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arXiv:1011.4452

Entanglement with restricted measurement setups

Sebastian Meznaric

Published Nov 19, 2010• 1 citation
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Abstract

We consider what is the effective amount of entanglement when the measurement operators one has at their disposal are restricted. Such a scenario occurs when superselection rules are in effect or when there are imperfections in our measurement setup. Given a quantum state and the restrictions on measurements, we consider the following scenario. Imagine we have an ideal, or non-restricted, measurement setup and a state $\rho_1$ and a non-ideal, or restricted, measurement setup and a state $\rho_2$. Then the minimum amount of entanglement in $\rho_1$ so that all quantum communication protocols still perform with the same fidelity as with $\rho_2$ is effectively the entanglement with restricted measurement setup. For indistinguishable particles, we find that any quantum communication protocol that can be performed with indistinguishable particles, can thus be performed with a ``normal'' state of no more than $E_P$ of entanglement. For the imperfect measurement apparatus we find an upper bound for the effective entanglement, which is a linear function of the usual entanglement, with the proportionality factor dependent only on the imperfections in the measurement device.

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We consider what is the effective amount of entanglement when the measurement operators one has at their disposal are restricted. Such a scenario occurs when superselection rules are in effect or when there are imperfections in our measurement setup. Given a quantum state and the restrictions on measurements, we consider the following scenario. Imagine we have an ideal, or non-restricted, measurement setup and a state ρ1\rho_1ρ1​ and a non-ideal, or restricted, measurement setup and a state ρ2\rho_2ρ2​. Then the minimum amount of entanglement in ρ1\rho_1ρ1​ so that all quantum communication protocols still perform with the same fidelity as with ρ2\rho_2ρ2​ is effectively the entanglement with restricted measurement setup. For indistinguishable particles, we find that any quantum communication protocol that can be performed with indistinguishable particles, can thus be performed with a ``normal'' state of no more than EPE_PEP​ of entanglement. For the imperfect measurement apparatus we find an upper bound for the effective entanglement, which is a linear function of the usual entanglement, with the proportionality factor dependent only on the imperfections in the measurement device.
There always exists an entanglement witness for every entangled quantum state. Negativity of the expectation value of an entanglement witness operator guarantees entanglement of the corresponding state, given that the measurement devices involved are perfect, i.e., the performed measurements actually constitute the witness operator for the state under consideration. In a realistic situation, there are two possible ways of measurements to drive the process away from the ideal one. First, wrong measurements may be performed, and, second, while the measurement operators are implemented correctly, the detection process may be noisy. Entanglement witnesses are prone to both of these imperfections. The concept of measurement-device-independent entanglement witnesses was introduced to remove the first problem. We analyze the ``detection loophole'' in the context of measurement-device-independent entanglement witnesses, which deal with the second problem of imprecise measurements. We obtain an upper bound, a threshold value, on the entanglement witness function in the measurement-device-independent entanglement witness scenario, below which entanglement is guaranteed for given nonideal detector efficiencies, that can involve both lost events and dark counts. We also present a threshold value on the function in the ``worst-case scenario'' where the threshold of the witness operator for guaranteeing entanglement assumes its maximal negative value.
Superselection rules severely constrain the operations which can be implemented on a distributed quantum system. While the restriction to local operations and classical communication gives rise to entanglement as a nonlocal resource, particle number conservation additionally confines the possible operations and should give rise to a new resource. In Phys. Rev. Lett. 92, 087904 (2004) we showed that this resource can be quantified by a single additional number, the superselection-induced variance (SIV)(\mathrm{SIV})(SIV), without changing the concept of entanglement. In this paper, we give the results on pure states in greater detail; additionally, we provide a discussion of mixed-state nonlocality with superselection rules where we consider both formation and distillation. Finally, we demonstrate that SIV\mathrm{SIV}SIV is indeed a resource, i.e., that it captures how well a state can be used to overcome the restrictions imposed by the superselection rule.
Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement, namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe these models as well as the motivations for studying them. For the bounded measurement model and the depolarization model, we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model, we prove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes communication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality of the BB84 protocol in terms of communication complexity. We emphasize that although some of the results appear intuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results. We believe that these techniques are of independent interests, too.
Abstract Entanglement is an essential resource for quantum com-munication and quantum computation, similar to shared random bits in the classical world. Entanglement distil-lation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity ofthese protocols is the minimal amount of classical information that needs to be exchanged for the conversion. In thispaper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., wherethe inputs are mixed states and/or prepared adversarially. We consider three models of imperfect entanglement,namely, the bounded measurement model, the depolarization model, and the fidelity model. We describe there mod-els as well as the motivations for studying them. For the bounded measurement model and the depolarization model,we prove tight and almost-tight bounds on the output quality of non-interactive protocols. For the fidelity model weprove a lower bound that matches the upper bound given by Ambainis et al., and thus completely characterizes commu-nication complexity of entanglement distillation protocols for this model. Our result also suggests the optimality ofthe BB84 protocol in terms of communication complexity. We emphasize that although some of the results appearintuitively straightforward, their proofs are not. In fact, two novel techniques are developed for proving these results.We believe that these techniques are of independent interests, too.