On the Conservation of Information in Quantum Physics

On the Conservation of Information in Quantum Physics Marco Roncaglia Physics Department an Research Center OPTIMAS, University of Kaiserslautern, Germany (Date: September 11, 2017 escribe the full informational content of quantum states. In this paper, we propose to treat separately the coherent an the incoherent contributions of the informational content of quantum states. Starting from the very basic principles of quantum mechanics, we will introuce the concept of coherent entropy, a quantity able to etect the information that quantum states convey in time. In this context, pure states contain more coherent information than mixe states, as the missing information has been converte into correlations with the environment. We will fin that the coherent information, associate to genuinely quantum phenomena, is inee conserve uner unitary processes. Informational content of quantum states A pure state is the eigenstate of some complete observable, whose measurement gives a fully preictable outcome. Hence, the associate zero entropy accounts for the absence of information obtainable by a repeate measurement over many copies. However, it appears reuctive to attribute a zero informational content to pure states. In fact, in contrast with single eterministic classical states, they represent the ieal resource for performing quantum tasks, like interference phenomena, quantum computation, an so on. An inicator beyon the (von-neumann entropy is neee to escribe such information. Let us take the textbook example of one qubit, i.e. a pure state in imension = 2 of the Hilbert space. Observables are represente by the set of Pauli matrices with eigenvalues {+1, 1}, where σ z has eigenstates {, }, an σ x has eigenstates { +, }. If our source emits quantum objects in the state = ( + + / 2, a measurement along σ x gives a ranom sequence of values +1 an 1, with equal probability p + = p = 1/2. In this case the entropy of information is one bit, i.e. the maximum obtainable for a ichotomic variable. Differently, if the observer measures along σ z, he obtains the constant sequence of +1, with zero entropy. This property of etecting ifferent entropies uner ifferent meaarxiv:1708.05727v2 [quant-ph] 7 Sep 2017 Accoring to quantum mechanics, the informational content of isolate systems oes not change in time. However, subaitivity of entropy seems to escribe an excess of information when we look at single parts of a composite systems an their correlations. Moreover, the balance between the entropic contributions coming from the various parts is not conserve uner unitary transformations. Reasoning on the basic concept of quantum mechanics, we fin that in such a picture an important term has been overlooke: the intrinsic quantum information encoe in the coherence of pure states. To fill this gap we are le to efine a quantity, that we call coherent entropy, which is necessary to account for the missing information an for re-establishing its conservation. Interestingly, the coherent entropy is foun to be equal to the information conveye in the future by quantum states. The perspective outline in this paper may be of some inspiration in several fiels, from founations of quantum mechanics to black-hole physics. PACS numbers: 03.67.-a, 03.65.Ta, 04.70.Dy Introuction Every physicist is confient with the principle of energy conservation an aware on its importance an implications. During time evolution in isolate systems, energy is converte from one form to another or transferre between ifferent subsystems, provie the total amount remains the same. However, when we consier information the picture is not so clear. In quantum physics, the conservation of information has been relate to no-cloning theorems [1], but apparently it has not been associate to a suitable conserve quantity. It is a well-known fact that the von-neumann entropy S(ρ = Tr(ρ log 2 ρ of any isolate quantum system with ensity operator ρ oes not change in time. This is a consequence of the fact that S(ρ epens only on the spectrum of ρ, an the unitarity of time evolution preserves the spectrum at the quantum level. In this sense, people say that any physical process governe by quantum mechanics information is never lost. However, this is a static vision that involves isolate quantum systems. So far, there is no complete theory able to clearly escribe how quantum information flows between interacting systems, accounting for a correct balance at any time. Whenever a system A, which is initially in a pure state ρ A = ψ A ψ A, is no more isolate because it interacts with another system B, we start to observe the increase of its mixeness, quantifie by the entropy S(ρ A, where ρ A = Tr B (ρ AB. If we look at the total entropy S(ρ A + S(ρ B, we see that it can only increase with time as a consequence of the subaitivity property S(ρ AB S(ρ A + S(ρ B [2, 3]. Moreover, interaction creates some correlations between A an B, with an aitional contribution of information, which can be measure by their mutual information I A:B. Such a escription is not governe by a balance equation of a conserve quantity, giving the illusion that information is create via interaction. Where oes all this information come from? It is clear that entropy is not the right quantity to

