Automatic hermiticity for mixed states

We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product $I_Q$ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator $Q$. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time $T_A$ but also a future state at the final time $T_B$ is given, we study a couple of candidates for it, and introduce a ``skew density matrix'' composed of both ensembles of the future and past states such that the trace of the product of it and an operator ${\cal O}$ matches a normalized matrix element of ${\cal O}$. We argue that the skew density matrix defined with $I_Q$ at the present time $t$ for large $T_B-t$ and large $t-T_A$ approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product $I_{Q_J}$ for large $t-T_A$.


Introduction
Quantum theory is described well via the Feynman path integral.In the Feynman path integral, the action in the integrand is considered to be path-dependent, while the measure is usually supposed to be path-independent.However, we could consider a theory such that not only the action but also the measure is path-dependent.This is the complex action theory (CAT) whose action is complex at a fundamental level but expected to look real effectively [1].
The CAT provides us with falsifiable predictions [1][2][3][4].Deeper understanding via the CAT have been tried even for the Higgs mass [5], quantum mechanical philosophy [6][7][8], some finetuning problems [9,10], black holes [11], de Broglie-Bohm particles and a cut-off in loop diagrams [12], a mechanism to obtain Hermitian Hamiltonians [13], the complex coordinate formalism [14], and the momentum relation [15,16].There are two types of CAT.One is the future-not-included theory, where only the past state |A(T A ) at the initial time T A is given and the time integration is performed over the period from T A to a reference time t.The other is the future-included one, where not only the past state |A(T A ) but also the future state |B(T B ) at the final time T B is given, and the time integration is performed over the whole period from T A to T B .We elucidated various interesting properties of the future-notincluded CAT [16].In Ref. [17] we argued that, if a theory is described with a complex action, then such a theory is suggested to be the future-included theory, rather than the future-notincluded one, as long as we respect objectivity.Even so, the future-not-included CAT itself still remains a fascinating theory, and a good playground to study various intriguing aspects of the CAT.
In the future-included theory, the normalized matrix element Ô BA ≡ B(t)| Ô|A(t) B(t)|A(t) of an operator Ô is expected to have the role of an expectation value [1]. 1 Indeed, if we regard it so, we can obtain nice properties such as the Heisenberg equation, Ehrenfest's theorem, and a conserved probability current density [21,22].However, Ô BA is generically complex even for Hermitian Ô, even though any observables are real.To resolve this problem, in Refs.[23,24], we proposed a theorem that states that, provided that an operator Ô is Q-Hermitian, i.e., Hermitian with regard to a modified inner product I Q that makes a given non-normal Hamiltonian 2 normal by using an appropriately chosen Hermitian operator Q, the normalized matrix element defined with I Q becomes real and time-develops under a Q-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude defined with I Q from the past state to the future state is maximized.We call this way of thinking the maximization principle.We proved the theorem in the case of non-normal Hamiltonians Ĥ [23] 3 and in the real action theory (RAT) [24].In addition, we studied the periodic CAT and proposed a variant type of the maximization principle, by which the period could be determined [32].
The maximization principle is based on the natural way of thinking and looks promising.Behind the principle, the automatic hermiticity mechanism [13] has a key role.In the CAT the imaginary parts of the eigenvalues λ i of a given non-normal Hamiltonian Ĥ are supposed to be bounded from above for the Feynman path integral e i S Dpath to converge.Then we can imagine that some Imλ i take the maximal value B, and denote the corresponding subset of {i} as A. After a long time development, only the subset A contributes most significantly, and on the subset a Q-Hermitian Hamiltonian effectively emerges .This is the automatic hermiticity mechanism that we proposed and studied explicitly for pure states time-developing forward [13].In Ref. [21], utilizing it for pure states time-developing forward and backward, we showed that the normalized matrix element of O at the present time t in the future-included theory for large T B − t and large t − T A corresponds to the expectation value of O in the future-not-included theory defined with a modified inner product I Q ′ for large t − T A .This study strongly suggests that the future-included CAT is not excluded phenomenologically, even though it looks very exotic.The automatic hermiticity mechanism has an essential role for the CAT to be viable, but so far we have studied it only for pure states, not for mixed states.Thus it would be natural to pose the question: how does it work for mixed states?Even though mixed states can always be expressed by pure states defined in a larger system that includes the mixed states in its subsystem, it is interesting and worthwhile to study how mixed states are defined and how they behave in the CAT.In particular, it is intriguing to study the automatic hermiticity mechanism for mixed states in the CAT, because the emergence of a Hermitian Hamiltonian is crucially important for the CAT to be sensible, and also because mixed states are generic quantum states along with pure states.
