On Existence of Quantum Trajectories for the Linear Deterministic Processes

Abstract

The method of “quantum trajectories”, i.e. transitions from a pure to a pure quantum state, is a useful tool in the open quantum systems theory and applications. This method relies on the nonlinear stochastic differential equations as a dynamical model. In contrast to this, we pose the question of existence of quantum trajectories for the pure states, each of which would be a solution to a linear, deterministic master equation. It turns out that this task is rather delicate. In its full generality, the task is practically intractable. On the other hand, we do not obtain a general answer even for certain well-established and widely used Markovian processes. Only for a few models for the case of the environment on the absolute zero temperature, we obtain existence of the desired quantum trajectories. In all other cases, there is not even a single such quantum trajectory. In conjunction with the standard method of quantum trajectories, our findings pose some nontrivial challenges for the foundations of the open systems theory and some interpretational and cosmological contexts.

Appendices

Appendix A: A Three-Level Atom

A three-level atom with the (non-degenerate) energies $E_1< E_2 < E_3$ is endowed by the dipole transitions which exclude the transitions between the two lower levels. Defining the operators $\hat{\sigma }_{ij}\equiv \vert i\rangle \langle j\vert , i\ne j=1,2,3$, the master equation (in the interaction picture) follows from the quantum-optical master equation [1, 7]:

$$\begin{aligned} \dot{\hat{\rho }}= & {} \gamma _1 (N_1+1) \left( \hat{\sigma }_{13}\hat{\rho }\hat{\sigma }_{31}-{1\over 2}\{\hat{\sigma }_{31}\hat{\sigma }_{13},\hat{\rho }\}\right) + \gamma _1 N_1 \left( \hat{\sigma }_{31}\hat{\rho }\hat{\sigma }_{13}-{1\over 2}\{\hat{\sigma }_{13}\hat{\sigma }_{31},\hat{\rho }\}\right) \nonumber \\{} & {} +\gamma _2 (N_2+1) \left( \hat{\sigma }_{23}\hat{\rho }\hat{\sigma }_{32}-{1\over 2}\{\hat{\sigma }_{32}\hat{\sigma }_{23},\hat{\rho }\}\right) + \gamma _2 N_2 \left( \hat{\sigma }_{32}\hat{\rho }\hat{\sigma }_{23}-{1\over 2}\{\hat{\sigma }_{23}\hat{\sigma }_{32},\hat{\rho }\}\right) .\nonumber \\ \end{aligned}$$

(18)

In (18): $N_i\equiv N(\omega _i)=(e^{\omega _i/k_BT}-1)^{-1}$, with the transition frequencies $\omega _i, i=1,2$; we assume ($\hbar =1$).

Therefore the four Lindblad operators (for the finite temperature): $\hat{\sigma }_{13}, \hat{\sigma }_{31}$, $\hat{\sigma }_{23}, \hat{\sigma }_{32}$; generalization to the degenerate case is straightforward. Then the pure-state condition (3) of the main text gives the equality:

$$\begin{aligned} \gamma _1N_1(p_1+p_3-2p_1p_3) + \gamma _1(p_3-p_1p_3) + \gamma _2N_2(p_2+p_3-2p_2p_3) + \gamma _2(p_3-p_2p_3)=0, \end{aligned}$$

(19)

where $p_i\equiv \vert \langle i\vert \psi \rangle \vert ^2$.

The case $T=0$. For $T=0$, $N_1 = 0 = N_2$ and (19) requires $p_3=0$, with arbitrary $p_i,i=1,2$. That is, every pure state $\vert \psi \rangle = c_1\vert 1\rangle + c_2\vert 2 \rangle$ is allowed. However, since $\hat{\sigma }_{i3}\vert \psi \rangle =0, \forall {i=1,2,3}$, placing $N_1=0=N_2$ into (18) reveals that every pure state $\vert \psi \rangle$ (i.e. for arbitrary $c_i, i=1,2$) is a stationary state, which does not evolve in time–no PPSD is possible, in principle.

