Assessing non-Markovian dynamics through moments of the Choi state

Isolated systems undergo unitary evolution. However, a general quantum evolution (or a quantum channel) can be represented by a completely-positive trace-reserving (CPTP) map $(\Lambda(t,t_{0}))$ which maps an element ($\rho(t_{0})$) of the set of density operators ($B(\mathcal{H})$) to another element of the set i.e, $\Lambda(t,t_{0}):\rho(t_{0})\mapsto\rho(t)$. The set of all such CPTP maps can be represented as $D$. We assume that the inverse $\Lambda^{-1}(t,t_{0})$ exists for all time from $t_{0}$ to $t$. One can thus write the dynamical map for any $t\geq s\geq t_{0}$, into a composition

Even though $\Lambda(t,t_{0})$ is always completely positive since it must correspond to a physically legitimate dynamics (and hence $\Lambda^{-1}(t,t_{0})$ is well defined) and $\Lambda(s,t_{0})$ is completely positive, the map $\Lambda(t,s)$ however, need not be completely positive. A dynamics acting on the system of interest is said to be divisible iff it can be written as Eq. (1) for any time $t\geq s\geq t_{0},$ where $t_{0}$ is the initial time of dynamics, and $\circ$ represents the composition between two maps. The dynamics $\Lambda(t,t_{0})$ is said to be positive divisible (P divisible) if $\Lambda(t,s)$ is a positive map for every $t\geq s\geq t_{0}$ satisfying the composition law. The dynamics $\Lambda(t,t_{0})$ is said to be completely positive divisible (CP divisible) if $\Lambda(t,s)$ is a CPTP map for every $t\geq s\geq t_{0}$ and satisfies the composition law.

The above mathematical characterization of a dynamical map $\Lambda(t,t_{0})$ in terms of ‘divisibility’, describing a memoryless evolution as a composition of physical maps leads to the definition of quantum Markovianity. According to the RHP criterion, a dynamics is said to be non-Markovian if it is not CP-divisible Rivas et al. (2010). Another way of characterizing non-Markovian dynamics is provided by Breuer et. al. Breuer et al. (2009); Laine et al. (2010) where distinguishability of quantum states after the action of dynamical map is considered. Due to the interaction of a quantum system with the noisy environment, two quantum states lose their state distinguishability gradually with time. However, if at any instant of time, the distinguishability increases, then there is backflow of information from the environment to the system leading to the signature of non-Markovianity. The former way of representing a dynamics to be non-Markovian is known as RHP-type non-Markovianity Rivas et al. (2010, 2014), whereas the latter one is known as BLP-type non-Markovianity Breuer et al. (2009, 2016). A dynamics which is Markovian in the RHP sense is also Markovian in the BLP sense, but the converse is not true in general. Therefore, CP divisibility breaking is a necessary but not sufficient condition for information backflow from the environment to the system. In this paper we adopt CP divisibility as the sole property of quantum Markovianity, and any deviation from CP-divisibility (indivisible) will be considered as the benchmark of non-Markovianity.

Now, for each of the dynamical maps $\Lambda(t,t_{0})\in D$, one can find a one-to-one correspondence to a state, called the Choi-state $\mathcal{C}_{\Lambda}(t,t_{0})\in\mathcal{F}$ (where $\mathcal{F}$ is the set of all Choi-states) via channel-state duality where the Choi state Choi (1975) is defined as

with $\ket{\phi}\bra{\phi}$ being a maximally entangled bipartite state of dimension $d\crossproduct d$. According to the Choi–Jamiolkowski isomorphism Choi (1975); Jamiołkowski (1972), for checking complete-positivity of $\Lambda(t,t_{0})$, it is sufficient to check the positive semidefiniteness of the corresponding Choi-state $(\mathcal{C}_{\Lambda}(t,t_{0}))$.

In the bipartite scenario, one of the most well-known detection schemes of entanglement is based on the PPT criterion which examines whether the partial transposed state $\rho^{T_{A}}_{AB}$ (where partial transposition is taken w.r.t subsystem A) is positive semi-definite (all eigenvalues are non-negative) or not. Violation of this criterion implies that the given state $\rho_{AB}$ is entangled. This criterion has been shown to be a necessary and sufficient condition for $2\otimes 2$, $2\otimes 3$ and $3\otimes 2$ systems and has many applications in theoretical works Castelnovo (2013); Eisler and Zimborás (2014); Wen et al. (2016); Ruggiero et al. (2016a); Blondeau-Fournier et al. (2016); Ruggiero et al. (2016b). However, the transposition map not being a physical one, is impossible to implement exactly in an experimental scenario. A useful measure using this PPT criterion is the negativity measure Vidal and Werner (2002), defined as:

where $\lambda_{i}$’s are the negative eigenvalues of $\rho^{T_{A}}_{AB}$. But this again requires an access to the full spectrum of $\rho^{T_{A}}_{AB}$, which is not obtainable through an experimental setup. To overcome this issue, the idea of moments of the partially transposed density matrix (PT-moments) was introduced to study the correlations in many-body systems in relativistic quantum field theory by Calabrese et al. in 2012 Calabrese et al. (2012).