We need to introduce density matrices to describe mixed states in the CAT.In the futurenot-included CAT, there is only one class of state vectors time-developing forward from the past, while in the future-included CAT there are two classes of state vectors time-developing not only forward but also backward from the future.Hence it would be more non-trivial to define density matrices and see the emergence of hermiticity for them in the future-included CAT rather than the future-not-included one.Therefore, in this letter, after reviewing the modified inner product I Q and automatic hermiticity mechanism for pure states, we first define density matrices to describe mixed states and study the emergence of hermiticity for them in the future-not-included CAT.Next, we investigate a couple of candidates for density matrices in the future-included CAT, and introduce a "skew density matrix" composed of both ensembles of the future and past states such that the trace of the product of it and an operator O becomes a normalized matrix element of O. Furthermore, we argue that the skew density matrix defined with I Q at the present time t for large T B − t and large t − T A approximately corresponds to another type of density matrix composed of only an ensemble of the past state and defined with another inner product I Q J for large t − T A .Finally we summarize the study in this letter and discuss the outlook of our theory.
2 Modified inner product and the automatic hermiticity mechanism for pure states In this section we briefly review the modified inner product I Q and the automatic hermiticity mechanism for pure states by following Refs.[13,14].The eigenstates of a given non-normal Hamiltonian Ĥ, |λ i (i = 1, 2, . . . ) obeying Ĥ|λ i = λ i |λ i , are not orthogonal to each other in the usual inner product I.Let us introduce a modified inner product I Q [13,14] 4 such that |λ i (i = 1, 2, . . . ) become orthogonal to each other with regard to it, i.e., for arbitrary kets |u and |v , I Q (|u , |v ) ≡ u| Q v ≡ u|Q|v , where Q is a Hermitian operator that obeys λ i | Q λ j = δ ij .Using the diagonalizing operator P = (|λ 1 , |λ 2 , . ..) such that P −1 ĤP = D = diag(λ 1 , λ 2 , . . .), we choose Q = (P † ) −1 P −1 .Next we define the We also introduce † Q for kets and bras as We can decompose Ĥ as Ĥ = ĤQh + ĤQa , where ĤQh = Q-Hermitian and anti-Q-Hermitian parts of Ĥ, respectively.Let us consider a state 5 |A i (t) , which obeys the Schrödinger equation We introduce a normalized state and an expectation value of an operator time-develops by a Q-Hermitian Hamiltonian, and Ehrenfest's theorem holds.This property is intriguing, but we will see the emergence of the Q-hermiticity even before considering the classical limit via the automatic hermiticity mechanism, which we explain below.
. Now we assume that the anti-Q-Hermitian part of Ĥ is bounded from above for the Feynman path integral e i S Dpath to converge.Based on this assumption we can imagine that some Imλ j take the maximal value B, and denote the corresponding subset of {j} as A. After a long time has passed, i.e., for large t − t 0 , the states with Imλ j | j∈A contribute most in the sum.Let us define a diagonalized Hamiltonian DR by and introduce Ĥeff ≡ P DR P −1 .Since ( DR and the expectation value of an operator O, Thus we have seen for pure states that the Q-hermitian Hamiltonian Ĥeff emerges.