The case $T>0$. Then $N_1\ne 0 \ne N_2$ and the pure-state condition reads:

$$\begin{aligned} \gamma _1(N_1+1)(p_3-p_1p_3) + \gamma _1N_1(p_1-p_1p_3)+\gamma _2(N_2+1)(p_3-p_2p_3)+\gamma _2N_2(p_2-p_2p_3)=0, \end{aligned}$$

(20)

with the constraint $p_1+p_2+p_3=1$. Without loss of generality, choose (like for the models of the atomic dark states and induced transparency [7, 8]) $\gamma _1=1=100\gamma _2$ and $N_1=0.4=1000N_2$. By inspection it can be seen that (20) does not return any relevant solutions. For example, place $p_3=1-p_1-p_2$ and solve (20) for $p_1$. For $p_3=0$, (20) implies a negative value for either $p_1$ or $p_2$. For nonzero $p_3$, Fig. 1 exhibits that the sum $p_1+p_2>1$; the minimum value for $p_2$ that returns a real value for $p_1$ is larger than 0.83.

Fig. 1

The plot of $p_1+p_2$ for the nonzero $p_3$. The dashed line is for the one, while the solid one is for the other solution for $p_1$ following from (20). It is presented that $p_1+p_2>1$ for every choice of $p_2>0.83$

Analogous elimination of the other terms (either $p_1$ or $p_2$) leads to the similar conclusions. Therefore it is found, that even a single pure state cannot be found to fulfill the pure-state condition.

Appendix B: A Multimode System

Consider a linear system of n harmonic oscillators or more generally modes in contact with independent thermal baths on the temperature T. In any case this system can be transformed into a set of mutually uncoupled normal coordinates, i.e. of uncoupled modes, here presented by the commuting Bose annihilation operators $\hat{a}_i, i=1,2,...,n$; $[\hat{a}_i,\hat{a}_j^{\dagger }]=\delta _{ij}$. Then a generalization of (11) of the main text is straightforward:

$$\begin{aligned} \dot{\hat{\rho }} = \sum _{i=1}^n \gamma _i \left( (N(\omega _i)+1)\left( \hat{a}_i\hat{\rho }\hat{a}_i^{\dagger }-{1\over 2}\{\hat{a}_i^{\dagger }\hat{a}_i,\hat{\rho }\}\right) + N(\omega _i) \left( \hat{a}_i^{\dagger }\hat{\rho }\hat{a}_i-{1\over 2}\{\hat{a}_i\hat{a}_i^{\dagger },\hat{\rho }\}\right) \right) . \end{aligned}$$

(21)

The Lindblad operators are all the Bose operators, $\hat{a}_i$ and $\hat{a}_i^{\dagger }$. Then (3) takes the form:

$$\begin{aligned} \sum _i \gamma _i \left( (N(\omega _i)+1) \left( \langle \hat{a}_i^{\dagger } \hat{a}_i \rangle - \langle \hat{a}_i\rangle \langle \hat{a}_i^{\dagger }\rangle \right) + N(\omega _i) \left( \langle \hat{a}_i \hat{a}_i^{\dagger }\rangle - \langle \hat{a}_i^{\dagger }\rangle \langle \hat{a}_i\rangle \right) \right) =0. \end{aligned}$$

(22)

Due to the Cauchy-Schwarz inequality (cf. Section 3.2), as well as to $\gamma _i\ge 0, N(\omega _i) \ge 0, \forall {i}$, all the terms in (22) are non-negative. Therefore the only possibility to satisfy (22) is already recognized in Section 3.2 of the main text:

$$\begin{aligned} N(\omega _i) = 0, \quad \langle \hat{a}_i^{\dagger }\hat{a}_i \rangle - \langle \hat{a}_i\rangle \langle \hat{a}_i^{\dagger }\rangle =0, \end{aligned}$$

(23)

for every mode i. Mutual independence of the modes implies the conclusion drawn in Section 3.2 for every mode i as well as for the solution, $\otimes _i\vert \alpha _i\rangle$, of (22) (where $\alpha _i$ states for a “coherent state” for the ith mode).

