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arXiv:1801.08102 · Physical Review A

Energy-constrained two-way assisted private and quantum capacities of quantum channels

Noah Davis, Максим Евгеньевич Широков, Mark M. Wilde

Published Jun 7, 2018• 24 citations
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Abstract

With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential for their effective implementation. A particular communication setting that has received increased focus is related to quantum key distribution and distributed quantum computation. In this setting, a quantum channel connects a sender to a receiver, and their goal is to distill either a secret key or entanglement, along with the help of arbitrary local operations and classical communication (LOCC). In this work, we establish a general theory of energy-constrained, LOCC-assisted private and quantum capacities of quantum channels, which are the maximum rates at which an LOCC-assisted quantum channel can reliably establish a secret key or entanglement, respectively, subject to an energy constraint on the channel input states. We prove that the energy-constrained squashed entanglement of a channel is an upper bound on these capacities. We also explicitly prove that a thermal state maximizes a relaxation of the squashed entanglement of all phase-insensitive, single-mode input bosonic Gaussian channels, generalizing results from prior work. After doing so, we prove that a variation of the method introduced by Goodenough et al. [New J. Phys. 18, 063005 (2016)] leads to improved upper bounds on the energy-constrained secret-key-agreement capacity of a bosonic thermal channel. We then consider a multipartite setting and prove that two known multipartite generalizations of the squashed entanglement are in fact equal. We finally show that the energy-constrained, multipartite squashed entanglement plays a role in bounding the energy-constrained LOCC-assisted private and quantum capacity regions of quantum broadcast channels.

DOI
10.1103/physreva.97.062310

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With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential for their effective implementation. A particular communication setting that has received increased focus is related to quantum key distribution and distributed quantum computation. In this setting, a quantum channel connects a sender to a receiver, and their goal is to distill either a secret key or entanglement, along with the help of arbitrary local operations and classical communication (LOCC). In this work, we establish a general theory of energy-constrained, LOCC-assisted private and quantum capacities of quantum channels, which are the maximum rates at which an LOCC-assisted quantum channel can reliably establish a secret key or entanglement, respectively, subject to an energy constraint on the channel input states. We prove that the energy-constrained squashed entanglement of a channel is an upper bound on these capacities. We also explicitly prove that a thermal state maximizes a relaxation of the squashed entanglement of all phase-insensitive, single-mode input bosonic Gaussian channels, generalizing results from prior work. After doing so, we prove that a variation of the method introduced by Goodenough et al. [New J. Phys. 18, 063005 (2016)] leads to improved upper bounds on the energy-constrained secret-key-agreement capacity of a bosonic thermal channel. We then consider a multipartite setting and prove that two known multipartite generalizations of the squashed entanglement are in fact equal. We finally show that the energy-constrained, multipartite squashed entanglement plays a role in bounding the energy-constrained LOCC-assisted private and quantum capacity regions of quantum broadcast channels.
In order to efficiently put quantum technologies into action, we must know the characteristics of the underlying quantum systems and effects. An interesting example is the use of the secret-key-agreement capacity of a quantum channel as a guide and measure for the implementation of quantum key distribution (QKD) and distributed quantum computation. We define the communication task of establishing a secret key over a quantum channel subject to an energy constraint on the input state and while allowing for unlimited local operations and classical communication (LOCC) between a sender and receiver. We then use the energy-constrained squashed entanglement to bound the capacity of the channel for secret-key agreement, and we show that a thermal state input maximizes a relaxation of this bound for phase-insensitive, single-mode Gaussian channels. We also establish improved upper bounds on the energy-constrained secret-key-agreement capacity for a bosonic thermal channel that is not entanglement breaking. We then generalize our results to the multipartite setting and show that the energy-constrained multipartite squashed entanglement bounds the LOCC-assisted private capacity region for a quantum broadcast channel. Next, we define the broadcast amplitude damping channel. In the setting of QKD, we discuss a communication task using the broadcast amplitude damping channel and give bounds on its achievable rate region.
We study the LOCC-assisted quantum capacity of a bosonic dephasing channel with energy constraint on the input states. We start our analysis by focusing on the energy-constrained squashed entanglement of the channel, which is an upper bound for the energy-constrained LOCC-assisted quantum capacity. As computing energy-constrained squashed entanglement of the channel is challenging due to a double optimization (over the set of density matrices and the isometric extensions of a squashing channel), we first derive an upper bound for it, and then, we discuss how tight that bound is for the energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing channel. In doling so, we prove that the optimal input state is diagonal in the Fock basis. Then, we analyze two explicit examples of squashing channels through which we derive explicit upper and lower bounds for the energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing channel in terms of its quantum capacity with different noise parameters. As the difference between upper and lower bounds becomes smaller by increasing the dephasing parameter, the bounds become tighter.
We define the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed entanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Christandl, Winter, et al., establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1 + η)=(1 - η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1=(1 - η)). Thus, in the high-loss regime for which η ≪ 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal.
The squashed entanglement of a quantum channel is an additive function of quantum channels, which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication. This quantity has led to an upper bound of log((1+η)/(1-η)) on the capacity of a pure-loss bosonic channel for such a task, where η is the average fraction of photons that make it from the input to the output of the channel. The purpose of this paper is to extend these results beyond the single-sender single-receiver setting to the more general case of a single sender and multiple receivers (a quantum broadcast channel). We employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel, along the way developing several new properties of these measures. We apply our results to the case of a pure-loss broadcast channel with one sender and two receivers.
The squashed entanglement of a quantum channel is an additive function of quantum channels, which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication. This quantity has led to an upper bound of log((1+eta)/(1−eta))\\log ((1+\\eta )/(1-\\eta ))log((1+eta)/(1−eta)) on the capacity of a pure-loss bosonic channel for such a task, where $\\eta $ is the average fraction of photons that make it from the input to the output of the channel. The purpose of this paper is to extend these results beyond the single-sender single-receiver setting to the more general case of a single sender and multiple receivers (a quantum broadcast channel). We employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel, along the way developing several new properties of these measures. We apply our results to the case of a pure-loss broadcast channel with one sender and two receivers.
We determine the exact quantum communication and entanglement-distribution capacities of three fundamental bosonic channels, namely thermal loss, thermal amplification and additive Gaussian noise, assisted by operations that completely preserve positive partial transpose (PPT). These operations are more powerful than local operations and classical communication (LOCC), but cannot generate distillable entanglement on their own. For thermal loss, the unconstrained capacity coincides with the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound throughout the non-entanglement-breaking regime. The corresponding upper bounds are also attained for thermal amplification and additive noise. We extend these results to energy-constrained communication, establishing exact capacities for all three channels at every prescribed mean input energy. Our results solve a class of bosonic capacity problems under PPT assistance and establish exact benchmarks for determining whether the same communication rates can be achieved via LOCC alone.