Inequalities in Rényi's quantum thermodynamics
Natália Bebiano, João da Providência, João Pinheiro da Providência
aa r X i v : . [ qu a n t - ph ] J un Inequalities in R´enyi’s quantum thermodynamics
N. Bebiano ∗ , J. da Providˆencia † and J.P. da Providˆencia ‡ March 7, 2018
Abstract
A theory of thermodynamics has been recently formulated and derived onthe basis of R´enyi entropy and its relative versions. In this framework, we definethe concepts of partition function, internal energy and free energy, and funda-mental quantum thermodynamical inequalities are deduced. In the context ofR´enyi’s thermodynamics, the variational Helmholtz principle is stated and thecondition of equilibrium is analyzed. The R´enyi maximum entropy principleis formulated and the equality case is discussed. The obtained results reduceto the von Neumann ones when the R´enyi entropic parameter α approaches 1.The Heisenberg and Schr¨odinger uncertainty principles on the measurementsof quantum observables are revisited. The presentation is self-contained andthe proofs only use standard matrix analysis techniques. Keywords:
R´enyi entropy, R´enyi relative entropy, partition function, Helmholtzfree energy, uncertainty principles,
A complete theory of thermodynamics has been recently formulated and derivedon the basis of the R´enyi entropy and its relative version [14], which are crucial, for ∗ CMUC, University of Coimbra, Department of Mathematics, P 3001-454 Coimbra, Portugal([email protected]) † CFisUC, University of Coimbra, Department of Physics, P 3004-516 Coimbra, Portugal (provi-dencia@teor.fis.uc.pt) ‡ Department of Physics, Univ. of Beira Interior, P-6201-001 Covilh˜a, Portugal([email protected]) H, defined in acomplex Hilbert space with finite dimension, and being described by an arbitrary density matrix ρ , i.e., a positive definite matrix with trace 1. In statistical physics,isolated systems are described by microcanonical ensembles and systems in equilib-rium with a heat bath are described by canonical ensembles . The canonical ensembleis not adequate for the statistical description of systems with a small number ofparticles compared with Avogadro’s number, such as a DNA molecule, while the mi-crocanonical ensemble is hard to handle. This fact explains the interest on statisticaldescriptions based on different definitions of entropy from the von Neumann entropy,such as the Tsallis or the R´enyi’s entropies.According to classical thermodynamics, the entropy of a thermally isolated systemis maximal for the equilibrium state ( maximum entropy principle ). The Helmholtzfree energy of a system in thermal contact with its environment, or with a heat bathcharacterized by a temperature T , is minimal for the equilibrium state ( minimumfree energy principle ).This paper is organized as follows. In Section 2 we define the R´enyi internalenergy and the R´enyi entropy of a physical system in terms of the density matrix ρ ,and, in accordance with the principles of thermodynamics, we determine the state ofequilibrium of the system by minimizing, at constant temperature, the Helmholtz freeenergy. In section 3, the close relation between the R´enyi relative entropy and theHelmholtz free energy is discussed. Since the state described by the density matrix ρ iscompletely arbitrary, it is not characterized by a well defined temperature. In Section4, we investigate the relation between the partition function and the internal energy,for arbitrary temperature. In Section 5, the R´enyi maximum entropy principle isformulated. In Section 6, the uncertainty principle on the measurements of quantumobservables is revisited. In Section 7, the obtained results are discussed and someopen problems are formulated. Let M n be the matrix algebra of n × n matrices with complex