Appendix C: The Phase Damped Harmonic Oscillator

This model [7] is comprised of a single harmonic oscillator in contact with a thermal bath such that the bosonic number operator, $\hat{N}=\hat{a}^{\dagger }\hat{a}$, is coupled with the bath’s variable(s). The effective master equation (in the Schrödinger picture) is ($\hbar =1$):

$$\begin{aligned} \dot{ \hat{\rho }}= -\imath \omega _{\circ } [\hat{N},\hat{\rho }] +\gamma \left( \hat{N}\hat{\rho }\hat{N} - {1\over 2}\{{\hat{N}}^2,\hat{\rho }\}\right) \end{aligned}$$

(24)

The only Lindblad operator, i.e. the number operator, $\hat{N}$ commutes with the system Hamiltonian and hence the pure-state condition:

$$\begin{aligned} (\Delta \hat{N})^2=\langle \hat{N}^2\rangle - \langle \hat{N}\rangle ^2=0, \end{aligned}$$

(25)

determines its eigenstates $\vert n\rangle$, $\hat{N}\vert n\rangle = n\vert n\rangle$, as the solutions. However, those states are the exact “pointer basis” states [2] that do not evolve in time. Thus there is not a single pure state whose dynamics could describe the decoherence dynamics (24). This conclusion applies to all the similar Markovian decoherence-models for a qubit or a continuous-variable (CV) systems [1,2,3].

Appendix D: The One-Qubit Generalized Depolarizing Channel

The so-called generalized one-qubit depolarizing channel is modeled by the following master equation (in the interaction picture) [34]:

$$\begin{aligned} \dot{\hat{\rho }} = \sum _{i=x}^z \gamma _i (\hat{\sigma }_i\hat{\rho }\hat{\sigma }_i - \hat{\rho }), \end{aligned}$$

(26)

where appear the standard Pauli operators $\hat{\sigma }_i, i = x,y,z$.

Therefore there are three Lindblad operators, $\hat{L}_i=\hat{\sigma }_i$, which give rise to the PPSD condition (3):

$$\begin{aligned} \sum _i \gamma _i (1-\langle \hat{\sigma }_i\rangle ^2) = 0. \end{aligned}$$

(27)

It is obvious that (27) implies $\langle \hat{\sigma }_i\rangle =1,\forall {i}$, which cannot be fulfilled for any pure state. This result may be expected due to the fact that this is a unital channel, i.e., dynamics preserving the fully mixed state, $\hat{I}/2$, where $\hat{I}$ is the identity operator . This is a situation also for all the unital maps for which the Lindblad operators do not have even a single common eigenstate [1].

Appendix E: Decay of a Two-Level Atom into a Squeezed Field Vacuum

This is another standard model in quantum optics for a two-level atom that is described by the following master equation [1, 7]:

$$\begin{aligned} \dot{\hat{\rho }} = \gamma _{\circ } \left( \hat{C}\hat{\rho }\hat{C}^{\dagger } - {1\over 2} \{\hat{C}^{\dagger }\hat{C},\hat{\rho }\} \right) , \end{aligned}$$

(28)

where $\hat{C} = \cosh (r) \hat{\sigma }_- + e^{\imath \theta }\sinh (r) \hat{\sigma }_+$, and the environmental squeeze parameters r and $\theta$, while $\hat{\sigma }_-=\vert g\rangle \langle e\vert$ for the excited (e) and the ground (g) atomic states.

