[Paper Review] Transmitting quantum information by superposing causal order of mutually unbiased measurements

Manish K. Gupta, Ujjwal Sen|arXiv (Cornell University)|Sep 28, 2019

Quantum Information and Cryptography4 citations

TL;DR

This paper demonstrates that superposing the causal order of two mutually unbiased projective measurements via a quantum switch preserves quantum information in the output state, contrary to the incoherent case where such measurements fully erase input information. The coherence of the control qubit enables non-trivial off-diagonal elements in the output density matrix, enabling quantum information transfer despite the non-commutativity and mutual unbiasedness of the measurements.

ABSTRACT

Two quantum measurements sequentially acting one after the other, if they are mutually unbiased, will lead to a complete removal of information encoded in the input quantum state. We find that if the order of the two sequential measurements can be superposed, with a quantum switch, then the information encoded in the input can still be retained in the output state.

Motivation & Objective

• To investigate whether coherent superposition of causal orders in sequential measurements can preserve quantum information when the measurements are mutually unbiased.
• To contrast the quantum information content of the output state under coherent (quantum switch) versus incoherent (classical mixture) ordering of measurements.
• To generalize the effect from qubits (d=2) to qudits (d>2) using mutually unbiased bases (MUBs).
• To explore the implications for quantum communication protocols relying on MUBs, such as quantum key distribution.

Proposed method

• The authors use a quantum switch to coherently superpose two sequential measurement orders: first M1 then M2, and first M2 then M1, using a control qubit.
• For a d-dimensional Hilbert space, mutually unbiased bases (MUBs) are constructed, with measurement operators defined as projectors onto MUB vectors.
• The Kraus operators for the combined process are defined as $\widetilde{\mathcal{W}}_{ij} = |0\rangle\!\langle0| \otimes M^{k}_{i}M^{l}_{j} + |1\rangle\!\langle1| \otimes M^{l}_{j}M^{k}_{i}$, encoding the two causal orders.
• The output state is computed as $\widetilde{\mathcal{N}}(\rho_c \otimes \tilde{\rho}_s) = \sum_{i,j} \widetilde{\mathcal{W}}_{ij} (\rho_c \otimes \tilde{\rho}_s) \widetilde{\mathcal{W}}_{ij}^\dagger$, with the control qubit $\rho_c$ in a superposition state.
• Analytical calculations are performed for small dimensions (d=2, d=3) to show that the output state retains off-diagonal terms when the control qubit is coherent.
• The incoherent case is modeled by setting $\rho_c = p|0\rangle\!\langle0| + (1-p)|1\rangle\!\langle1|$, showing that off-diagonal terms vanish, leading to a maximally mixed state.

Experimental results

Research questions

• RQ1 Can coherent superposition of causal orders in sequential measurements preserve quantum information when the measurements are mutually unbiased?
• RQ2 How does the quantum switch enable information retention despite the complete randomness of outcomes in mutually unbiased measurements?
• RQ3 What is the role of control qubit coherence in maintaining quantum information in the output state?
• RQ4 Does the preservation of quantum information via causal superposition hold beyond qubits (d=2) in higher-dimensional Hilbert spaces?
• RQ5 Can this mechanism provide a quantum advantage in protocols relying on MUBs, such as quantum key distribution?

Key findings

• When the causal order of two mutually unbiased projective measurements is coherently superposed via a quantum switch, the output state retains non-zero off-diagonal elements in its density matrix, indicating preserved quantum information.
• In contrast, when the two measurement orders are applied incoherently (classical mixture), the output state becomes maximally mixed, with all off-diagonal elements vanishing.
• The preservation of quantum information is directly linked to the coherence of the control qubit; when the control qubit is in a mixed state, the output state becomes diagonal and information-erased.
• For d=2, the output state under the quantum switch is not diagonal, even though each individual measurement sequence would fully randomize the input state due to mutual unbiasedness.
• The result generalizes to d-dimensional qudits: the output state remains non-diagonal and dependent on input coherence when the control qubit is in superposition, confirming the robustness of the effect in higher dimensions.
• The mechanism demonstrates a form of 'super-activation' where non-commutativity and causal superposition jointly enable information retention despite the destructive nature of sequential mutually unbiased measurements in the incoherent case.