2 BS1 "i mirror BS2 "i which we efine as S( ρ, where ρ ij = δ ij ρ ij is the ensity operator where all the off-iagonal entries have been set to zero. Now, we efine the coherent entropy as S c (ρ = max σ U ρ [S( σ] min σ U ρ [S( σ], (1 #i mirror Figure 1. In a Mach-Zehner interferometer single photons after the beam splitter BS1 prouce a signal with maximum entropy S = 1. In fact, a photon etection just after BS1 gives ranom sequences of +1 an 1 (photons in the upper an the lower branch, respectively, with equal probability. Instea after BS2, the photons are all foun in the state, which yiel a signal with zero entropy. Such a recombination into a single state is a genuinely quantum phenomenon, since it is ue to interference between the two coherent beams coming from BS1. surements is genuinely quantum, as it is ultimately ue to interference: in the present example, istinct states ( +, coherently recombine into a single state (, thanks to their well-efine relative phases. In quantum optics, this example is realize by the Mach-Zehner interferometer (see Fig.1, where the path of a single-photon beam is split in two ifferent irections by a 50% reflective mirror (the beam splitter BS1 an then constructively recombine into a single path by a secon beam splitter BS2. A measure of the presence of the photon in the upper ( or lower branch ( after BS1 gives a ranom sequence of +1 an 1, with equal probability. At the output after BS2, the photon etection prouces a steay sequence of +1 s, which yiel a signal with zero entropy. As the role of the beam splitter is to rotate the basis of measurement, we euce that the entropy of the etecte signal epens in essence on the observable we choose. In the case of mixe states the effect of interference is reuce, so in every measurement basis we ect to have a resiual ranomness with a nonvanishing entropy; the limit case is the completely mixe state, where the entropy of the outcome is maximal for every measurement. Notice that an observer who etects the signal after BS1 is not able to istinguish between the pure case an the totally mixe one as they have the same statistics. However, it is important to introuce a measure able to account for the information carrie the in the former coherent case, ifferent from the entropy S(ρ which only quantifies the incoherent information in the latter case. Coherent entropy Once the measurement basis is fixe, the probability of obtaining a given output is encoe in the iagonal elements ρ ii, i = 1,..., of the ensity operator in that basis. The entropy of the output measurements is given by the iagonal entropy of ρ, where U ρ = {UρU : U M, UU = I} is the set of all matrices which are unitarily equivalent to ρ. In other wors, S c (ρ measures the ifference between the maximal an the minimal entropy of the outputs obtaine by measuring ρ over any possible observable. As we have seen, this ifference accounts for all the interference effects, so it has to be intene as a measure [4] of the coherent informational content of ρ. As it shoul be for a proper intrinsic property of a quantum state, S c (ρ is inepenent of any choice of measurement mae by the erimenter, i.e. it is invariant uner local unitary transformations. The apparently har optimization problem of evaluating Eq.(1 eventually leas to a very simple result: S c (ρ = log 2 S(ρ. (2 Proof - For every ensity operator σ U ρ, we have S( σ = Tr( σ log σ = Tr(σ log σ, as σ is the iagonal part of σ. The ifference S( σ S(σ = Tr[σ(log σ log σ] = S( σ σ 0 ue to the non negativity of the relative entropy S( σ σ, an the equality hols if σ = σ [5]. Since S(σ = S(ρ, σ U ρ, we get min σ Uρ [S( σ] = S(ρ. Intuitively, the operation of eleting the off-iagonal elements is a ecoherence operation (i.e. entropy increasing, which has no effect only in the basis where ρ is alreay iagonal. Regaring the first term in Eq.(1, we can say it is equal to the entropy of the totally mixe state, namely log 2. Inee, it is a less well-known fact that uner the most general unitary group, every ensity operator can be transforme into the matrix with iagonal elements uniformly equal to 1/ [8]. The assignment of a purely quantum entropic measure S c (ρ to a state through Eq. (2 says that the informational content of a pure state is all coherent, while its (von-neumann entropy is zero. On the opposite sie, in a totally mixe state the information is entirely incoherent. Notice that the sum of S c (ρ an S(ρ is always equal to log 2 for every state, meaning that every quantum state (at variance with classical ones prouces a constant unavoiable maximal ranomness in the outcomes. Though the ression in the r.h.s. of (2 has alreay appeare in the literature [6] as the amount of thermoynamic work that ρ can extract from a heat bath or the number of pure state istillable from ρ [7], it was not obtaine an interprete in the present way. In the following section, we provie another striking interpretation of the same formula.