Since there is only one Lindblad operator, $\hat{L}=\hat{C}$, the pure-state condition reads:

$$\begin{aligned} \cosh ^2(r) p_e + \sinh ^2(r)(1-p_e) - p_e(1-p_e)(\cosh (2r)+\sinh (2r)\cos (\omega )) = 0; \end{aligned}$$

(29)

in (29): $p_e=\vert \langle e\vert \psi \rangle \vert ^2$, $\omega = \theta +2\delta$, with the arbitrary phase $\delta$.

Solutions to (29) are readily obtained: $(1 + e^{\imath \omega } \coth (r))^{-1}$ and $1-(1+e^{\imath \omega }\tanh (r))^{-1}$. The complex term can be eliminated for either $\omega =0$ or $\omega =\pi$. For the latter the negative values for $p_e$ are obtained. Therefore the only possibility is $\omega =0$ (i.e. $\delta =-\theta /2$) when the two solutions become equal. Hence the unique solution for $p_e = (1 + \coth (r))^{-1}$, which is independent of $\theta$. Thus, for the environmental state with the fixed r and $\theta$ parameters, the unique state of the two-level system is found to read:

$$\begin{aligned} \vert \psi \rangle = (1 + \coth (r))^{-1/2} \vert e\rangle + e^{-\imath \theta /2} (1-(1 + \coth (r))^{-1})^{1/2} \vert g\rangle , \end{aligned}$$

(30)

which is the eigenstate of the operator $\hat{C}$ with the eigenvalue $e^{\imath \theta /2}\sqrt{\sinh (2r)/2}$, as it can be easily checked.

Fig. 2

The plot of the sum, $P=n_1^2+n_2^2+n_3^2$, for: (a) $t=0.00001$ and the parameter range $r\in [0,1]$ and $\theta \in [0,2\pi ]$, and (b) $\theta =\pi /2$ and the parameter range $r\in [0,1]$ and $t\in [0,3]$

Certainly, the single pure state (30) cannot define a trajectory in the Hilbert state space of the system. Going to the matrix representation of the dissipator, $\mathcal {D}$, in (28) and therefore to the map $e^{\mathcal {D}t}$, it can be shown that dynamics of the state (30) takes the form of $e^{\mathcal {D}t}[\vert \psi \rangle \langle \psi \vert ] = (\hat{I} + \sum _i n_i(t) \hat{\sigma }_i)/2$, where (for simplicity we choose $\gamma _{\circ }=1$):

$$\begin{aligned} n_1(t)= & {} \sqrt{(1 - p) p} \cos [\theta /2] (e^{ t (-\cosh [2 r] - \sinh [2 r])/4} + e^{ t (-\cosh [2 r] + \sinh [2 r])/4} \nonumber \\{} & {} - e^{ t (-\cosh [2 r] - \sinh [2 r])/4} \cos [\theta ] + e^{ t (-\cosh [2 r] + \sinh [2 r])/4} \cos [\theta ])\nonumber \\{} & {} - (e^{ t (-\cosh [2 r] - \sinh [2 r])/4} + e^{t (-\cosh [2 r] + \sinh [2 r])/4}) \sqrt{(1 - p) p} \sin [\theta ] \sin [\theta /2], \nonumber \\ n_2(t)= & {} -\sqrt{(1 - p) p} (e^{ t (-\cosh [2 r] - \sinh [2 r])/4} + e^{t (-\cosh [2 r] \sinh [2 r])} \nonumber \\{} & {} + e^{ t (-\cosh [2 r] - sinh[2 r])/4} \cos [\theta ] - e^{ t (-\cosh [2 r] + \sinh [2 r])/4} \cos [\theta ]) \sin [\theta /2] \nonumber \\{} & {} - (e^{ t (-\cosh [2 r] - \sinh [2 r])/4} - e^{t (-\cosh [2 r] + \sinh [2 r])/4}) \sqrt{(1 - p) p} \cos [\theta /2] \sin [\theta ], \nonumber \\ n_3(t)= & {} e^{- t \cosh [2r]/2} (2p-1)- (1-e^{- t \cosh [2r]/2} ) {{\,\textrm{sech}\,}}[2r],\nonumber \end{aligned}$$