entries and H n thevector space of Hermitian matrices, named in physics as observables . By H n, + wedenote the cone of Hermitian positive definite matrices and H n, + , consists of positive2ermitian matrices with unit trace, called the state space . This set coincides with theclass of density matrices acting on an n × n quantum system, and we use the termsstate and density matrix synonymously. Matrices in H n, + with rank one describe pure states and those with rank greater than one represent mixed states.Throughout we use the conventions 0 log 0 = 0 , log 0 = −∞ and log ∞ = ∞ . Fora density matrix ρ with eigenvalues ρ ≥ . . . ≥ ρ n , the α - R´enyi entropy [14] is definedas S α ( ρ ) =: log Tr ρ α − α = log P ni =1 ρ αi − α , α ∈ (0 , ∪ (1 , ∞ ) . (1)If α >
1, then Tr ρ α < ρ α < . If α <
1, we have Tr ρ α > ρ α > . Hence, S α ( ρ ) ≥ ρ , and equality holds if and onlyif ρ is a pure state. For ρ = . . . = ρ n = 1 /n , we obtain S α ( ρ ) = log n , which is themaximum possible value of S α ( ρ ). Therefore,0 ≤ S α ( ρ ) ≤ log n. To avoid dividing by zero in (1), we consider α = 1 , but l’Hˆopital rule shows thatthe α -R´enyi entropy approaches the Shannon entropy S [21] as α approaches 1: S ( ρ ) = lim α → S α ( ρ ) = − Tr ρ log ρ. The special cases α = 0 and α = ∞ may be defined by taking the limit. Inphysics, many uses of R´enyi entropy involve the limiting cases S ( ρ ) = lim α → S α ( ρ )and S ∞ ( ρ ) = lim α →∞ S α ( ρ ), known as “max-entropy” and “min-entropy”, as S α ( ρ ) isa monotonically decreasing function of α : S α ( ρ ) ≤ S α ′ ( ρ ) for α < α ′ . Min-entropy is the smallest entropy measure in the class of R´enyi entropies and itis the strongest measure of information content of a discrete quantum variable. It isnever larger than the Shannon entropy S .A function g : H n → R , is concave if, for A , A ∈ H n , 0 ≤ p ≤
1, the followingholds, g ( pA + (1 − p ) A ) ≥ pg ( A ) + (1 − p ) g ( A ) . Theorem 2.1
R´enyi’s entropy map S α : H n, + , → R for < α < is concave. Proof.
This is a simple consequence of the concavity of both x α , for α <
1, andlog x . 3or α > x α is convex, so S α ( ρ ) is neither purely convex nor concave.The α - R´enyi relative entropy ( α -RRE) [18] between two quantum states ρ ∈ H n, + , and σ ∈ H n, + is defined by D α ( ρ k σ ) = log Tr( ρ α σ − α ) α − , α ∈ (0 , ∪ (1 , ∞ ) . The special cases α = 1 and α = ∞ are defined taking the limit.The α -RRE satisfies D α ( U ∗ ρU k U ∗ σU ) = D α ( ρ k σ )for all unitary matrices U . If ρ and σ commute they are simultaneously diagonalizableand so D α ( ρ k σ ) = P ni =1 ρ αi σ − αi α − , where ρ i and σ i are respectively the eigenvalues (with simultaneous eigenvectors) of ρ and σ .Computing Tr( ρ α σ − α ) for small values of 1 − α, we findTr( ρ α σ − α ) = Tre α log ρ e (1 − α ) log σ = Tre log ρ e ( α −
1) log ρ e (1 − α ) log σ = Tr ρ (1 + ( α − ρ − log σ ) + O ((1 − α ) ))= 1 + ( α − ρ (log ρ − log σ ) + O ((1 − α ) ) . Thus, D α ( ρ k σ ) = Tr ρ (log ρ − log σ ) + O ((1 − α )) , and so when α → , one obtainsthe von Neumann relative entropy [16]: D ( ρ k σ ) = Tr ρ (log ρ − log σ ) . A map g : H n × H n → R , is jointly convex , if, for A , A , B , B ∈ H n , 0 ≤ λ ≤ g ( λA + (1 − λ ) A , λB + (1 − λ ) B ) ≤ λg ( A , B ) + (1 − λ ) g ( A , B ) , and g is jointly concave if − g is jointly convex.The joint convexity of α -RRE for α ∈ (0 ,
1) is one of its most important properties.This result was obtained obtained in [7] in a more general context, and next we givea simple proof. For this purpose, Lieb’s joint concavity Theorem [20] stated in thefollowing Lemma, is needed. 4 emma 2.1
For all matrices K ∈ M n , A, B ∈ H n, + and all q, r such that ≤ q ≤ , ≤ r ≤ with q + r ≤ , the real valued function Tr K ∗ A q KB r is jointly concave in A, B.