3 s1 Env t = t 1 t = t 2 1 2 Figure 2. Scheme of measurements uring time. The state ρ conveys information between past s 1 an future s 2 measurement outputs. Some information has leake into the environment uring ecoherence, but the total information is conserve. Time correlations In this section we want to show that the coherent entropy of a given a state ρ resse in Eq.(2 is exactly equal to the amount of information conveye between past an future measurements, ue to quantum self-correlations in time. We consier the scheme epicte in Fig.2: the state of interest ρ is prepare by the measurement of some observable on ρ 1 at time t 1 an a subsequent ecoherence through interaction with the environment. At a later time t 2, the quantum state unergoes another measurement. The time correlation between the two measurement signals s 1, s 2, with probabilities p(s 1 an p(s 2, is estimate by their mutual information I 1:2 = ( p(s1, s 2 p(s 1, s 2 log 2 (3 p(s s 1 p(s 2 1,s 2 where p(s 1, s 2 is the joint probability. The quantum state can be viewe as a channel, whose capacity is obtaine by maximizing I 1:2 over all inputs. For the sake of clarity, we present here a etaile calculation of I 1:2 in the case of one qubit an a epolarizing channel as a moel of ecoherence. Assume that initially we have the state ρ 1 = 1 2 (I 2 + r 1 σ represente by the vector r 1 insie the Bloch sphere. At time t 1, we ecie to perform a projective measurement along the irection escribe by the unit vector ˆn 1. The outcome s 1 = ±1 will correspon the state P s1 ˆn 1 = 1 2 (I 2 + s 1ˆn 1 σ with probability Tr(ρ 1 P s1 ˆn 1 = 1 2 (1 + s 1r 1 ˆn 1. The subsequent epolarizing channel will simply reuce the length of the Bloch vector ˆn 1 n 1, without changing its irection. Finally, at time t 2, we perform a secon projective measurement along ˆn 2. The outcome s 2 = ±1 will be relate to the state P s2 ˆn 2 = 1 2 (I 2 + s 2ˆn 2 σ with probability Tr(Pn s1 1 P s2 ˆn 2 = 1 2 (1 + s 1s 2 n 1 ˆn 2. Hence we have p(s 1, s 2 = 1 2 (1 + s 1r 1 ˆn 1 1 2 (1 + s 1s 2 n 1 ˆn 2 p(s 1 = p(s 2 = s 2=±1 s 1=±1 s2 p(s 1, s 2 = 1 2 (1 + s 1r 1 ˆn 1 p(s 1, s 2 = 1 2 [1 + s 2(r 1 ˆn 1 (n 1 ˆn 2 ] The mutual information (3 is ( ( 1 + (r1 ˆn 1 (n 1 ˆn 2 1 + n1 ˆn 2 I 1:2 = H 2 H 2, 2 2 where H 2 (x = x log 2 x (1 x log 2 (1 x is the binary entropy. The maximum of I 1:2 is achieve for r 1 ˆn 1 = 0 (i.e., the first measurement is orthogonal to the initial state, or simply the initial state is totally mixe with r 1 = 0 an n 1 ˆn 2 = n 1 (i.e., the secon measurement is collinear to the first one. This gives the value ( 1 + n1 I 1:2 = log 2 2 H 2 = 1 S(ρ, (4 2 which is the = 2 version of S c (ρ as resse in Eq.(2. Notice that the two possibilities s 1 = ±1 of intermeiate quantum state ρ = 1 2 (I 2 + s 1 n 1 σ are unitarily equivalent, so they have the same entropy S(ρ. It is possible to prove the exact match between S c an the maximal I 12 also in arbitrary imension [8]. Conservation of quantum information Assuming that the whole universe is in a pure state, then ρ an its environment can be written in Schmit ecomposition an the entanglement entropy between them is exactly S(ρ [2, 3]. Hence, for every quantum state ρ of imension the sum of the mutual information sent in time quantifie by its coherent entropy S c (ρ in Eq.(2 an the entanglement entropy S(ρ with the rest of the universe turns out to be the constant log 2. As a consequence, uring a unitary evolution any loss of coherence is compensate by an equal increase of entanglement with the environment, an vice versa. This fact constitutes the basic statement for a conservation law of quantum information. If we interpret S c (ρ as a measure of coherence of ρ, we obtain that Eq.