Abstract
In this work, we study the circuit complexity for generalized coherent states in thermal systems by adopting the covariance matrix approach. We focus on the coherent thermal (CT) state, which is non-Gaussian and has a nonvanishing one-point function. We find that even though the CT state cannot be fully determined by the symmetric two-point function, the circuit complexity can still be computed in the framework of the covariance matrix formalism by properly enlarging the covariance matrix. Now the group generated by the unitary is the semiproduct of translation and the symplectic group. If the reference state is Gaussian, the optimal geodesic is still be generated by a horizontal generator such that the circuit complexity can be read from the generalized covariance matrix associated to the target state by taking the cost function to be $F_2$. For a single harmonic oscillator, we discuss carefully the complexity and its formation in the cases that the reference states are Gaussian and the target space is excited by a single mode or double modes. We show that the study can be extended to the free scalar field theory.
Highlights
Complexity has been a focus in the recent study of the AdS=CFT correspondence [1] and black hole physics
We find that even though the coherent thermal (CT) state cannot be fully determined by the symmetric two-point function, the circuit complexity can still be computed in the framework of the covariance matrix formalism by properly enlarging the covariance matrix
There are two proposals put forward by Susskind and his collaborators to quantify the size of the Einstein-Rosen bridge (ERB): one is the “complexity 1⁄4 volume”(CV) conjecture [5], which states that the holographic complexity is given by the volume of the codimension-1 maximal spacelike surface in the bulk connecting the left and right sides; the other is the “complexity 1⁄4 action”(CA) conjecture [6,7], which states that the holographic complexity is captured by the gravitational action of the bulk region known as the
Summary
Complexity has been a focus in the recent study of the AdS=CFT correspondence [1] and black hole physics. The optimal circuit was determined geometrically by the minimal geodesic in the space of unitaries Uwith a suitable metric, as developed by Nielsen and his collaborators [39] This approach has been applied to free fermionic theories in [40,41]. The study of the complexity has been generalized to the TFD state in free scalar field theory [49,50,51,52]. We extend our study on the complexity of the CT state to the free scalar field theory by fixing the reference state to be the Dirac vacuum state.
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