Theorem 2.2
The map D α : H n, + , × H n, + → R is jointly convex for α ∈ (0 , . Proof.
Consider in Lemma 2.1, r = 1 − α , q = α , α ∈ (0 ,
1) and K = I n . For ρ , ρ ∈ H n, + , , σ , σ ∈ H n, + , ≤ λ ≤
1, and the real valued function g ( ρ, σ ) = Tr ρ α σ − α , the lemma ensures that g ( λρ + (1 − λ ) ρ , λσ + (1 − λ ) σ ) ≤ λg ( ρ , σ ) + (1 − λ ) g ( ρ , σ ) . Since log x/ ( α −
1) for α ∈ (0 ,
1) is a decreasing and convex function of x , we getlog( g ( λρ + (1 − λ ) ρ , λσ + (1 − λ ) σ )) α − ≤ log( λg ( ρ , σ ) + (1 − λ ) g ( ρ , σ )) α − ≤ λ log g ( ρ , σ ) α − − λ ) log g ( ρ , σ ) α − , and the result follows. Corollary 2.1
The von Neumann map D ( ρ k σ ) : H n, + , × H n, + , → R is jointlyconvex. Proof.
The result follows taking the limit α → α -RRE The following result extends the well known non negativity property of von Neu-mann relative entropy: D ( ρ k σ ) ≥ ρ, σ such that Tr ρ = Tr σ = 1 . Theorem 2.3
Let σ ∈ H n, + . Then, for ρ ranging over H n, + , , D α ( ρ k σ ) ≥ − log Tr σ, α ∈ (0 , ∪ (1 , ∞ ) . Equality occurs if and only if ρ = σ/ Tr σ. roof. For α <
1, minimizing D α ( ρ k σ ) for a fixed σ is equivalent to minimizing T = Tr( ρ α σ − α ) . For α >
1, minimizing D α ( ρ k σ ) for a fixed σ is equivalent to maximizing T . Next, weoptimize T . Suppose that the matrices ρ, σ are such that T is optimal. Since the traceis unitarily invariant, without loss of generality, we can take σ in diagonal form. Then,for ǫ > S arbitrary in H n , we have e iǫS = I n + iǫS + O ( ǫ ) , and so dd ǫ Tr( ρ α e iǫS σ − α e − iǫS ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = i Tr S [ σ − α , ρ α ] = 0 , where [ σ − α , ρ α ] = σ − α ρ α − ρ α σ − α . implying that [ ρ α , σ − α ] = [ ρ, σ ] = 0 . As a consequence, the Hermitian matrices ρ, σ are simultaneously unitarily diago-nalizable. Since the trace is unitarily invariant, without loss of generality we mayassume ρ, σ in diagonal form, ρ = diag( ρ , . . . , ρ n ), σ = diag( σ , . . . , σ n ). As log x isan increasing function of the argument x , we find T ≤ n X i =1 e α log ρ i +(1 − α ) log σ i = n X i =1 ρ αi σ (1 − α ) i , α ≤ . and T ≥ n X i =1 e α log ρ i +(1 − α ) log σ i = n X i =1 ρ αi σ (1 − α ) i , α ≥ . Thus, D α ( ρ k σ ) ≥ log P ni =1 ρ αi σ (1 − α ) i α − ≥ − log n X j =1 σ j . Next, we optimize P ni =1 ρ αi σ (1 − α ) i under the constraint P ni =1 ρ i = 1, using Lagrangemultipliers techniques. We consider the function ψ = n X i =1 ρ αi σ (1 − α ) i − λ n X i =1 ρ i − ! , λ ∈ R . The extremum condition leads to ∂ψ∂ρ i = αρ α − i σ − αi − λ = 0 , so that ρ i = (cid:18) λα (cid:19) / ( α − σ i . The Lagrange multiplier λ is determined observing that P ni =1 ρ i = 1 . Thus,( λ/α ) / ( α − = 1 / P ni =1 σ i and so ρ i = σ i / P nj =1 σ j . The asserted result is finallyobtained. 6 The R´enyi-Peierls-Bogoliubov inequality
In statistical mechanics, the absolute temperature is usually denoted by T , andits inverse, 1 /T , by β . The internal energy is defined as the expectation value of theHamiltonian H in the state ρ , i.e., Tr ρH . Here, we are assuming that β = 1 /T = 1.The α - expectation value of the Hermitian operator H is defined and denoted as h H i α := 1 α − ρ α e ( α − H Tr ρ α , α ∈ (0 , ∪ (1 , ∞ ) . where ρ ∈ H n, + , . We define, for β =1, the α - R´enyi internal energy ( α -RIE) as the α -expectationvalue of H in the state ρ , E α ( ρ, H ) := h H i α = 1 α − ρ α e ( α − H Tr ρ α , α ∈ (0 , ∪ (1 , ∞ ) . (2)We remark that some authors define differently the α -RIE, according toTr ρ α H Tr ρ α , that is, as the average of H in the state ρ α . In the definition we are proposing, thelogarithm