(2 is an exact relation between coherence an entanglement. Let us now consier the case of two spatially separate systems A an B, escribe by an overall pure state ρ AB. In general the state of a system is more coherent than the sum of its parts (subaitivity of entropy, S c (ρ AB = S c (ρ A + S c (ρ B + I A:B (5 where the excess of coherent entropy amounts to the non-negative quantity I A:B = S(ρ A + S(ρ B S(ρ AB, which is the (spatial mutual information between the two systems A an B. Curiously, the coherent entropy of ρ AB excees the sum of the contributions coming from its parts A an B even when I A:B receives contribution only from classical correlations. Notice that accoring to Eq.(5 S c obeys the monotonicity property, at variance with the entropy S. Moreover, S c is a convex function in the space of ensity matrices. Assuming also that A an B are isolate from the rest, so that unitary operations o not change the value of S c (ρ AB, then we observe that any variation of the

4 space-like mutual information I A:B is compensate by an opposite variation of the time-like mutual information quantifie by the coherent entropy S c (ρ A + S c (ρ B. Specifying further to the case of pure ρ AB, we fall in the situation where B is the environment of A, an vice versa. Now, the mutual information I A:B = 2S(ρ A = 2S(ρ B quantifies the entanglement between A an B, an oes not contain any contribution from classical correlations. The total quantum information consists of S c (ρ A bits localize in A, the same amount in B while the remainer I A:B is encoe in the Hilbert space that escribes both A an B. Every unitary process will alter the balance of these quantities, without changing their sum, which is equal to the constant log A + log B, i.e. the coherent entropy of the overall pure state. Let us assume that A is localize in a well-efine region in space, elimite by a close surface Σ. The information store in I A:B can be assigne to virtual egrees of freeom assigne to the bons connecting the real iniviual subsystems in A an B. Since all these bons cross the surface Σ, such information can be topologically locate on it. In other wors, uring the ecoherence of A, also B ecoheres, an the consequent lost information flows from both sies towar the surface: a sort of complementary of the holographic principle known in quantum gravity [9]. In this picture, a change in I A:B yiels no net flow of coherence through the surface. At variance with energy, information is a scalar, so it is relativistically invariant. In 1D lattice moels, a well-known realization of such mechanism occurs when we escribe matrix-prouct states (MPS where the mutual information between two bipartition A an B of a chain is encoe in the matrices which escribe the bon variables at the borer between A an B [10]. Multi-partitions After having analyze the case of two systems, it is interesting to unerstan how the quantum information carrie by a quantum state of a given system is istribute when we consier its partition in several parts [11]. In the case of a tripartition ABC, the overall coherent entropy is given by S c (ρ ABC = S c (ρ A + S c (ρ B + S c (ρ C + I A:B + I AB:C (6 or cyclic permutations of subscripts A, B, C. The avantage of having an ression like Eq.