and $\hat{\sigma }_i, i=1,2,3$, represent the standard Pauli operators. The condition of purity of state, $P=n_1^2+n_2^2+n_3^2=1$, is fulfilled for the initial instant of time $t=0$, and, from Fig. 2, for $t>0$ only for $r=0$, which is the case $\gamma _1=0$ investigated in Section 3.1. Therefore there is not even a single pure state satisfying (3), i.e., remaining pure in the course of the dynamics (28).

Appendix F: A Nonadiabatic Markovian Model

For an externally driven damped harmonic oscillator, the following Markovian master equation applies (in the interaction picture) [35]:

$$\begin{aligned} \dot{\hat{\rho }} = \vert \xi (t)\vert ^2 \gamma (t) \left( \hat{F}_+\hat{\rho }\hat{F}_- - {1\over 2}\{\hat{F}_- \hat{F}_+,\hat{\rho }\} + e^{-\hbar \alpha (t)/k_BT} (\hat{F}_-\hat{\rho }\hat{F}_+ - {1\over 2}\{\hat{F}_+\hat{F}_-,\hat{\rho }\}) \right) , \end{aligned}$$

(31)

where $\hat{F}_+= A \hat{x} + B \hat{p} = \hat{F}_-^{\dagger }$ and $A=(1+\imath \mu /\kappa )/2, B=\imath /m\omega (0)\kappa$, with all the positive parameters, $m,\omega (0),\kappa >0$ as well as $\gamma (t), \alpha (t)\ge 0, \forall {t}$.

From (31), the two Lindblad operators are found: $\hat{L}_1 = \hat{F}_+$ and $\hat{L}_2=e^{-\hbar \alpha (t)/2k_BT}\hat{F}_-$. Then applying (3) from the main text gives:

$$\begin{aligned} \left( \langle \hat{F}_-\hat{F}_+\rangle - \langle \hat{F}_-\rangle \langle \hat{F}_+\rangle \right) (1+ e^{-\hbar \alpha (t)/k_BT}) + e^{-\hbar \alpha (t)/k_BT} \langle [F_+,F_-]\rangle =0. \end{aligned}$$

(32)

With the use of the commutator, $[\hat{F}_+,\hat{F}_-]=\hbar /m\omega (0)\kappa$, (32) becomes a sum of the non-negative terms:

$$\begin{aligned} \left( \langle \hat{F}_-\hat{F}_+\rangle - \langle \hat{F}_-\rangle \langle \hat{F}_+\rangle \right) (1+ e^{-\hbar \alpha (t)/k_BT}) + {\hbar e^{-\hbar \alpha (t)/k_BT}\over m\omega (0)\kappa }=0. \end{aligned}$$

(33)

Since the second term on the rhs of (33) cannot equal zero, we conclude that there does not exist even a single pure state for the master equation (31) satisfying the PPSD condition (3).

Appendix G: The Walls-Collet-Milburn Model

Consider the dynamical model of quantum measurement performed on a two-mode system (a system S with the mode $\hat{a}$ and the “meter” system M with the mode $\hat{b}$) [36], such that only one of the coupled modes is in contact with the thermal bath of modes $\hat{c}_i$ on some temperature T. Assuming that interaction of the object of measurement with the meter system is given by a four-wave-mixing interaction:

$$\begin{aligned} \hat{H}_{SM} = - {\imath \over 2} \hat{a}^{\dagger } \hat{a} \otimes (\epsilon ^{*}\hat{b} - \epsilon \hat{b}^{\dagger }) \end{aligned}$$

