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arXiv:1502.07030 · Physical Review A

Recovering full coherence in a qubit by measuring half of its environment

Filippo M. Miatto, Kevin Piché, Thomas Brougham, Robert W. Boyd

Published Dec 18, 2015• 7 citations
arXivHTMLPDF

Abstract

When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than $\sqrt{D}$, where $D$ is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about $\sqrt{D}$ degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources.

DOI
10.1103/physreva.92.062331

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When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than D\sqrt{D}D​, where DDD is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about D\sqrt{D}D​ degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources.
In practical applications of quantum information science, quantum systems can have non-negligible interactions with the environment, and this generally degrades the power of quantum protocols as it introduces noise. Counteracting this by appropriately measuring the environment (and therefore projecting its state) would require access all the necessary degrees of freedom, which in practice can be far too hard to achieve. To better understand one's limitations, we calculate the upper bound of optimal quantum erasure (i.e. the highest recoverable visibility, or "coherence"), when erasure is realistically limited to an accessible subspace of the whole environment. In the particular case of a two-dimensional accessible environment, the bound is given by the sub-fidelity of two particular states of the \emph{inaccessible} environment, which opens a new window into understanding the connection between correlated systems. We also provide an analytical solution for a three-dimensional accessible environment. This result provides also an interesting operational interpretation of sub-fidelity. We end with a statistical analysis of the expected visibility of an optimally erased random state and we find that 1) if one picks a random pure state of 2 qubits, there is an optimal measurement that allows one to distill a 1-qubit state with almost 90\% visibility and 2) if one picks a random pure state of 2 qubits in an inaccessible environment, there is an optimal measurement that allows one to distill a 1-qubit state with almost twice its initial visibility.
Quantum correlations between two systems can be impaired by the presence of an environment, as it can make systems decohere. We wish to evaluate how much coherence has been irreversibly lost. To do so, we study two qubits which leak correlations into an arbitrarily large environment and we find that our ability to steer one of them to the most coherent set of states possible by acting on the other is given by the sub-fidelity of two conditional states of the environment. Sub-fidelity is a quantity that was already known, but that until now lacked an operational interpretation. Furthermore, we conjecture the existence of a family of measures, of which fidelity and sub-fidelity are the first two, which measure the irreversibility of decoherence as we gain access to more and more degrees of freedom of the enviroment.
Macroscopic devices that probe and control elementary quantum systems are subjected to environment-induced decoherence and typically consist of a large number of particles that cannot be precisely controlled and individually accessed. These two factors limit the control of quantum systems by an open macroscopic device. Here, we investigate a faithful and robust implementation of specific elementary quantum operations under such conditions. We propose a procedure for quantum restoration of a single qubit that is resilient to particle loss and decoherence of the macroscopic device and compare it with a similarly robust procedure for indirect quantum detection of the qubit state by measurement on the macroscopic device. Complementary to detection, the restoration recovers the original state of the system qubit by quantum erasing which decouples the system from the device. Perfect restoration is possible even if the particles of the device are only classically correlated. We experimentally witness the robustness of quantum detection and restoration for a two-qubit device represented by the quantum state of photons.
We have in mind a register of qubits for an quantum information system, and consider its decoherence in an idealized but typical situation. Spontaneous decay and other couplings to the far environment, considered as the world outside the quantum apparatus, will be neglected, while couplings to quantum states within the apparatus, i.e., to a near environment, are assumed to dominate. Thus the central system couples to the near environment, which in turn couples to a far environment. Considering that the dynamics in the near environment is not sufficiently well known or controllable, we shall use random matrix methods to obtain analytic results. We consider a simplified situation where the central system suffers weak dephasing from the near environment, which in turn is coupled randomly to the far environment. We find the anti-intuitive result that increasing the coupling between the near and far environment actually protects the central qubit.
This work presents Version 2.0 of Quantum Decoherence and Conditional Recoherence as Loss and Restoration of Shared Phase Influence, a clarification and extension of standard decoherence theory within orthodox quantum mechanics.The paper reframes quantum decoherence as the physical loss of shared phase influence between components of a quantum system due to unavoidable entanglement with environmental degrees of freedom. Using standard density-matrix and system–environment coupling formalisms, decoherence is shown to correspond to dispersal of phase information into the environment rather than destruction of quantum states or invocation of fundamental wavefunction collapse.Conditional recoherence is treated within the same framework, demonstrating that restoration of coherence is possible only under constrained physical conditions where phase information has not irreversibly propagated into uncontrolled environmental modes. Known mechanisms such as spin-echo refocusing, dynamical decoupling, and quantum error correction are shown to be consistent with this interpretation.This Version 2.0 manuscript emphasizes structural clarity over speculative interpretation. It preserves unitary quantum dynamics, avoids observer-dependent or consciousness-based claims, and remains fully compatible with open-quantum-system formalisms including Lindblad-type master equations. The work provides a clearer physical account of measurement, irreversibility, and classical emergence without introducing new forces or modifying established quantum theory.The manuscript is intended for researchers and educators working in quantum foundations, open quantum systems, and quantum information theory, and is suitable for citation and pedagogical use.
Effective quantum-information processing requires coherent control of large numbers of qubits on a time scale that is short compared to the decoherence time of the system. It is therefore important to extrapolate and measure decoherence times for large quantum registers and to determine the effect of different couplings between system and environment on the decoherence rate. For this purpose, we have experimentally realized a system that allows one to generate model quantum registers with more than 100 qubits and measure the decay of the information in these states while adjusting the strength of the interaction between the quantum register and the environment. Our results indicate a power-law dependence of the decoherence rate on the number of qubits in the system, with an exponent of the order of 0.5. This behavior remains qualitatively unchanged when the coupling strength to the environment is reduced by about an order of magnitude.