3 Density matrices for mixed states in the future-not-included CAT In this section we define density matrices to describe mixed states and study the automatic hermiticity mechanism for them in the future-not-included CAT.For a given ensemble 6   {|A i (t) }, each of which obeys Eq.( 1), let us consider a mixed state that is composed of |A i (t) N with the probability q i for each index i (q i ≥ 0, i q i = 1).We define the density matrix 7 and expectation value of an operator Ô for it by ρAA,mixed Ô ρAA,mixed where We note that ρAA,mixed is suppressed, Ehrenfest's theorem holds.Now, let us consider the long time development.Then, since |A i (t) N ≃ | Ãi (t) N obeys Eq.(3), we obtain the following relations for ρAA,mixed , and Ô 6 We note that each |A i (t) does not need to be orthogonal to each other, and that the number of elements does not have to match the order of the Hilbert space. 7When the density matrix is composed of only one component, ρA,mixed describes a pure state, and satisfies ρA,mixed and tr ρA,mixed where Ûeff (t obeys the von Neumann equation with the Q-Hermitian Hamiltonian Ĥeff and Ehrenfest's theorem holds.Thus we have confirmed that the automatic hermiticity mechanism works for mixed states as well as for pure states in the future-not-included CAT. 4 Density matrices for mixed states in the future-included CAT In this section we attempt to introduce density matrices to describe mixed states and study their properties in the future-included CAT.In addition we investigate the automatic hermiticity mechanism for the mixed states.The future-included theory is described not only by the state vector |A i (t) that time-develops forward from the initial time T A according to the Schrödinger equation ( 1) but also by the other one8 |B i (t) that time-develops backward from the final time T B according to the other Schrödinger equation: is a good candidate for an expectation value of an operator O in the future-included CAT, because, if it is viewed as such, then we can obtain the Heisenberg equation, Ehrenfest's theorem, and a conserved probability current density [21,22].
In the future-included CAT, let us consider the other ensemble 11 {|B i (t) } besides the ensemble {|A i (t) }.Now we have a simple question: what kind of mixed states can be considered in the future-included theory?One possible candidate would be the same type of mixed states as we considered in the previous section.Since |A i (t) time-develops according to Eq.( 1) in the same way as before, let us consider the same mixed state described by the density matrix ρAA,mixed Q (t) defined in Eq.( 5) for |A i (t) , and consider similar ones for |B i (t) .Let us introduce a normalized state and an expectation value of an operator Next let us consider a mixed state that is given by |B i (t) N with the probability r i for each index i (r i ≥ 0, i r i = 1).We define the density matrix to describe the mixed state and the expectation value of O for it by ρBB,mixed , which are almost the same as those for ρAA,mixed The only difference is that the sign in front of ĤQa is opposite.Therefore, if we use the automatic hermiticity mechanism for | Bi (t) ≡ | Bi (t) N for large T B − t, we find that the various relations for ρBB,mixed  (11).In the future-included CAT, we have a philosophy such that it is not that has a role of an expectation value of Ô.Therefore, ρA i A i Q (t) and ρB i B i Q (t) might not be good density matrices in this sense.Then what should we adopt as a density matrix in the future-included CAT if we wish to respect the philosophy?
We are now motivated to consider the other kind of density matrix such that the trace of the product of each component with an index i and Ô corresponds to Ô ρBA,mixed where the weight s i for each In addition, ρBA,mixed Q (t) can be expressed as ρBA,mixed , where Û (t − t r ) ≡ e − i Ĥ(t−tr) is neither unitary nor Q-unitary, and t r is a reference time.
They time-develop as follows: , we find that ρBA,mixed time-develop with an effectively obtained Q-Hermitian Hamiltonian Ĥeff as follows: However, ρ B Ã,mixed To resolve this problem, we will consider it in another way.