(34)

and that the interaction of the meter with the interaction is bilinear:

$$\begin{aligned} \hat{H}_{ME}= \hat{b}\otimes \hat{C}^{\dagger } + \hat{b}^{\dagger } \otimes \hat{C}, \end{aligned}$$

(35)

where $\hat{C}^{\dagger } = \sum _j\kappa _j \hat{c}_j$. Then it can be shown that, for longer times, the master equation for the object of measurement (in the interaction picture) is of the Markovian form:

$$\begin{aligned} \dot{\hat{\rho }}_S = {\vert \epsilon \vert ^2\over \gamma }\left( \hat{N}\hat{\rho }\hat{N}-{1\over 2} \{\hat{N},\hat{\rho }_S\}\right) \end{aligned}$$

(36)

where $\hat{N} \equiv \hat{a}^{\dagger }\hat{a}$ and the real parameter $\gamma >0$.

In (36) appears only one Lindblad operator, $\hat{L}\equiv \hat{N}$, which placed in (3) gives rise to the following condition for the pure-state dynamics of the object of measurement:

$$\begin{aligned} (\Delta \hat{N})^2 \equiv \langle \hat{N}^2\rangle - \langle \hat{N}\rangle ^2 = 0. \end{aligned}$$

(37)

Of course, every eigenstate, but no linear combination of the eigenstates, of the number operator $\hat{N}$ satisfies the condition (37). However, as it can be easily seen, those states also represent the stationary states for the process, thus not being subject of a dynamical change in time. In other words, a pure (sub)ensemble in the state $\vert n\rangle$ does not evolve in time.

Appendix H: Quantum Mechanics with Spontaneous Localization Model

Consider the Ghirardi-Rimini-Weber master equation [11]:

$$\begin{aligned} \dot{\hat{\rho }} = \lambda \left( \sqrt{{\alpha \over \pi }} \int ds e^{-\alpha (\hat{x}-s)/2}\hat{\rho }e^{-\alpha (\hat{x}-s)/2} - \hat{\rho }\right) \end{aligned}$$

(38)

It is worth emphasizing that (38) is assumed to describe dynamics of a closed, not of an open, system.

There is a Lindblad operator, $\hat{L}_s\equiv e^{-\alpha (\hat{x}-s)/2}$, for every value of the continuous real parameter s. Then the condition (3) obviously implies:

$$\begin{aligned} \Delta (e^{-\alpha (\hat{x}-s)^2/2}) = 0, \forall {s} \end{aligned}$$

(39)

for the standard deviation of the Lindblad operators.

Needless to say, there is not even a single pure state (in the Hilbert space) that could fulfill (39) and hence nonexistence of PPSD for the model.

Appendix I: Continuous Spontaneous Localization Model

The master equation of interest that is assumed to describe dynamics of a closed system reads [12]:

$$\begin{aligned} \dot{\hat{\rho }} = \lambda \sum _{i,k} \left( \hat{N}_i^{(k)} \hat{\rho }\hat{N}_i^{(k)} - {1\over 2} \{\hat{N}_i^{(k)2},\hat{\rho }\} \right) , \end{aligned}$$

(40)

with the Hermitian Lindblad operators $\hat{N}_i^{(k)}$, which represent appropriate bosonic number operators for the model. Then (3) leads to:

$$\begin{aligned} \sum _{i,k} (\Delta \hat{N}_i^{(k)})^2 = 0, \end{aligned}$$

(41)

where appear the standard deviations for $\hat{N}_i^{(k)}$s. Certainly, (41) can be only identically fulfilled, i.e. only if $\Delta \hat{N}_i^{(k)}=0, \forall {i,k}$. Therefore, the only pure states satisfying the PPSD condition (3) are the common eigenstates for all $\hat{N}_i^{(k)}$s. However, it can be easily shown than the eigenstates for those operators, denoted $\vert n_1^{(k)} n_2^{(k)} n_3^{(k)} ... \rangle$, are the stationary states for the process–again emphasizing impossibility of PPSD.