP − EFSB -R after Uˆ SP = EFS -R before Uˆ SP − EFS -R after Uˆ SP ⎛ ⎞ 2(1−e−2β ) 1 + 1 − (e β +e−β )2 eβ ⎠, =h β − h⎝ e + e−β 2
(B3) (B4) (B5)
where h(x) ≡ −x ln x − (1 − x) ln(1 − x). After the unitary interaction Uˆ SP , we perform the following projective measurement {Pˆ(k) } on P : 1 1 1 1 1 −1 ˆ ˆ , P(1) = . (B6) P(0) = 2 1 1 2 −1 1 Then, we perform the following feedback control: (1) When the result of the measurement {Pˆ(k) } is k = 0, we first set (λx ,λy ,λz ) to (1,0,0) from t = 0 to t = π . Second, we quasistatically vary (λx ,λy ,λz ) from (0,0,1) √ to (0,0,β −1 Arctanh[ tanhβ]). The work which is extracted from √S during√ this feedback is W (0) = −tanh[β] + β −1 Arctanh[ tanhβ] tanhβ. (2) When the result of the measurement {Pˆ(k) } is k = 1, we quasistatically vary (λx ,λy ,λz ) from (0,0,1) √ to (0,0,β −1 Arctanh[ tanhβ]). The work which is extracted from √S during√ this feedback is W (1) = −tanh[β] + β −1 Arctanh[ tanhβ] tanhβ. At the end of the feedback control, the system S is in the canonical distribution √ of temperature T with the Hamiltonian −β −1 Arctanh[ tanhβ]σˆ z . Thus, the difference of the Helmholtz free energy is 1 ln[2coshβ] β 1 + ln{2cosh[Arctanh( tanhβ)]}. β
−F = −
(B7)
Because of W (0) = W (1) , the average work Wext , on the other hand, is 1 Wext = −tanh[β] + Arctanh[ tanhβ] tanhβ. (B8) β With a straightforward algebra, we have Wext − (−F ) 1 = −tanhβ + ln [2coshβ] β 1 − (−Arctanh[ tanhβ] tanhβ β + ln{2cosh[Arctanh( tanhβ)]}) ⎛ ⎡ β 1 + 1− 1 e ⎝ = ⎣h β − h β e + e−β 2
2(1−e−2β ) (eβ +e−β )2
⎞⎤ ⎠⎦ . (B9)
FIG. 2. The entanglement transfer from S to P during the interaction USP . The straight line denotes the concurrence and the circle denotes the tangle. Note that the sum of the concurrence and the tangle is conserved.
The right-hand side of Eq. (B9) is equal to −kB T EF . Thus, the equality of (2) holds in the present information thermodynamic process; the extracted work can exceed the upper bound of the conventional thermodynamics just by the amount of the entanglement extracted by the measurement.
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The above achievement of the upper limit of Eq. (2) is given by the best interpretation of the information; if we perform a projective measurement other than {Pˆ(k) }, for example, the measurement {Pˆ(k) }, 1 0 Pˆ(0) = , 0 0
0 0 Pˆ(1) = , 0 1
(B10)
we cannot achieve the upper limit of Eq. (2) with any feedback. For the present S, we can choose the best projective
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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measurement with which we can achieve the upper limit of Eq. (2) for arbitrary USP , by using Eqs. (A13)–(A17). } is an example of We finally see that the above {Pˆ(k) unachievableness of the Sagawa-Ueda inequality (1). Note that the general measurement which consists of Uˆ SP and {Pˆ(k) } is not deterministic. Namely, the state which corresponds to the results k = 0 is not LU equivalent to that of k = 1. Thus, because of the proof of Theorem 5, we cannot achieve the equality of (1) by using any finite heat bath; we cannot achieve the equality of (1) with any feedback controls.
[11] J. M. Horowitz and J. M. R. Parrondo, New J. Phys. 13, 123019 (2011). [12] S. De Liberato and M. Ueda, Phys. Rev. E 84, 051122 (2011). [13] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [14] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). [15] H. Tajima, Ann. Phys. (NY) 329, 1 (2013). [16] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [17] A. Ac´ın, A. Andrianov, L. Costa, E. Jan´e, J. I. Latorre, and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000).
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