of the average of e ( α − H in state ρ α , multiplied by 1 / ( α − , is considered.This option considerably simplifies the formalism involved in the thermodynamicalconsiderations. On the other hand, it may be easily shown that, for α → E α ( ρ, H )approaches the standard expectation value of the Hamiltonian and of the internalenergy arising in statistical thermodynamics, E ( ρ, H ) = lim α → E α ( ρ, H ) = Tr ρH. We define (for β = 1) the α - R´enyi free energy ( α -RFE) as F α ( ρ, H ) := E α ( ρ, H ) − S α ( ρ ) = log Tr ρ α e ( α − H α − , α ∈ (0 , ∪ (1 , ∞ ) . Notice that F α ( ρ, H ) is closely related to the α -RRE, as F α ( ρ, H ) = D α ( ρ k e − H ) . According to the principles of thermodynamics, the state of equilibrium of a sys-tem is the one for which the free energy is minimized, at constant temperature.7he
Helmholtz state , which is the equilibrium state, is obtained by minimizing theHelmholtz free energy (for fixed temperature).The next Theorem characterizes, from the knowledge of H , the state which mini-mizes the α -RFE, the so called the equilibrium state of the system. This result is alsoknown as the Helmholtz free energy variational principle. Theorem 3.1 (R´enyi-Peierls-Bogoliubov inequality) Let H ∈ H n be given and ρ ∈ H n, + , be arbitrary. Then, F α ( ρ, H ) ≥ − log Tre − H , α ∈ (0 , ∪ (1 , ∞ ) . Equality occurs if and only if ρ = e − H / Tre − H . Proof.
Replacing in Theorem 2.3 σ by e − H , the result follows.If the state of equilibrium ρ is known, then the Hamiltonian of the system isobtained as H = − log ρ − log Tre − H I n , where I n ∈ M n is the identity matrix.Consider H as a perturbation of the Hamiltonian H . So, H may be regarded asa convenient approximation of H. The following result provides useful information onTre − H from Tre − H . Corollary 3.1
For
H, H ∈ H n , we have α − − αH e ( α − H Tre − H ≥ − log Tre − H Tre − H . Proof.
Considering, in Theorem 3.1, ρ = e − H / Tre − H , the result follows by a trivialcomputation. The partition function Z β = Tre − βH , (3)where β = 1 /T denotes the inverse of the absolute temperature and H is the Hamilto-nian of the physical system, plays a fundamental role in standard statistical thermo-dynamics. The discussion of some issues requires the consideration of the parameter β , so we will relax the restriction β = 1, which has been adopted up to now. Instandard statistical thermodynamics, the equilibrium properties of the system are8ncapsulated into the logarithm of the partition function. In particular, the internalenergy E β = Tr H e − βH Tre − βH is related to the derivative of log Z β with respect to β as E β = − d log Z β d β . So, the following question naturally arises. What is the relation between the internalenergy and the partition function in the context of R´enyi thermodynamics? Noticethat in R´enyi thermodynamics the partition function is as meaningful as in standardstatistical mechanics, because the expression of the equilibrium state in the R´enyithermodynamics coincides with the corresponding expression in the von Neumannsetting, ρ = ρ := e − H / Tre − H .Next we derive a relation between the internal energy and log Z β , in R´enyi’s ther-modynamics. For this purpose we define the α - derivative of the function ψ : R → R as the quotient ψ ( βα ) − ψ ( β ) β ( α − . Replacing ρ by e − H / Tre − H in (2), we conclude, from Corollary (3.1), that the R´enyi equilibrium internal energy (for β = 1) reduces to E α ( ρ , H ) = 1 α − − H − log Tre − αH ) . (4) Proposition 4.1
For β = 1 , the R´enyi equilibrium internal energy is the α derivativeof − log Z β , taken at β = 1 . Proof.
