Tag: Pseudo-density operator

Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography

Recently, we post the paper “temporal kirkwood-dirac quasiprobability distribution and unification of temporal state formalisms through temporal bloch tomography” on arXiv. This is joint work with Kavan Modi and Dagomir Kaszlikowski, in which we extend the Kirkwood–Dirac (KD) quasiprobability distribution to the temporal setting and develop a unified framework for temporal states.

The traditional (spatial) KD quasiprobability distribution is defined for density operators, which serve as a phase-space–like representation of a quantum state (see this post for a brief introduction). In this work, we generalize the KD formalism to temporal quantum processes, which consist of an initial state together with a sequence of quantum evolutions, \mathfrak{P} = (\rho_{t_0}, \mathcal{E}_{t_1 \leftarrow t_0}, \ldots, \mathcal{E}_{t_n \leftarrow t_{n-1}}), where \rho_{t_0} is the initial state and each \mathcal{E}_{t_j \leftarrow t_{j-1}} denotes a completely positive trace-preserving (CPTP) map describing the evolution from t_{j-1} to t_j. The system state at an intermediate time t_k is then obtained recursively as \rho_{t_k} = \mathcal{E}_{t_k \leftarrow t_{k-1}} \circ \cdots \circ \mathcal{E}_{t_1 \leftarrow t_0}(\rho_{t_0}).

Within this framework, we define the multi-time doubled temporal KD quasiprobability distribution by

\overleftrightarrow{Q}_{KD}(a_n,\ldots,a_0; b_n,\ldots,b_0)

= \mathrm{Tr}[ \Pi_{a_n|A_n}^{t_n} \mathcal{E}_{t_n \leftarrow t_{n-1}}(\cdots \Pi_{a_1|A_1}^{t_1}( \mathcal{E}_{t_1 \leftarrow t_0}( \Pi_{a_0|A_0}^{t_0} \rho_{t_0} \Pi_{b_0|B_0}^{t_0})) \Pi_{b_1|B_1}^{t_1} \cdots) \Pi_{b_n|B_n}^{t_n}]

By taking the left and right marginals of this doubled distribution, we recover the corresponding temporal KD quasiprobability distributions \overleftarrow{Q}_{KD}(a_n,\ldots,a_0) and \overrightarrow{Q}_{KD}(b_n,\ldots,b_0). These two distributions are related by complex conjugation, reflecting the intrinsic temporal structure encoded in the doubled formalism. The temporal Margenau-Hill (MH) quasiprobability distribution is defined as real part of the temporal KD distribution.

The temporal KD quasiprobability may take values outside the interval [0,1]. This phenomenon reflects the intrinsically quantum nature of multi-time processes, in direct analogy with the negativity and nonclassical features of the spatial KD quasiprobability distribution. Using the Heisenberg picture, we provide a general and unified explanation for the origin of this quantumness.

A crucial application of temporal KD quasiprobability distribution is that it provides a general framework for temporal state.

There exist several formalisms of temporal states. In our paper, we show that using the temporal KD quasiprobability distribution, we can compute correlators of Pauli operators at different time steps. Then, by applying the Bloch representation of the state, we can derive several distinct temporal state formalisms. This provides a unified picture of temporal states.

For example, with doubled temporal KD distribution, we obtain the doubled KD temporal state:

\overleftrightarrow{\Upsilon}= \frac{1}{d^{n+1}}\sum_{\mu_0,\ldots,\mu_n \atop \nu_0,\ldots,\nu_n=0}^{d^2-1}\overleftrightarrow{T}^{\mu_n,\ldots,\mu_0;\,\nu_n,\ldots,\nu_0}\left(\bigotimes_{i=0}^n \sigma_{\mu_i}\right)\otimes\left(\bigotimes_{j=0}^n \sigma_{\nu_j}\right)

This is the most information-complete temporal state, and we show that it coincides with the doubled density operator. The temporal states obtained from the left or right temporal KD distribution correspond to the left and right reduced states of the doubled KD temporal state. Similarly, the left and right MH distributions coincide and give the left/right MH temporal states. For the two-time case, the left and right MH temporal states are identical to the pseudo-density operator; however, for general multi-time processes, they are no longer the same. In general, the MH temporal states can be understood as the Hermitianization of the corresponding KD temporal state.