For a bipartite state $\rho_{AB}$, these PT-moments are given by,

for n=1,2,3,…. One may note that $P_{1}=\text{Tr}[\rho_{AB}^{T_{A}}]=1$ while $P_{2}=\text{Tr}[(\rho_{AB}^{T_{A}})^{2}]$ is related to the purity of the state. Therefore, $P_{3}$ is the first non-trivial moment which is necessary to capture additional information related to the partial transposition. Using only these first three PT moments, a simple but powerful entanglement detection criterion was proposed in ref. Elben et al. (2020a). This suggests that if a state $\rho_{AB}$ is PPT, then ${P_{2}}^{2}\leq P_{3}P_{1}$. Therefore, from the contrapositivity of this statement, it follows that if a state $\rho_{AB}$ violates this condition, then it must be entangled which is the $p_{3}$-PPT criterion. Just like the PPT condition, this $p_{3}$-PPT condition is also applicable to mixed states and is a state independent criterion unlike entanglement witness Horodecki et al. (2009); Gühne and Tóth (2009). While entanglement witness provides a stronger criterion for entanglement detection, some prior knowledge about the state is required for the implementation of entanglement witness.

Even though this $p_{3}$-PPT criterion is weaker than the general PPT criterion, the former involves simple functionals which are easy to realise in a real experiment by a method called shadow tomography Aaronson (2018); Aaronson and Rothblum (2019); Huang et al. (2020). For Werner states, the $p_{3}$-PPT criterion and the full PPT criterion are equivalent, and hence, the $p_{3}$-PPT criterion is a necessary and sufficient criteria for bipartite entanglement of Werner states. The PT-moments can be obtained experimentally with the help of shadow tomography without actually performing full state tomography, thus making it more efficient in terms of resources consumed. For a detailed discussion on shadow tomography and its advantage over general tomography, interested readers are referred to Refs. Aaronson (2018); Aaronson and Rothblum (2019); Huang et al. (2020); Elben et al. (2020a). The technique of PT moments offers unparalleled advantage in NISQ and in many-body systems where a single qubit is used as a control and many distinct PT moments can be estimated from the same data unlike using random global unitaries for randomized measurements Huang et al. (2020); Elben et al. (2020a). Furthermore, the $p_{3}$-moment, in addition to detecting mixed state entanglement Elben et al. (2020a); Neven et al. (2021); Yu et al. (2021) is also used to study entanglement dynamics in many-body quantum systems Brydges et al. (2019); Elben et al. (2020b).

It might be noted here that the $p_{3}$-PPT condition provides a necessary condition for separability. However, each higher order moment ($n\geq 4$) gives rise to an independent and different entanglement detection criterion, and evaluating all the higher order moments provides a necessary and sufficient criterion for NPT entanglement Neven et al. (2021). But this is very challenging from an experimental point of view and hence, moments up to third order are used in order to provide an entanglement detection criteria and this simplifies the task. Motivated by the above considerations, in the next section we explore whether a moment based detection scheme can be developed for non-Markovian dynamics, since in realistic scenarios, many different categories of dynamics consist of non-Markovian memory. We define $\Lambda$-moments $(r_{n})$, and based on it we develop a formalism for detection of non-Markovian dynamics characterized via indivisibility.

We would now like to present two examples in support of Theorem 1. It may be noted that here we will consider the set of operations which have Lindblad type generators. For system density matrix $\rho$, the Lindblad master equation can be written as

where the unitary aspects of the dynamics is described by the Hamiltonian $H$, $\gamma_{i}$ are the Lindblad coefficients, and $L_{i}$ are the Lindblad operators which describe the dissipative part of the dynamics Lindblad (1976).

Example 1: We consider a qubit system that interacts with an amplitude damping environment which is modeled by another qubit system. The non-Markovian character of this model studied earlier Mukherjee et al. (2015), is motivated by the experimental realization of such non-Markovian dynamics through the violation of temporal Bell-like inequalities in a controllable Nuclear Magnetic Resonance system Souza et al. (2013) .

The master equation is given by

The Lindblad coefficients are taken as $\gamma_{1}=\gamma_{2}=\gamma_{3}=e^{-t^{\prime}}\cos{t^{\prime}}$ (for all $i$) and $t^{\prime}=kt$, with $k$ being a constant having the dimension of $[T^{-1}]$. The corresponding dynamical map is given by $\Lambda(\rho)=e^{\mathcal{L}}(\rho)$. For small time approximation (i.e, $|\epsilon\gamma_{i}|<<1$), the Choi state corresponding to $\Lambda(\rho)$ is given by

with $\ket{\phi^{+}}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$. Here, we consider $\epsilon=0.001$ and $k=1$. It is known that the above dynamics shows its non-Markovian nature when $\gamma_{i}<0$. Therefore, for $\gamma_{i}<0$, we should have ${r}_{2}^{2}-{r}_{3}>0$, which is evident from Fig. 1.