Having in mind (4) and that the logarithm of the partition function, forarbitrary β , reads log Z β = log Tre − βH , we get E α ( ρ , H ) = − log Z α − log Z α − − log Z βα − log Z β β ( α − (cid:12)(cid:12)(cid:12)(cid:12) β =1 , and the result follows.Since − log Z = E α ( ρ , H ) − S α ( ρ ) , the relation between the partition functionand the internal energy also determines the entropy S α ( ρ ).The discussion in this Section is analogous to the arguments in [1].9 R´enyi maximum entropy principle
In order to formulate the maximum entropy principle (MaxEnt) in the context ofR´enyi thermodynamics we introduce the concept of R´enyi internal energy for β = 1,as a generalization of (2) E α,β ( ρ, H ) := 1 β h βH i α = log Tr ρ α e ( α − βH − log Tr ρ α β ( α − . (5)The parameter β controls, or tunes, the internal energy. Proposition 5.1
For arbitrary β , the R´enyi equilibrium internal energy is the α derivative of − log Z β . Proof.
For β = 1, the R´enyi equilibrium internal energy reduces to E α,β ( ρ , H ) = log Tre − βH − log Tre − αβH β ( α −
1) = log Z αβ − log Z β β ( α − , and the result follows. Proposition 5.2
The R´enyi equilibrium internal energy is a monotonously decreas-ing function of β . Proof.
We have that − log Z β is a convex function of β as log Z β is concave, becaused log Z β d β = Tr H e − βH Tre − βH − (cid:18) Tr H e − βH Tre − βH (cid:19) ≥ . Now, observing that equality occurs only in the limit β → ∞ , we conclude that E α,β ( ρ , H ), being, according to Proposition 5.1, the slope of the secant line throughthe points ( β, − log Z β ) and ( αβ, − log Z αβ ) , decreases as β increases.We remark that E α,β ( ρ , H ) lies in the interval defined by the lowest and thehighest eigenvalue of H . This follows, observing that for β = ±∞ these eigenvaluesare reached, λ min ( H ) ≤ E α,β ( ρ , H ) ≤ λ max ( H ) . For ρ = ρ , E α,β ( ρ, H ) may not be in that interval. Proposition 5.3
For ρ the β -dependent equilibrium state, and for λ min ( H ) , λ max ( H ) the lowest and the highest eigenvalue of H , respectively, we have lim β →∞ E α,β ( ρ , H ) = λ min ( H ) , lim β →−∞ E α,β ( ρ , H ) = λ max ( H ) . roof. The result follows keeping in mind the convexity of − log Z β and that λ min ( H ), λ max ( H ) are the slopes of the asymptotes to − log Z β .For β = 1, we define the α - R´enyi free energy F α,β ( ρ, H ) as F α,β ( ρ, H ) := E α,β ( ρ, H ) − β S α ( ρ ) = log Tr ρ α e ( α − βH β ( α − . The maximum entropy principle states that the equilibrium state ρ is obtained bymaximizing − βF α,β ( ρ, H ) with respect to ρ , under the constraint P ni =1 ρ i = 1, which,for β ≥ , is equivalent to minimizing F α,β ( ρ, H ) under the same constraint. Replacingin Theorem 3.1 σ by e − βH , we obtain Theorem 5.1 (R´enyi-Peierls-Bogoliubov-inequality) Let H ∈ H n be given and ρ ∈ H n, + , be arbitrary. Then, βF α,β ( ρ, H ) ≥ − log Tre − βH , α ∈ (0 , ∪ (1 , ∞ ) , β ∈ R . Equality occurs if and