(6, is that it involves only entropies an mutual informations, which are non negative objects quantifying amounts of information. The generalization to n partitions orere from 1 to n is n S c (ρ 1 n = S c (ρ k + I 1:2 + I 12:3 + + I 1 (n 1:n k=1 which can be mae symmetric with respect to any label orering. Locally achievable coherence The simple result (2 is obtaine when the optimization problem (1 is solve in the space of all the possible unitary transformations U ρ. However, one may be intereste to restrict the calculation to the family of local transformations with respect of a given partition. For a bipartite state ρ AB we can efine [ ] [ ] Sc loc (ρ AB = max S(σ min S(σ, (7 σ Uρ loc σ U loc AB ρ AB where where Uρ loc AB is the set of all matrices which are equivalent to ρ AB uner local unitaries U A U B. The result of such an optimization is not guarantee to give the same clean ression as in Eq.(2; instea we ect a lesser value which must be calculate numerically. It is appropriate to efine the coherence gap G(ρ AB = S c (ρ AB Sc loc (ρ AB, namely the information which cannot be accesse by local operations. The quantity G(ρ AB accounts for nonlocal correlations between A an B (not necessarily the entanglement in a similar fashion as the quantum iscor [12], or the eficit [7]. The remaining local correlations between A an B are quantifie by L(ρ AB = I A:B G(ρ AB. Examples In orer to familiarize with the concepts iscusse in this paper, we analyze the repartition of information in some quantum states. The Bell state Ψ + = ( 00 + 11 / 2 is a pure state with = 4, i.e. S c (ρ AB = 2, meaning 2 bits of information. After partial trace we get ρ A = Tr B ( Ψ + Ψ + = 1 2 ( 0 0 + 1 1, so both the subsystems are totally mixe, with S c (ρ A = S c (ρ B = 0. The two bits are store in the mutual information is I A:B = 2, which we can figure out as localize on the whole system AB, while A an B are separately incoherent. Notice that one bit is ue to entanglement entropy, S(ρ A = 1, while the remaining bit involves the classical parity correlations [3]. Remarkably, G(ρ AB = 1 is the same as the entanglement entropy. The three-site GHZ state Ψ GHZ = ( 000 + 111 / 2, a paraigmatic example where tripartite entanglement is present, while the pairwise one is zero. The single quantities are summarize in the following table: Ψ GHZ ρ A ρ B ρ C ρ AB ρ AC ρ BC ρ ABC S 1 1 1 1 1 1 0 S c 0 0 0 1 1 1 3 G 0 0 0 1 L 1 1 1 I 1 1 1 E f 0 0 0 where we have also inclue a row for the entanglement of formation E f, which is exactly computable

5 for pairs of qubits [13]. The single subsystems A, B an C are all totally incoherent. Three qubits are store in the mutual information between pairs, all mae of local correlations. Both the nonlocal inicators G an E f are vanishing between pairs. Interestingly, G can be compute also for the tripartite case, resulting in one nonlocally achievable bit. The three-site W state Ψ W = ( 001 + 010 + 100 / 3. In this case, we have Ψ W ρ A ρ B ρ C ρ AB ρ AC ρ BC ρ ABC S 0.918 0.918 0.918 0.918 0.918 0.918 0 S c 0.082 0.082 0.082 1.082 1.082 1.082 3 G 0.667 0.667 0.667 1.667 L 0.252 0.252 0.252 I 0.918 0.918 0.918 E f 0.550 0.550 0.550 Now, some information is carrie by single sites, while 0.918 bits is store in pairwise correlations: 0.252 local an 0.667 nonlocal. The presence of nonlocal correlations is confirme also by 0.550 bits of entanglement of formation. Conclusions This paper illustrates some arguments which lea to the efinition of an entropic value coming from coherent information in quantum states. Such a quantity, here calle coherent entropy, is inee physical as it quantifies the (mutual information conveye in time by quantum states; so it is necessary in orer to give a complete escription of their informational content. By means of this quantity an orinary mutual information between ifferent systems, it is possible to write equations of conservation of information in multipartite states, uring unitary processes. Looking at a specific part of an interacting system, we observe that time-like information is transforme into space-like one: the overall information is conserve an the flow through a close surface is governe by a holographic principle. As the universe is believe to obey to quantum mechanics where time evolution is unitary (issipation an ecay processes are not, because they are only partial escriptions it is imperative to elevate conservation of information to a funamental concept an taking avantage of it, like it happens with any other conserve quantity. A remarkable consequence is that no information has been generate or lost since creation of the universe, but it has only sprea out ue to ansion an interactions. The space-time symmetric treatment of mutual information suggests a possible use in general relativity. For instance, it coul help to she some light in solving the famous paraox of information loss in black holes [9, 14]. The change of metric signature after crossing the event horizon coul be responsible of the transformation of space-like information into time-like, i.e. a purification of quantum states. This is notoriously connecte with the interpretation of the measurement postulate in quantum mechanics which invokes a collapse of the wavefunction after extracting some information about the original state. On the contrary, in the present framework the consequence of a projective measurement is to inject quantum (coherent information into a state, as the output quantum state is pure. Finally, it woul be interesting to lore other possible consequences of conservation of coherent information in founations of quantum mechanics. We believe that the vision escribe in the present work coul yiel some interesting implications also in fiel theories an statistical mechanics. Acknowlegements Many thanks to Lorenzo Campos Venuti per very helpful iscussions. This paper is eicate to the memory of my frien Roberto Gheini, who was the most sincere person I ever knew. marco.roncaglia.it@gmail.com [1] M. Horoecki, R. Horoecki, A. Sen(De, an U. Sen, Foun. Phys. 35, 2041 (2005. [2] M. A. Nielsen an I. L. Chuang, Quantum Computation an Quantum Information (Cambrige University Press, 2000. [3] J. Preskill, Quantum Information an Computation (Lecture Notes for Physics 229, California Institute of Technology, 1998. [4] J. Åberg, arxiv:quant-ph/0612146; T. Baumgratz, M. Cramer, an M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014. [5] A. Wehrl, Rev. Mo. Phys. 50, 221 (1978. [6] J. Oppenheim, M. Horoecki, P. Horoecki, an R. Horoecki, Phys. Rev. Lett. 89, 180402 (2002. [7] M. Horoecki, K. Horoecki, P. Horoecki, R. Horoecki, J. Oppenheim, A. Sen, an U. Sen, Phys. Rev. Lett. 90, 100402 (2003. [8] The proof is illustrate in the Supplemental Material. [9] L. Susskin, J. Math. Phys. 36, 6377 (1995. [10] M. Fannes, B. Nachtergaele an R.F. Werner, Commun. Math. Phys. 144, 443 (1992. [11] A.C.S. Costa, R.M. Angelo, an M.W. Beims, Phys. Rev. A 90, 012322 (2014. [12] H. Ollivier an W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2001. [13] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998. [14] S. L. Braunstein an A. K. Pati, Phys. Rev. Lett. 98, 080502 (2007.

