Entanglement cost and quantum channel simulation

This paper proposes a revised definition for the entanglement cost of a quantum channel $\mathcal{N}$. In particular, it is defined here to be the smallest rate at which entanglement is required, in addition to free classical communication, in order to simulate $n$ calls to $\mathcal{N}$, such that the most general discriminator cannot distinguish the $n$ calls to $\mathcal{N}$ from the simulation. The most general discriminator is one who tests the channels in a sequential manner, one after the other, and this discriminator is known as a quantum tester [Chiribella et al., Phys. Rev. Lett., 101, 060401 (2008)] or one who is implementing a quantum co-strategy [Gutoski et al., Symp. Th. Comp., 565 (2007)]. As such, the proposed revised definition of entanglement cost of a quantum channel leads to a rate that cannot be smaller than the previous notion of a channel's entanglement cost [Berta et al., IEEE Trans. Inf. Theory, 59, 6779 (2013)], in which the discriminator is limited to distinguishing parallel uses of the channel from the simulation. Under this revised notion, I prove that the entanglement cost of certain teleportation-simulable channels is equal to the entanglement cost of their underlying resource states. Then I find single-letter formulas for the entanglement cost of some fundamental channel models, including dephasing, erasure, three-dimensional Werner--Holevo channels, epolarizing channels (complements of depolarizing channels), as well as single-mode pure-loss and pure-amplifier bosonic Gaussian channels. These examples demonstrate that the resource theory of entanglement for quantum channels is not reversible. Finally, I discuss how to generalize the basic notions to arbitrary resource theories.Comment: 28 pages, 7 figure

Exact entanglement cost of quantum states and channels under PPT-preserving operations

This paper establishes single-letter formulas for the exact entanglement cost of generating bipartite quantum states and simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we establish that the exact entanglement cost of any bipartite quantum state under PPT-preserving operations is given by a single-letter formula, here called the $\kappa$-entanglement of a quantum state. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum states. Notably, this is the first time that an entanglement measure for general bipartite states has been proven not only to possess a direct operational meaning but also to be efficiently computable, thus solving a question that has remained open since the inception of entanglement theory over two decades ago. Next, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. The entanglement cost in both cases is given by the same single-letter formula and is equal to the largest $\kappa$-entanglement that can be shared by the sender and receiver of the channel. It is also efficiently computable by a semidefinite program.Comment: 54 pages, 8 figures; comments are welcome

Extendibility limits the performance of quantum processors

Resource theories in quantum information science are helpful for the study and quantification of the performance of information-processing tasks that involve quantum systems. These resource theories also find applications in other areas of study; e.g., the resource theories of entanglement and coherence have found use and implications in the study of quantum thermodynamics and memory effects in quantum dynamics. In this paper, we introduce the resource theory of unextendibility, which is associated to the inability of extending quantum entanglement in a given quantum state to multiple parties. The free states in this resource theory are the $k$-extendible states, and the free channels are $k$-extendible channels, which preserve the class of $k$-extendible states. We make use of this resource theory to derive non-asymptotic, upper bounds on the rate at which quantum communication or entanglement preservation is possible by utilizing an arbitrary quantum channel a finite number of times, along with the assistance of $k$-extendible channels at no cost. We then show that the bounds we obtain are significantly tighter than previously known bounds for both the depolarizing and erasure channels.Comment: 39 pages, 6 figures, v2 includes pretty strong converse bounds for antidegradable channels, as well as other improvement

Entanglement Cost of Quantum Channels

The entanglement cost of a quantum channel is the minimal rate at which entanglement (between sender and receiver) is needed in order to simulate many copies of a quantum channel in the presence of free classical communication. In this paper we show how to express this quantity as a regularised optimisation of the entanglement formation over states that can be generated between sender and receiver. Our formula is the channel analog of a well-known formula for the entanglement cost of quantum states in terms of the entanglement of formation; and shares a similar relation to the recently shattered hope for additivity. The entanglement cost of a quantum channel can be seen as the analog of the quantum reverse Shannon theorem in the case where free classical communication is allowed. The techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse Shannon theorem and feature the one-shot formalism for quantum information theory, the post-selection technique for quantum channels as well as Sion's minimax theorem. We discuss two applications of our result. First, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only improves the range of parameters where security can be shown, but also allows us to prove security for storage devices for which no results were known before. Second, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.Comment: v3: error in proof of Lemma 13 corrected, corrected Figure 5, 24 pages, 5 figure

Resource theory of unextendibility and nonasymptotic quantum capacity ()

In this paper, we introduce the resource theory of unextendibility as a relaxation of the resource theory of entanglement. The free states in this resource theory are the -extendible states, associated with the inability to extend quantum entanglement in a given quantum state to multiple parties. The free channels are -extendible channels, which preserve the class of -extendible states. We define several quantifiers of unextendibility by means of generalized divergences and establish their properties. By utilizing this resource theory, we obtain nonasymptotic upper bounds on the rate at which quantum communication or entanglement preservation is possible over a finite number of uses of an arbitrary quantum channel assisted by -extendible channels at no cost. These bounds are significantly tighter than previously known bounds for both the depolarizing and erasure channels. Finally, we revisit the pretty strong converse for the quantum capacity of antidegradable channels and establish an upper bound on the nonasymptotic quantum capacity of these channels

Computable lower bounds on the entanglement cost of quantum channels

A class of lower bounds for the entanglement cost of any quantum state was recently introduced in [arXiv:2111.02438] in the form of entanglement monotones known as the tempered robustness and tempered negativity. Here we extend their definitions to point-to-point quantum channels, establishing a lower bound for the asymptotic entanglement cost of any channel, whether finite or infinite dimensional. This leads, in particular, to a bound that is computable as a semidefinite program and that can outperform previously known lower bounds, including ones based on quantum relative entropy. In the course of our proof we establish a useful link between the robustness of entanglement of quantum states and quantum channels, which requires several technical developments such as showing the lower semicontinuity of the robustness of entanglement of a channel in the weak*-operator topology on bounded linear maps between spaces of trace class operators.Comment: 24 pages. Technical companion paper to [arXiv:2111.02438], now published as [Nat. Phys. 19, 184-189 (2023)]. In v2, which is close to the published version, we improved the presentation and corrected a few typo

No second law of entanglement manipulation after all

We prove that the theory of entanglement manipulation is asymptotically irreversible under all non-entangling operations, showing from first principles that reversible entanglement transformations require the generation of entanglement in the process. Entanglement is thus shown to be the first example of a quantum resource that does not become reversible under the maximal set of free operations, that is, under all resource non-generating maps. Our result stands in stark contrast with the reversibility of quantum and classical thermodynamics, and implies that no direct counterpart to the second law of thermodynamics can be established for entanglement -- in other words, there exists no unique measure of entanglement governing all axiomatically possible state-to-state transformations. This completes the solution of a long-standing open problem [Problem 20 in arXiv:quant-ph/0504166]. We strengthen the result further to show that reversible entanglement manipulation requires the creation of exponentially large amounts of entanglement according to monotones such as the negativity. Our findings can also be extended to the setting of point-to-point quantum communication, where we show that there exist channels whose parallel simulation entanglement cost exceeds their quantum capacity, even under the most general quantum processes that preserve entanglement-breaking channels. The main technical tool we introduce is the tempered logarithmic negativity, a single-letter lower bound on the entanglement cost that can be efficiently computed via a semi-definite program.Comment: 16+30 pages, 3 figures. v2: minor clarification