only if ρ = e − βH / Tre − βH . This result is in agreement with the corresponding expression in von Neumann statis-tical mechanics. We observe that, in conventional thermodynamics, β ≥ . However,if n is finite, it is also meaningful to consider β < . The equilibrium state depends only on the value of the parameter β , which isdetermined by the required value of the internal energy. The uncertainty principle was formulated by Heisenberg in 1927 and states that itis not possible to measure simultaneously, with absolute precision, the position oper-ator x and the momentum operator p of a particle. These operators are considered inthe one dimensional context. The product of the uncertainties in the respective mea-surements ∆ x and ∆ p , is of the order of Plank’s constant ~ . We consider units suchthat ~ = 1 . This indeterminacy relation may be formulated in precise mathematicalform as ∆ x ∆ p ≥ ~ . The Heisenberg-Robertson uncertainty principle, firstly proposed by Heisenbergand then generalized by Robertson [19] gives a lower bound for the product of thestandard deviation of two observables. To state it, we introduce some useful concepts.11or A ∈ H n , the expectation value of the measurement of the observable A in the state ρ ∈ H n, + , is h A i = Tr ρA Tr ρ . The variance in the measurement of A is defined as σ A = 1Tr ρ Tr ρ ( A − h A i ) . The uncertainty in the measurement of A is defined as the standard deviation σ A . Asusually, we denote the anticommutator of A, B as { A, B } = AB + BA.
The covariance of A, B ∈ H n is determined asCov( A, B ) = 1Tr ρ Tr ρ (cid:18) { A, B } − h A ih B i (cid:19) . Observe that Cov(
A, A ) = σ A , i.e., the variance is a particular case of the covariance,and Cov( A, B ) = Cov(
B, A ) . The Heisenberg-Robertson uncertainty relation statesthat σ A σ B ≥ |h [ A, B ] i| , and was improved by Schr¨odinger as σ A σ B ≥ |h [ A, B ] i| + 14 h{ A, B } − h A ih B ii . The following theorem gives a lower bound for the product of the standard devia-tions of two quantum obsevables:
Theorem 6.1
Let A and B be Hermitian matrices and ρ ∈ H n, + , . Then, σ A σ B ≥ Cov(
A, B ) + (cid:18) h i [ A, B ] i (cid:19) . (6) Equality occurs if and only if A is a multiple of B . Proof.
We observe that i [ A, B ] is Hermitian, as ( i [ A, B ]) ∗ = i [ A, B ]. Let A ′ ρ := ρ / (Tr ρ ) / ( A − h A i ) , B ′ ρ := ρ / (Tr ρ ) / ( B − h B i ) . We easily find σ A = Tr( A ′ ρ A ′∗ ρ ) , σ B = Tr( B ′ ρ B ′∗ ρ )12ndTr( A ′ ρ B ′∗ ρ ) = 1Tr ρ Tr ρ ( A − h A i )( B − h B i ) = 1Tr ρ Tr ρ (cid:18) { B, A } + 12 [ B, A ] − h A ih B i (cid:19) . On the other hand,Tr( B ′ ρ A ′∗ ρ ) = 1Tr ρ Tr ρ (cid:18) { B, A } −
12 [
B, A ] − h A ih B i (cid:19) , so that Tr( A ′ ρ B ′∗ ρ ) = 12 h{ B, A }i − h A ih B i + 12 i h i [ B, A ] i , and Tr( B ′ ρ A ′∗ ρ ) = 12 h{ B, A }i − h A ih B i − i h i [ B, A ] i . According to the matricial Schwartz inequality, we haveTr( A ′ ρ A ′∗ ρ )Tr( B ′ ρ B ′∗ ρ ) ≥ Tr( A ′ ρ B ′∗ ρ )Tr( B ′ ρ A ′∗ ρ ) . Equality occurs if and only if A ′ ρ is a multiple of B ′ ρ , that is, if and only if A is amultiple of B .We present the relation (6) in a form susceptible of extension. Let us introducethe covariance matrix σ ( A, B ) = (cid:20) σ A Cov(
