arXiv subjects

Min-Hsiu Hsieh

Publications and source records attributed to Min-Hsiu Hsieh.

At least 163 records · Page 9Linked to original sources

arXiv2015

Exponential Decay of Matrix $Φ$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles

In the study of Markovian processes, one of the principal achievements is the equivalence between the $Φ$-Sobolev inequalities and an exponential decrease of the $Φ$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $Φ$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum ensembles. We consider two special cases of quantum unital channels, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix $Φ$-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix $Φ$-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.

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arXiv2015

Quantum Versus Classical Advantages in Secret Key Distillation (and Their Links to Quantum Entanglement)

We consider the extraction of shared secret key from correlations that are generated by either a classical or quantum source. In the classical setting, two honest parties (Alice and Bob) use public discussion and local randomness to distill secret key from some distribution $p_{XYZ}$ that is shared with an unwanted eavesdropper (Eve). In the quantum settings, the correlations $p_{XYZ}$ are delivered to the parties as either an \textit{incoherent} mixture of orthogonal quantum states or as \textit{coherent} superposition of such states; in both cases, Alice and Bob use public discussion and local quantum operations to distill secret key. While the power of quantum mechanics increases Alice and Bob's ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. In this paper we first demonstrate that the classical key rate is equivalent to the quantum key rate when the correlations are generated incoherently in the quantum setting. For coherent sources, we next show that the rates are incomparable, and in fact, their difference can be arbitrarily large in either direction. However, we identify a large class of non-trivial distributions $p_{XYZ}$ that possess the following properties: (i) Eve's advantage is always greater in the quantum source than in its classical counterpart, and (ii) for the quantum entanglement shared between Alice and Bob in the coherent source, the so-called entanglement cost/squashed entanglement/relative entropy of entanglement can all be computed. With property (ii), we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.

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arXiv2015

Passive $\mathcal{PT}$-symmetric couplers without complex optical potentials

In addition to the implementation of parity-time ($\mathcal{PT}$)-symmetric optical systems by carefully and actively controlling the gain and loss, we show that a $2\times 2$ $\mathcal{PT}$-symmetric Hamiltonian has a unitarily equivalent representation without complex optical potentials in the resulting optical coupler. Through the Naimark dilation in operator algebra, passive $\mathcal{PT}$-symmetric couplers can thus be implemented with a refractive index of real values and asymmetric coupling coefficients. This opens up the possibility to implement general $\mathcal{PT}$-symmetric systems with state-of-the-art asymmetric slab waveguides, dissimilar optical fibers, or cavities with chiral mirrors.

physics.optics↗

arXiv2015

Compatibility of state assignments and pooling of information

We say that two (or more) state assignments for one and the same quantum system are compatible if they could represent the assignments of observers with differing information about the system. A criterion for compatibility was proposed in [Phys. Rev. A 65, 032315 (2002)]; however, this leaves unanswered the question of whether there are degrees of compatibility which could be represented by some quantitative measure, and whether there is a straightforward procedure whereby the observers can pool their information to arrive at a unique joint state assignment. We argue that such measures are only sensible given some assumption about what kind of information was used in making the state assignments in the first place, and that in general state assignments do not represent all of the information possessed by the observers. However, we examine one particular measure, and show that it has a straightforward interpretation, assuming that the information was acquired from a particular type of measurement, and that in this case there is a natural rule for pooling information. We extend this measure to compatibility of states for k observers, and show that the value is the solution to a semidefinite program. Similar compatibility measures can be defined for alternative notions of state compatibility, including Post-Peierls (PP) and Equal Support (ES) compatibility.

quant-ph↗

arXiv2015

Fully Quantum Source Compression with a Quantum Helper

We study source compression with a helper in the fully quantum regime, extending our earlier result on classical source compression with a quantum helper [arXiv:1501.04366, 2015]. We characterise the quantum resources involved in this problem, and derive a single-letter expression of the achievable rate region when entanglement assistance is available. The direct coding proof is based on a combination of two fundamental protocols, namely the quantum state merging protocol and the quantum reverse Shannon theorem (QRST). This result demonstrates an unexpected connection between distributed source compression with the QRST protocol, a quantum protocol that consumes noiseless resources to simulate a noisy quantum channel.

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arXiv2015

Distributions Attaining Secret Key at a Rate of the Conditional Mutual Information

In this paper we consider the problem of extracting secret key from an eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a "helping" Eve offers Alice and Bob no greater advantage for obtaining secret key than a fully adversarial one. For each of the above scenarios, strong necessary conditions are derived on the structure of distributions attaining a secret key rate of $I(X:Y|Z)$. In obtaining our results, we completely solve the problem of secret key distillation under scenario (i) and identify $H(S|Z)$ to be the optimal key rate using shared randomness, where $S$ is the Gács-Körner Common Information. We thus provide an operational interpretation of the conditional Gács-Körner Common Information. Additionally, we introduce simple example distributions in which the rate $I(X:Y|Z)$ is achievable if and only if two-way communication is allowed.

