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arXiv:1609.06994 · Physical Review A

Deconstruction and conditional erasure of quantum correlations

Mario Berta, Fernando G. S. L. Brandão, Christian Majenz, Mark M. Wilde

Published Oct 15, 2018• 29 citations
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Abstract

We define the deconstruction cost of a tripartite quantum state on systems $ABE$ as the minimum rate of noise needed to apply to the $AE$ systems, such that there is negligible disturbance to the marginal state on the $BE$ systems, while the system $A$ of the resulting state is locally recoverable from the $E$ system alone. We refer to such actions as deconstruction operations and protocols implementing them as state deconstruction protocols. State deconstruction generalizes Landauer erasure of a single-party quantum state as well the erasure of correlations of a two-party quantum state. We find that the deconstruction cost of a tripartite quantum state on systems $ABE$ is equal to its conditional quantum mutual information (CQMI) $I(A;B|E)$, thus giving the CQMI an operational interpretation in terms of a state deconstruction protocol. We also define a related task called conditional erasure, in which the goal is to apply noise to systems $AE$ in order to decouple system $A$ from systems $BE$, while causing negligible disturbance to the marginal state of systems $BE$. We find that the optimal rate of noise for conditional erasure is also equal to the CQMI $I(A;B|E)$. State deconstruction and conditional erasure lead to operational interpretations of the quantum discord and squashed entanglement, which are quantum correlation measures based on the CQMI. We find that the quantum discord is equal to the cost of simulating einselection, the process by which a quantum system interacts with an environment, resulting in selective loss of information in the system. The squashed entanglement is equal to half the minimum rate of noise needed for deconstruction and/or conditional erasure if Alice has available the best possible system $E$ to help in the deconstruction and/or conditional erasure task.

DOI
10.1103/physreva.98.042320

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We define the deconstruction cost of a tripartite quantum state on systems ABEABEABE as the minimum rate of noise needed to apply to the AEAEAE systems, such that there is negligible disturbance to the marginal state on the BEBEBE systems, while the system AAA of the resulting state is locally recoverable from the EEE system alone. We refer to such actions as deconstruction operations and protocols implementing them as state deconstruction protocols. State deconstruction generalizes Landauer erasure of a single-party quantum state as well the erasure of correlations of a two-party quantum state. We find that the deconstruction cost of a tripartite quantum state on systems ABEABEABE is equal to its conditional quantum mutual information (CQMI) I(A;B∣E)I(A;B|E)I(A;B∣E), thus giving the CQMI an operational interpretation in terms of a state deconstruction protocol. We also define a related task called conditional erasure, in which the goal is to apply noise to systems AEAEAE in order to decouple system AAA from systems BEBEBE, while causing negligible disturbance to the marginal state of systems BEBEBE. We find that the optimal rate of noise for conditional erasure is also equal to the CQMI I(A;B∣E)I(A;B|E)I(A;B∣E). State deconstruction and conditional erasure lead to operational interpretations of the quantum discord and squashed entanglement, which are quantum correlation measures based on the CQMI. We find that the quantum discord is equal to the cost of simulating einselection, the process by which a quantum system interacts with an environment, resulting in selective loss of information in the system. The squashed entanglement is equal to half the minimum rate of noise needed for deconstruction and/or conditional erasure if Alice has available the best possible system EEE to help in the deconstruction and/or conditional erasure task.
Insights from quantum information theory show that correlation measures based on quantum entropy are fundamental tools that reveal the entanglement structure of multipartite states. In that spirit, Groisman, Popescu, and Winter [Phys. Rev. A 72, 032317 (2005)PLRAAN1050-294710.1103/PhysRevA.72.032317] showed that the quantum mutual information I(A;B) quantifies the minimal rate of noise needed to erase the correlations in a bipartite state of quantum systems AB. Here, we investigate correlations in tripartite systems ABE. In particular, we are interested in the minimal rate of noise needed to apply to the systems AE in order to erase the correlations between A and B given the information in system E, in such a way that there is only negligible disturbance on the marginal BE. We present two such models of conditional decoupling, called deconstruction and conditional erasure cost of tripartite states ABE. Our main result is that both are equal to the conditional quantum mutual information I(A;B|E)-establishing it as an operational measure for tripartite quantum correlations.
Erasure of information stored in a quantum state requires energy cost and is inherently an irreversible operation. If quantumness of a system is physical, does erasure of quantum correlation as measured by discord also need some energy cost? Here, we show that change in quantum correlation is never larger than the total entropy change of the system and the environment. The entropy cost of erasing correlation has to be at least equal to the amount of quantum correlation erased. Hence, quantum correlation can be regarded as genuinely physical. We show that the new bound leads to the Landauer erasure. The physical cost of erasing quantum correlation is well respected in the case of bleaching of quantum information, thermalization, and can have potential application for any channel leading to erasure of quantum correlation.
We show that the minimal rate of noise needed to catalytically erase the entanglement in a bipartite quantum state is given by the regularized relative entropy of entanglement. This offers a solution to the central open question raised in [Groisman et al., Phys. Rev. A 72, 032317 (2005)PLRAAN1050-294710.1103/PhysRevA.72.032317] and complements their main result that the minimal rate of noise needed to erase all correlations is given by the quantum mutual information. We extend our discussion to the tripartite setting where we show that an asymptotic rate of noise given by the regularized relative entropy of recovery is sufficient to catalytically transform the state to a locally recoverable version of the state.
In this brief report we revisit the concept of dissension, which was introduced as a natural extension of quantum discord for three qubit system. Here we bring in new expression for quantum dissensions and more interestingly we name one such expression as \textit{residual correlation}. The basic objective behind the introduction of such a quantity is to capture the extra amount of correlation generated by doing measurement in a correlated system from a situation where we do not bring in a correlated system in the measurement process. Apart from this we also present an operational interpretation of this correlation in context of state merging. Remarkably, we find that for three qubit system if one discards relevant prior information, the change in the cost of state merging ( merging the quantum information of two parties into one) is captured by the \textit{residual correlation}. In addition to that we found that this quantity can be negative for mixed states. This indeed opens up a new dimension in the tripartite scenario where we can observe situations where there is a decrease in the cost of state merging on discarding relevant prior information. We claim that this result establishes a re conceptualization of information processing tasks in tripartite situations where we can use suitable measurement and states to bring down the cost of the protocol.
Erasing memory is a fundamental operational task in quantum information processing, governed by Landauer's principle, which links information loss to thermodynamic work. We introduce and analyze assisted quantum erasure, where correlations with a remote system reduce the energetic cost of resetting a memory. We identify exclusive control of erasure as the central operational requirement: only a designated party should be able to achieve the minimal cost, whereas any adversary must fail. In the device-dependent regime, we show that entanglement of formation exactly characterizes exclusivity, establishing entanglement as the decisive thermodynamic resource. Moving to a one-sided device-independent scenario, in which only the memory holder's device is trusted, we develop an operational erasure protocol based on random dephasing and conditional operations. Finally, in a fully device-independent setting, we show how Bell nonlocality and self-testing translate observed violations into lower bounds on any adversary's erasure capability, yielding a device-independent notion of exclusive thermodynamic control. Taken together, these results elevate quantum erasure from a thermodynamic constraint to an operational primitive: the erasure work cost quantifies secure, exclusive control over quantum memory, ensuring that an unauthorized agent cannot fully erase information under a bounded work budget.
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.