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Fidelity preserving and decoherence for mixed unitary quantum channels

Kai Liu, Deguang Han

Abstract

Distinguishable and non-distinguishable quantum states are fundamental resources in quantum mechanics and quantum technologies. Interactions with the environment often induce decoherence, impacting both the distinguishability and non-distinguishability between quantum states. In this paper, we investigate mixed unitary quantum channels and the conditions under which fidelity, a measure of quantum state closeness, is preserved. More precisely, for quantum channels in the form $Φ(ρ) = \sum_{i=1}^N p_i U_i ρU_i^*$, we analyze their effect on quantum state $|\varphi\rangle$ through the associated purification $|Ψ\rangle$, explore the structure of such quantum channels that preserve either distinguishable or non-distinguishable states and then discuss the challenges of maintaining fidelity, particularly under the influence of phase damping.

Fidelity preserving and decoherence for mixed unitary quantum channels

Abstract

Distinguishable and non-distinguishable quantum states are fundamental resources in quantum mechanics and quantum technologies. Interactions with the environment often induce decoherence, impacting both the distinguishability and non-distinguishability between quantum states. In this paper, we investigate mixed unitary quantum channels and the conditions under which fidelity, a measure of quantum state closeness, is preserved. More precisely, for quantum channels in the form , we analyze their effect on quantum state through the associated purification , explore the structure of such quantum channels that preserve either distinguishable or non-distinguishable states and then discuss the challenges of maintaining fidelity, particularly under the influence of phase damping.

Paper Structure

This paper contains 15 sections, 20 theorems, 91 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Fidelity preserving for distinguishable quantum states
  3. Preserve distinguishable states in single Qubit system
  4. Preserve distinguishable states in higher dimensional space or multi-qubits system
  5. Quantum system as a d-dimensional space
  6. Quantum system as a Multi-qubits system
  7. Fidelity preserving for Non-distinguishable states
  8. Decoherence and General phase damping channel
  9. General Phase damping channel
  10. Decoherence under the general phase damping channel
  11. Quantum circuit implementation
  12. Appendices
  13. A. Unitary matrix preserves distinguishable states in Qutrit system are block diagonal
  14. B. Controlled phase damping channel is not a tensor channel
  15. C. Dephasing rate and decoherence factor of phase damping channel

Key Result

Theorem 1

gupta2015 Suppose that $\Phi: B(H)\rightarrow B(K)$ is a quantum channel with Kraus representation $\Phi(\rho) = \sum_{j=1}^{r}A_{j}\rho A_{j}^{*}$ for some $A_{j}\in B(H,K)$, $1\leq j\leq r$. Then $\Phi(\rho) = \sum_{j=1}^{m}B_{j}\rho B_{j}^{*}$ for operators $B_{j}\in B(H,K)$, $1\leq j\leq m$ if a

Figures (5)

  • Figure 4.1: Coherence decay for Phase damping channel
  • Figure 4.2: Coherence decay under a rank 2 general phase damping error with relative phase $\theta$ and mix parameter p
  • Figure 4.3:
  • Figure 4.4:
  • Figure 4.5: Quantum circuit for general phase damping.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof
  • Remark 1
  • Remark 2
  • Lemma 6
  • Proposition 7
  • ...and 22 more