Example 2: As a second example, we consider a pure dephasing non-Markovian dynamics. A qubit system interacts with a thermal reservoir which is modeled by an infinite set of harmonic oscillators in the vacuum state. The Hamiltonian corresponding to the system-reservoir interaction is

where $\omega_{\sigma},\omega_{k}$ are the energy gap of the system and the frequency of $k$-th mode of reservoir, respectively, $a_{k}^{\dagger}(a_{k})$ are creation (annihilation) operators of the harmonic oscillator, $\alpha_{k}$ is the coupling constant for the $k$-th mode, and $\theta_{k}$ is the corresponding phase. The master equation is given by

The time dependent decay rate $\gamma$ corresponding to a Lorentzian spectral density is (Salimi et al., 2016; Mukherjee et al., 2015)

with $g=\sqrt{{\lambda}^{2}-2\gamma_{0}\lambda}$, $t^{\prime}=kt$. Here $k$ is a constant having the dimension of $[T^{-1}]$, $\lambda$ is the spectral width and $\gamma_{0}$ is the coupling strength.

In the small time approximation (i.e, $|\epsilon\gamma|<<1$), the Choi state corresponding to the dynamical map, $\Lambda(\rho)=e^{\mathcal{L}}(\rho)$ is given by $\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$ where $\ket{\phi^{+}}$ is the maximally entangled state defined earlier. It is known that the above dynamics shows its non-Markovian nature when $\gamma<0$ which is possible only when $\gamma_{0}>\lambda/2$ Mukherjee et al. (2015). So, for $\gamma<0$, we should have ${r}_{2}^{2}-{r}_{3}>0$ which is again evident from Fig. 2. We consider here $\lambda=1.5$, $\gamma_{0}=1$ and $k=1$ for the figure.

For a detailed comparison between our proposed measure ($\mathcal{M}$) and the RHP measure ($\mathcal{I}$), we consider pure dephasing channel as presented in Example 2. The time dependent dephasing rate is given by Salimi et al. (2016)

where, $\omega$ and $J(\omega)$ represents the frequency of the reservoir modes and spectral function of the reservoir respectively. For Ohmic spectral density, the spectral function is given by

where, $\omega_{c}$ is the cut-off frequency of the reservoir.

In the small time approximation (i.e, $|\epsilon\gamma|<<1$), the Choi state corresponding to the dynamical map, $\Lambda(\rho)=e^{\mathcal{L}}(\rho)$ is given by $\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$ where $\ket{\phi^{+}}$ is the maximally entangled state.

RHP Measure: From the Choi–Jamiolkowski isomorphism, we know that the dynamical map $\Lambda_{\epsilon}=\Lambda(t+\epsilon,t)$ is CP iff $(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}\geq 0$ $\forall\epsilon$. From the trace preserving condition, $||(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}||_{1}=1$ iff $\Lambda_{\epsilon}$ is completely positive and $||(\mathbb{I}\otimes\Lambda_{\epsilon})\ket{\phi^{+}}\bra{\phi^{+}}||_{1}>1$, if $\Lambda_{\epsilon}$ is not completely positive. Considering these facts, one can define

where $\Lambda_{\epsilon}=(\mathbb{I}+\epsilon\mathcal{L})$. It is clear that $g(t)\geq 0$ and $g(t)=0$ iff $\Lambda_{\epsilon}$ is CP. The integral

is the RHP measure Rivas et al. (2014, 2010).

In case of pure dephasing channel with Ohmic spectral density, $\mathcal{C}_{\Lambda}=(\mathbb{I}\otimes(\mathbb{I}+\epsilon\mathcal{L}))\ket{\phi^{+}}\bra{\phi^{+}}$ has eigen values $\lambda_{1}=0$, $\lambda_{2}=0$, $\lambda_{3}=\gamma\epsilon$, $\lambda_{4}=1-\gamma\epsilon$.
Therefore,

and

Our Proposed Measure: For the same example, the value of $f(t)$ defined in Eq.(20) turns out to be

Therefore, our proposed measure becomes

Comparing Eqs.(43) and (45), it turns out that the two measures are related by $\mathcal{I}=2\mathcal{M}$.

It is worth noting that a straightforward calculation for other well-known channels, such as the depolarizing channel, reveals that the RHP measure is twice that of our proposed measure. However, establishing a general relationship between the two measures for arbitrary channels remains a subject of further investigation.