12 Ô ρBA,mixed for Q = 1 corresponds to the weak value for the generalized state introduced in Ref. [19], but is different from the generalized weak value tr( ρf Ô ρi) tr( ρf ρi) introduced in Refs.[34,35].The latter expression is more general since the numbers of ensembles of initial and final states for the density matrices ρi and ρf are taken independently, while, in our skew density matrix, the numbers of ensembles are supposed to be equal.This is because we are keeping in mind the maximization principle, by which a pair of initial and final states is generically chosen such that the absolute value of the transition amplitude is maximized.Then, in a situation such that a pair {|A i , |B i } and each weight {s i } are given, our skew density matrix enables us to calculate and simulate the "expectation value" of O.
5 Hermiticity and reality for ρBA,mixed In Ref. [21], utilizing the automatic hermiticity mechanism for pure states time-developing forward and backward, we obtained the following correspondence: based on the Schrödinger equations ( 1) and is a matrix element of an operator O defined with a usual inner product (Q = 1) in the future-included theory, while is a usual expectation value of O defined with a modified inner product I Q ′ in the future-not-included theory.We showed this correspondence by improving the method used in Ref. [1], which first multiplies O BA by 1 = A(t)|B(t) A(t)|B(t) and then evaluates |B(t) B(t)|.This correspondence strongly suggests that the future-included CAT is not excluded phenomenologically, even though it looks very exotic.Utilizing this method, let us estimate O BA Q and ρBA Q (t), based on the Schrödinger equations ( 1) and . The expansion of |B(T B ) used in Ref. [21], |B(T B ) = i b i |λ i B in terms of the eigenstate of Ĥ † , |λ B = Q|λ , is found to produce too many Q, so it does not seem to be an appropriate choice in the present study.Hence we adopt another expansion: where in the third line we have smeared the present time t a little bit, and the off-diagonal elements wash to 0. In the last line we have used the automatic hermiticity mechanism for large T B − t, and introduced Here, supposing that Reλ i are not degenerate, we have introduced Λ A ≡ for large T B − t and large t − T A .Next, let us consider the expectation value in the future-not-included theory: t) , where Q J ≡ J( Ĥ) † QJ( Ĥ) = (P J −1 −1 ) † P J −1 −1 , and P J −1 ≡ J( Ĥ) −1 P diagonalizes Ĥ: (P J −1 ) −1 ĤP J −1 = P −1 ĤP = D. We introduce |λ i J −1 ≡ J( Ĥ) −1 |λ i , so that |λ i J −1 is Q J -orthogonal, i.e., We use the automatic hermiticity mechanism for large t − T A .|A(t) behaves as | Ã(t) = i∈A a i (t)|λ i , and Q J is estimated as follows: Then the expectation value in the future-not-included theory is expressed as for large t − T A .Thus we have obtained the following correspondence: which suggests that the future-included theory is not excluded, although it looks very exotic.O Ã Ã Q J is real for Q J -Hermitian O, and time-develops according to the Q J -Hermitian Hamiltonian Ĥeff .We can apply this correspondence to each i-component Next let us evaluate the skew density matrix ρBA t) .Utilizing the above evaluation of |B(t) B(t)| Q , we obtain the correspondence: Here t r is a reference time, We can apply this correspondence to each i-component Therefore, though our skew density matrix ρB i A i Q (t) is not Q-Hermitian by its definition, after a long time development it results in a usual expression of density matrix ρ Ãi Ãi Q J (t) that is Q J -Hermitian.Application to ρBA,mixed is rather straightforward and we easily see that it time-develops similarly.Indeed, applying this correspondence to each component ρB i A i Q (t), we find that the expectation value of O for ρB which is real for Q J -Hermitian O. Finally, ρBA,mixed which show that ρBA,mixed

Discussion
We first reviewed the modified inner product I Q that makes a given non-normal Hamiltonian normal with regard to it, and the automatic hermiticity mechanism [13,14,21], which we previously proposed and studied for pure states in the CAT.Next, in the case of the future-not-included CAT, we defined a density matrix ρAA,mixed Q (t) to describe a mixed state and an expectation value of an operator O for it, and studied their properties.In the classical limit, Ô A i A i Q (t) time-develops by a Q-Hermitian Hamiltonian, and Ehrenfest's theorem holds.In addition, we showed that, if we consider a long time development, eigenvectors having the largest imaginary part of the eigenvalues of Ĥ dominate most.On the subspace spanned by such eigenvectors, a Q-Hermitian Hamiltonian effectively emerges, the expectation value of O becomes real for Q-Hermitian O, and the density matrix obeys the von Neumann equation with the Q-Hermitian Hamiltonian.Thus we confirmed that the automatic hermiticity mechanism works for mixed states in the future-not-included theory.