A, B )Cov(
A, B ) σ B (cid:21) . The inequality in (6) can be expressed asdet σ ( A, B ) ≥ (cid:18) h i [ A, B ] i (cid:19) . For m observables { X k } mk =1 , let X ′ jρ := (Tr ρ ) − / ρ / ( X j − h X j i ) , j = 1 , . . . , m. Then Tr X ′ jρ X ′ k ∗ ρ = 1Tr ρ Tr ρ ( X j X k − h X j ih X k i ) = Cov( X j , X k ) − i h i [ X j , X k ] i where h i [ X j , X k ] i = 1Tr ρ Tr ρ ( i [ X j , X k ]) . Notice that Cov( X j , X k ) = 12 (Tr X ′ j ρ X ′ k ∗ ρ + Tr X ′ kρ X ′ j ∗ ρ ) (7)13nd − i h i [ X j , X k ] i = 12 (Tr X ′ jρ X ′ k ∗ ρ − Tr X ′ kρ X ′ j ∗ ρ ) . We consider the m × m covariance matrix σ ( X , . . . , X m ) = Cov( X , X ) . . . Cov( X , X m )... . . . ...Cov( X m , X ) . . . Cov( X m , X m ) (8)and the matrix formed by the measurements of the commutators of the observables, δ ( X , . . . , X m ) = − i h i [ X , X ] i . . . − i h i [ X , X m ] i ... . . . ... − i h i [ X m , X ] i . . . − i h i [ X m , X m ] i . (9)The m × m matrix τ = Tr X ′ ρ X ′ ∗ ρ . . . Tr X ′ X ′ m ∗ ρ ... . . . ...Tr X ′ mρ X ′ ∗ ρ . . . Tr X ′ mρ X ′ m ∗ ρ . is positive semidefinite, as z ∗ τ z ≥ z ∈ C m . In fact, τ may be seen as theGram matrix of the operators X kl with respect to the Hilbert-Schmidt inner product h Y, X i = Tr X ∗ Y. Obviously, τ = σ + δ. Theorem 6.2
For σ ( X , . . . , X m ) , δ ( X , . . . , X m ) in (8), (9), such that σ ( X , . . . , X m ) + δ ( X , . . . , X m ) is positive definite and m an even number, wehave det σ ( X , . . . , X m ) > det iδ ( X , . . . , X m ) . (10)To prove this result we present an auxiliary Lemma. Lemma 6.1
For C a positive definite matrix with even dimension, with A = ( C + C T ) / and B = ( C − C T ) / , we have det A > det iB.
Proof.
By hypothesis, C is positive definite, so it is clear that B ∈ H n and A ispositive definite. We consider the characteristic polynomialdet( λA + B ) . A is symmetric and B antisymmetric, the condition det( λA + B ) = 0 impliesdet( λA − B ) = det( λ A − B ) = 0, so that the characteristic roots occur in symmetricpairs. Let U be a unitary matrix such that U ∗ A − / BA − / U = diag( λ , . . . λ m ) . Then B = A / U diag( λ , . . . λ m ) U ∗ A / , A = A / U U ∗ A / , and C = A + B = A / U diag(1 + λ , . . . λ m ) U ∗ A / implying that λ , . . . , λ m ∈ [ − , . Thus,det( U ∗ A − / iBA − / U ) = det iB det A = ( − m/ λ . . . λ m < . Observing that ( − m/ λ . . . λ m >
0, the result follows.The proof of Theorem 6.2 is a simple consequence of Lemma 6.1, observing that σ ( X , . . . , X m ) = ( τ + τ T ) / δ ( X , . . . , X m ) = ( τ − τ T ) / Corollary 6.1
For δ ( X , . . . , X m ) in (9) and σ j = Cov( X j , X j ) in (7), m Y j =1 σ j ≥ det iδ ( X , . . . , X m ) . Proof.