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arXiv2015

A Classical Analog to Entanglement Reversibility

In this letter we introduce the problem of secrecy reversibility. This asks when two honest parties can distill secret bits from some tripartite distribution $p_{XYZ}$ and transform secret bits back into $p_{XYZ}$ at equal rates using local operation and public communication (LOPC). This is the classical analog to the well-studied problem of reversibly concentrating and diluting entanglement in a quantum state. We identify the structure of distributions possessing reversible secrecy when one of the honest parties holds a binary distribution, and it is possible that all reversible distributions have this form. These distributions are more general than what is obtained by simply constructing a classical analog to the family of quantum states known to have reversible entanglement. An indispensable tool used in our analysis is a conditional form of the Gács-Körner Common Information.

quant-ph↗

arXiv2015

Source Compression with a Quantum Helper

We study classical source coding with quantum side-information where the quantum side-information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct part of our result is proved via the measurement compression theory by Winter. Our result reveals that a helper's scheme that separately conducts a measurement and a compression is suboptimal, and the measurement compression is fundamentally needed to achieve the optimal rate region.

quant-ph↗

arXiv2015

The Learnability of Unknown Quantum Measurements

Quantum machine learning has received significant attention in recent years, and promising progress has been made in the development of quantum algorithms to speed up traditional machine learning tasks. In this work, however, we focus on investigating the information-theoretic upper bounds of sample complexity - how many training samples are sufficient to predict the future behaviour of an unknown target function. This kind of problem is, arguably, one of the most fundamental problems in statistical learning theory and the bounds for practical settings can be completely characterised by a simple measure of complexity. Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert space. Learning an unknown quantum state becomes a dual problem to ours, and as a byproduct, we can recover Aaronson's famous result [Proc. R. Soc. A 463:3089-3144 (2007)] solely using a classical machine learning technique. In addition, other famous complexity measures like covering numbers and Rademacher complexities are derived explicitly. We are able to connect measures of sample complexity with various areas in quantum information science, e.g. quantum state/measurement tomography, quantum state discrimination and quantum random access codes, which may be of independent interest. Lastly, with the assistance of general Bloch-sphere representation, we show that learning quantum measurements/states can be mathematically formulated as a neural network. Consequently, classical ML algorithms can be applied to efficiently accomplish the two quantum learning tasks.

quant-ph↗

arXiv2014

Asymptotic State Discrimination and a Strict Hierarchy in Distinguishability Norms

In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With a simple two-qubit product state ensemble, we demonstrate a strict majorization of the two-way LOCC norm over the one-way norm.

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arXiv2014

The Private and Public Correlation Cost of Three Random Variables with Collaboration

In this paper we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties -- called Alice and Bob -- share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution $P_{XYZ}$. We obtain a single-letter characterization of the trade-off between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider is also closely related to the task of secrecy formation in which $P_{XYZ}$ is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.

quant-ph↗

arXiv2014

An upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution

State redistribution is the protocol in which, given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol of coherent state merging as a primitive.

quant-ph↗

arXiv2013

Local PT symmetry violates the no-signaling principle

Bender et al. have developed PT-symmetric quantum theory as an extension of quantum theory to non-Hermitian Hamiltonians. We show that when this model has a local PT symmetry acting on composite systems it violates the non-signaling principle of relativity. Since the case of global PT symmetry is known to reduce to standard quantum mechanics, this shows that the PT-symmetric theory is either a trivial extension or likely false as a fundamental theory.

quant-ph↗

arXiv2013

When do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?

In this paper we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the \emph{perfect} distinguishability problem of two-qubit ensembles - both for separable operations and LOCC - by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then for the well-known task of minimum error discrimination, it is shown that \textit{almost all} two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that \textit{any} two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as "non-locality without entanglement". We next consider an example of $N$ parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant "trine ensemble", and we establish a tight connection between the $N$-copy ensemble and Shor's "lifted" single-copy ensemble. For any finite $N$, we prove that optimal identification of the states cannot be achieved by LOCC; however as $N\to\infty$, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum-error probability is considered.

quant-ph↗

arXiv2013

Quantum rate distortion coding with auxiliary resources

We extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.

quant-ph↗

arXiv2013

A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ($\ep$-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot $\ep$-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the $\ep$-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

quant-ph↗

arXiv2013

Entanglement-assisted quantum turbo codes

An unexpected breakdown in the existing theory of quantum serial turbo coding is that a quantum convolutional encoder cannot simultaneously be recursive and non-catastrophic. These properties are essential for quantum turbo code families to have a minimum distance growing with blocklength and for their iterative decoding algorithm to converge, respectively. Here, we show that the entanglement-assisted paradigm simplifies the theory of quantum turbo codes, in the sense that an entanglement-assisted quantum (EAQ) convolutional encoder can possess both of the aforementioned desirable properties. We give several examples of EAQ convolutional encoders that are both recursive and non-catastrophic and detail their relevant parameters. We then modify the quantum turbo decoding algorithm of Poulin et al., in order to have the constituent decoders pass along only "extrinsic information" to each other rather than a posteriori probabilities as in the decoder of Poulin et al., and this leads to a significant improvement in the performance of unassisted quantum turbo codes. Other simulation results indicate that entanglement-assisted turbo codes can operate reliably in a noise regime 4.73 dB beyond that of standard quantum turbo codes, when used on a memoryless depolarizing channel. Furthermore, several of our quantum turbo codes are within 1 dB or less of their hashing limits, so that the performance of quantum turbo codes is now on par with that of classical turbo codes. Finally, we prove that entanglement is the resource that enables a convolutional encoder to be both non-catastrophic and recursive because an encoder acting on only information qubits, classical bits, gauge qubits, and ancilla qubits cannot simultaneously satisfy them.

quant-ph↗

arXiv2013

A Return to the Optimal Detection of Quantum Information

In 1991, Asher Peres and William Wootters wrote a seminal paper on the nonlocal processing of quantum information [\textit{Phys. Rev. Lett.} \textbf{66} 1119 (1991)]. We return to their classic problem and solve it in various contexts. Specifically, for discriminating the "double trine" ensemble with minimum error, we prove that global operations are more powerful than local operations with classical communication (LOCC). Even stronger, there exists a finite gap between the optimal LOCC probability and that obtainable by separable operations (SEP). Additionally we prove that a two-way, adaptive LOCC strategy can always beat a one-way protocol. Our results provide the first known instance of "nonlocality without entanglement" in two qubit pure states.

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