The situation becomes more non-trivial in the future-included theory, because, in the future-included theory, there are two classes of ensembles of state vectors, {|A i (t) } and {|B i (t) }, that time-develop forward from the initial time T A and backward from the final time T B , respectively.So it seems that there are at least a couple of candidates for density matrices in the future-included theory.As the first candidate, we investigated a pair of density matrices ρAA,mixed Q (t) and ρBB,mixed Q (t), which are composed of only either {|A i (t) } or {|B i (t) }, and argued that, though the pair has nice properties, it has a common disadvantage in the future-included theory.In general, the trace of the product of a density matrix and an operator O has to match an expectation value of O, but this is not the case for this pair, because it is the matrix element of O, O BA Q , that is expected to work as an expectation value of O in the future-included theory.To resolve this problem, we introduced a "skew density matrix" ρBA,mixed Q (t), which is composed of both {|A i (t) } and {|B i (t) }.The skew density matrix has a nice property such that the trace of the product of it and an operator O matches the matrix element O BA Q (t).It also obeys the the von Neumann equation as it is.In addition, utilizing the automatic hermiticity mechanism, we showed that the skew density matrix ρBA,mixed Q (t) and matrix element O BA Q (t) defined with an inner product I Q in the future-included theory for large T B − t and large t − T A approximately correspond to another density matrix ρAA,mixed Q J (t) and an expectation value O AA Q J (t) defined with another inner product I Q J in the future-not-included theory for large t − T A .Therefore, even though the skew density matrix is not a density matrix in a usual sense, it can effectively work as if it were a usual density matrix.Thus we argued that it is the skew density matrix that is expected to have a role of a density matrix in the future-included theory.In addition, we confirmed that the automatic hermiticity mechanism works for mixed states in the future-included theory.Now density matrices have been implemented in the CAT.What should we study by using them?It would be interesting to investigate the classical dynamics of the CAT in phase space via the Wigner function.For this purpose, it would be better to study further in detail the harmonic oscillator model that we previously formulated by introducing the twobasis formalism [36].Also, it would be intriguing to evaluate von Neumann entropy in the CAT.Furthermore, density matrices are necessary tools if we wish to investigate quantum measurement quantitatively in a composite system via the master equation.We would like to report such studies in the future.
are Q-Hermitian and real for Q-Hermitian Ô, respectively.They time-develop as follows: d dt ρBB,mixed Q and Ô Q (t) are Q-Hermitian and real for Q-Hermitian Ô, respectively.They time-develop as follows: d dt ρAA,mixed Q , which show that ρBA,mixed Q (t) and ρBB,mixed Q (t).If we consider the long time development, then for |A ).Now we use the automatic hermiticity mechanism for large t − T A .Then, since |A(t) ≡ i a i (t)|λ i behaves as | Ã(t) ≡ i∈A a i (t)|λ i , we obtain obeys the von Neumann equation with the Q J - ρ Ã Ã,mixed Q J (t) is real for Q J -Hermitian O. Thus we have seen that the problem with ρBA,mixed Q (t) and Ô ρBA,mixed Q (t) mentioned at the end of the previous section can be effectively resolved by considering the long time development for large T B − t and large t − T A .