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

Postselection threshold against biased noise

The highest current estimates for the amount of noise a quantum computer can tolerate are based on fault-tolerance schemes relying heavily on postselecting on no detected errors. However, there has been no proof that these schemes give even a positive tolerable noise threshold. A technique to prove a positive threshold, for probabilistic noise models, is presented. The main idea is to maintain strong control over the distribution of errors in the quantum state at all times. This distribution has correlations which conceivably could grow out of control with postselection. But in fact, the error distribution can be written as a mixture of nearby distributions each satisfying strong independence properties, so there are no correlations for postselection to amplify.Comment: 13 pages, FOCS 2006; conference versio

Strong and uniform convergence in the teleportation simulation of bosonic Gaussian channels

In the literature on the continuous-variable bosonic teleportation protocol due to [Braunstein and Kimble, Phys. Rev. Lett., 80(4):869, 1998], it is often loosely stated that this protocol converges to a perfect teleportation of an input state in the limit of ideal squeezing and ideal detection, but the exact form of this convergence is typically not clarified. In this paper, I explicitly clarify that the convergence is in the strong sense, and not the uniform sense, and furthermore, that the convergence occurs for any input state to the protocol, including the infinite-energy Basel states defined and discussed here. I also prove, in contrast to the above result, that the teleportation simulations of pure-loss, thermal, pure-amplifier, amplifier, and additive-noise channels converge both strongly and uniformly to the original channels, in the limit of ideal squeezing and detection for the simulations. For these channels, I give explicit uniform bounds on the accuracy of their teleportation simulations. I then extend these uniform convergence results to particular multi-mode bosonic Gaussian channels. These convergence statements have important implications for mathematical proofs that make use of the teleportation simulation of bosonic Gaussian channels, some of which have to do with bounding their non-asymptotic secret-key-agreement capacities. As a byproduct of the discussion given here, I confirm the correctness of the proof of such bounds from my joint work with Berta and Tomamichel from [Wilde, Tomamichel, Berta, IEEE Trans. Inf. Theory 63(3):1792, March 2017]. Furthermore, I show that it is not necessary to invoke the energy-constrained diamond distance in order to confirm the correctness of this proof.Comment: 19 pages, 3 figure

Quantum enigma machines and the locking capacity of a quantum channel

The locking effect is a phenomenon which is unique to quantum information theory and represents one of the strongest separations between the classical and quantum theories of information. The Fawzi-Hayden-Sen (FHS) locking protocol harnesses this effect in a cryptographic context, whereby one party can encode n bits into n qubits while using only a constant-size secret key. The encoded message is then secure against any measurement that an eavesdropper could perform in an attempt to recover the message, but the protocol does not necessarily meet the composability requirements needed in quantum key distribution applications. In any case, the locking effect represents an extreme violation of Shannon's classical theorem, which states that information-theoretic security holds in the classical case if and only if the secret key is the same size as the message. Given this intriguing phenomenon, it is of practical interest to study the effect in the presence of noise, which can occur in the systems of both the legitimate receiver and the eavesdropper. This paper formally defines the locking capacity of a quantum channel as the maximum amount of locked information that can be reliably transmitted to a legitimate receiver by exploiting many independent uses of a quantum channel and an amount of secret key sublinear in the number of channel uses. We provide general operational bounds on the locking capacity in terms of other well-known capacities from quantum Shannon theory. We also study the important case of bosonic channels, finding limitations on these channels' locking capacity when coherent-state encodings are employed and particular locking protocols for these channels that might be physically implementable.Comment: 37 page

Simulating Hamiltonians in Quantum Networks: Efficient Schemes and Complexity Bounds

We address the problem of simulating pair-interaction Hamiltonians in n node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair-interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interactions we present optimal schemes for inversion.Comment: 19 pages, LaTeX2

Classical Computation in the Quantum World

Quantum computation is by far the most powerful computational model allowed by the laws of physics. By carefully manipulating microscopic systems governed by quantum mechanics, one can efficiently solve computational problems that may be classically intractable; on the contrary, such speed-ups are rarely possible without the help of classical computation, since most quantum algorithms heavily rely on subroutines that are purely classical. A better understanding of the relationship between classical and quantum computation is indispensable, in particular in an era where the first quantum device exceeding classical computational power is within reach. In the first part of the thesis, we study some differences between classical and quantum computation. We first show that quantum cryptographic hashing is maximally resilient against classical leakage, a property beyond reach for any classical hash function. Next, we consider the limitation of strong (amplitude-wise) simulation of quantum computation. We prove an unconditional and explicit complexity lower bound for a category of simulations called monotone strong simulation and further prove conditional complexity lower bounds for general strong simulation techniques. Both results indicate that strong simulation is fundamentally unscalable. In the second part of the thesis, we propose classical algorithms that facilitate quantum computing. We propose a new classical algorithm for the synthesis of a quantum algorithm paradigm called quantum signal processing. Empirically, our algorithm demonstrates numerical stability and acceleration of more than one magnitude compared to state-of-the-art algorithms. Finally, we propose a randomized algorithm for transversally switching between arbitrary stabilizer quantum error-correcting codes. It has the property of preserving the code distance and thus might prove useful for designing fault-tolerant code-switching schemes.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149943/1/cupjinh_1.pd