Notice that σ ( X , . . . , X m ) in (8) is positive definite. From Theorem (6.1)and Hadamard determinantal inequality, we obtain m Y j =1 σ j ≥ det σ ( X , . . . , X m ) , and the result follows. α -variance The α - expectation value of the Hermitian operator A has been defined as h A i α = 1 α − ρ α e ( α − A Tr ρ α , where ρ ∈ H n, + , . If A >
0, then h A i α > . The α expectation value is strongly nonlinear. We observe that, for A, B ∈ H n , λ ∈ R , we have, in general, h γA i α = γ h A i α h A + B i α = h A i α + h B i α except for h γI n i α = γ = γ h I n i α and h A + γI n i α = h A i α + γ. Notice that h ( A − h A i α I n ) i α = 0 and that ( A − h A i α I n ) > . The α - variance in themeasurement of A may be naturally defined as σ A,α = Tr ρ ( A − h A i α I n ) . This definition is consistent with the one for α = 1, since lim α → σ A,α = σ A . Thederivation and physical interpretation of inequalities analogous to (6) and (10), for α ∈ (0 , Example 6.1
We consider the Hamiltonian H = diag(3 , , , , , , , , , , , ∈ M , and compute, in the equilibrium state, for β = 1 , the α -expectation value and the α -standard deviation for α ∈ { , / , , , ∞} . We have found the following values α = 0 , h H i α = 2 . , σ H,α = 1 . . α = 1 / , h H i α = 2 . , σ H,α = 1 . . α = 1 , h H i α = 1 . , σ H,α = 1 . . α = 2 , h H i α = 1 . , σ H,α = 1 . .α = ∞ , h H i α = 1 , σ H,α = 1 . .The equilibrium free energy F α ( ρ , H ) = 0 . does not depend on α , so, theentropy of the equilibrium state S α ( ρ ) = h H i α + F α ( ρ , H ) has also been determined.For α = ∞ , h H i α is equal to the lowest eigenvalue of H . We notice that h H i α decreases as α increases, and that the α -standard deviation of the measurement of H is highest for α = 1 . We have presented self-contained proofs of fundamental inequalities in the settingof R´enyi’s statistical thermodynamics, which is formulated through the replacements,of h βH i and of S ( ρ ), in the expression of the free energy, respectively, by h βH i α and S α ( ρ ), for α a parameter in (0 , ∪ (1 , ∞ ). Definitions for thermodynamical16uantities, such as free energy, entropy and partition function were given. We adoptedthe paradigm in [14, 22] for dealing with thermodynamical processes in the frameworkof quantum theory. By assuming the laws of thermodynamics, the equilibrium stateof a given system is determined. The R´enyi MaxEnt principle has been stated andthe equilibrium state has been determined.Uncertainty relations have been revisited in the present context. It has been shownthat the product of the uncertainties on the measurements of an even number of ob-servables can not be less than a certain function of their commutators. This extendsthe uncertainty principles of Heisenberg and its refinement by Schr¨odinger, who in-troduced the correlations of two observables. The statement of these principles inR´enyi’s statistical thermodynamics is an open problem.Different types of uncertainty relations have been considered. There are many waysto quantify the uncertainties of measurements. The lower bound in the Heisenberg-Robertson formulation can happen to be zero, and so having a global state indepen-dent lower bound may be desirable.Entropic uncertainty relations have significant importance within quantum infor-mation providing the foundation for the security of quantum cryptographic protocols.Using majorization techniques, explicit lower bounds for the sum of R´enyi entropiesdescribing probability distributions have been derived. Some results admit general-izations to arbitrary mixed states.For α ∈ (0 , ∪ (1 , ∞ ) and ρ, σ ∈ H n, + , the sandwiched α -RRE is defined as D α ( ρ k σ ) = 1 α − (cid:16) Tr (cid:16) σ (1 − α ) α ρσ (1 − α ) α (cid:17) α (cid:17) and reduces to the α -RRE when α and ρ commute. The problems we have discussedmay also be considered in the context of this entropy.A demanding avenue of research is the study of operator R´enyi entropic inequali-ties. Acknowledgments
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