6 SUPPLEMENTAL MATERIAL Maximal mutual information in time for states in arbitrary imension Referring to the scheme in Fig.2, the maximum of I 1:2 must be calculate with respect to every measurement at t = t 1 an t = t 2 of the complete observable A an B, respectively. The first measurement A on the initial state -imensional ρ in will generate an output signal s 1 = 1,...,, proucing a new state equal to the projector P s1 A = s 1 A s 1 A, with probability A. Successively, the system will be subjecte to some ecoherence, which, in general, can be moele in the operator sum formalism as ρ = m M m P s1 A M m with Kraus operators obeying the completeness relation m M mm m = I. It is important to consier that in general the intermeiate state ρ epens on the first outcome s 1. For our scopes, all these intermeiate states shoul be unitarily equivalent, in orer to correctly quantify the information carrie by the given state ρ. Below, we show that such situation is always mae possible by a choice for the M m s. Finally, at time t 2 the quantum state unergoes another measurement B, generating the output s 2, with outcoming state P s2 s2 A, with probability Tr(ρPA. In this process, the joint an marginal probability istributions are given by p(s 1, s 2 = A Tr(ρ s 1 P s2 B p(s 1 = A s 2 B = Tr(ρ inp s1 A H({s 2 } {s 1 }. These two terms can be optimize separately in orer to fin the maximum value of I 1:2 in the space of all the measurement basis A an B. Let us observe that B are just the iagonal elements (s 2 = 1,..., of ρ s1 in the B basis. If we inicate with ρ s1 the iagonal part of ρ s1 in the B basis, we obtain C 2 = s 1 A S( ρ s 1 which is clearly minimal when B is the basis of eigenvectors of all the ρ s1 s, giving C 2 = S(ρ, because all the ρ s1 s have the same eigenvalues. Denoting with λ i (i = 1,..., the eigenvalues of ρ 1, it is easy to see that the term C 1 reaches its theoretical maximum log 2, with the choice { B = λ (s 1+s 2 1mo, A = 1/, s (8 1 = 1,..., In other wors, the spectra of the ensity matrices ρ s1 are given by all the cyclic permutations of {λ 1, λ 2,..., λ }. In this way, we get s 1 B = 1, s 2. The licit form of the Kraus operators for obtaining the first row of Eq. (8 is M m = s=1 P m B Π s 1 B (Πs 1 A PA s with the conition Tr(PA 1P B m = λ m, which etermines what the basis A shoul be. The n-steps cyclicpermutation operator of vectors { i A } i=1 in a given - imensional orthogonal basis A is p(s 2 = s 1 A Tr(ρ s 1 P s2 B Π n A = k k + n A k A where we have obviously use the completeness relation s P α s = I, α = A, B. Moreover we have specifie the inex in the intermeiate state ρ s1, remembering that they have the same eigenvectors an eigenvalues, possibly in ifferent orers. So, the mutual information can be written as I 1:2 = ( p(s1, s 2 p(s 1, s 2 log 2 = C 1 C 2 p(s s 1 p(s 2 1,s 2 with C 1 = s 2 p(s 2 log 2 p(s 2 Moreover, in the secon row of (8 we have require that ρ in at t 1 gives all the possible outcomes with equal probability. In summary, we have obtaine that the maximal amount of information conveye between past an future measurements is I 1:2 = log 2 S(ρ namely equal to the coherent entropy of ρ resse in Eq.(2. C 2 = s 1 A s 2 B log 2 ( B Every quantum state amit a measurement with completely ranom outputs where C 1 is the Shannon entropy of the secon measurement H({s 2 }, an C 2 is the conitional entropy The statement can be reformulate through the following theorem.

7 Theorem. Let ρ be a -imensional square matrix with spectrum {r j, j = 1,..., } an relate orthonormalize eigenvectors r j. Then, it exists a basis of orthonormal vectors { φ k, k = 1,..., } where the all the iagonal elements of ρ are equal. Proof - We can procee in a constructive way by writing own the licit transformation φ k = 1 ( 2πijk r j. (9 j=1 Let us check that the φ k are orthonormal φ k φ k = 1 j,j =1 j=0 ( 2πij k ( 2πijk = 1 1 [ 2πi(k k ] j = δ kk r j r j which assures the unitarity of the transformation in Eq. (9. The iagonal terms are φ k ρ φ k = 1 = 1 j,j =1 ( 2πij k r j = 1 Tr(ρ j=1 ( 2πijk r j ρ r j for every k. In particular, if ρ is a ensity matrix, we have Tr(ρ = 1, so all the iagonal elements in the basis φ k are all equal to 1/. Notice that the choice (9 is not unique, because we have the freeom to perform the gauge transformation r j (iθ j r j. The result of the present theorem was iscusse as a guie exercise (Problem 3, Sect. 2.2 in the book by R. A. Horn an C. R. Johnson, Matrix Analysis, (Cambrige University Press, 1985 to be solve through an iterate inverse Jacobi proceure that maximizes the 2-norm of the off-iagonal part i j ρ ij 2.