arXiv subjects

Prabha Mandayam

Publications and source records attributed to Prabha Mandayam.

At least 37 records · Page 2Linked to original sources

arXiv2022

Achieving fault tolerance against amplitude-damping noise

With the intense interest in small, noisy quantum computing devices comes the push for larger, more accurate -- and hence more useful -- quantum computers. While fully fault-tolerant quantum computers are, in principle, capable of achieving arbitrarily accurate calculations using devices subjected to general noise, they require immense resources far beyond our current reach. An intermediate step would be to construct quantum computers of limited accuracy enhanced by lower-level, and hence lower-cost, noise-removal techniques. This is the motivation for our work, which looks into fault-tolerant encoded quantum computation targeted at the dominant noise afflicting the quantum device. Specifically, we develop a protocol for fault-tolerant encoded quantum computing components in the presence of amplitude-damping noise, using a 4-qubit code and a recovery procedure tailored to such noise. We describe a universal set of fault-tolerant encoded gadgets and compute the pseudothreshold for the noise, below which our scheme leads to more accurate computation. Our work demonstrates the possibility of applying the ideas of quantum fault tolerance to targeted noise models, generalizing the recent pursuit of biased-noise fault tolerance beyond the usual Pauli noise models. We also illustrate how certain aspects of the standard fault tolerance intuition, largely acquired through Pauli-noise considerations, can fail in the face of more general noise.

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arXiv2021

Characterizing incompatibility of quantum measurements via their Naimark extensions

We obtain a formal characterization of the compatibility or otherwise of a set of positive-operator-valued measures (POVMs) via their Naimark extensions. We show that a set of POVMs is jointly measurable if and only if there exists a single Naimark extension, specified by a fixed ancilla state on the same ancilla Hilbert space, that maps them to a set of commuting projective measurements (PVMs). We use our result to obtain an easily checkable sufficient condition for the compatibility of a pair of dichotomic observables in any dimension. This in turn leads to a characterization of the compatibility regions for some important classes of observables including a pair of unsharp qubit observables. Finally, we also outline as to how our result provides an alternate approach to quantifying the incompatibility of a general set of quantum measurements.

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arXiv2021

On optimal cloning and incompatibility

We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure $\mathcal{Q}_{c}$ is {\it faithful} since it vanishes only for commuting observables. We prove an upper bound for $\mathcal{Q}_{c}$ for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.

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arXiv2021

Queue-Channel Capacities with Generalized Amplitude Damping

The generalized amplitude damping channel (GADC) is considered an important model for quantum communications, especially over optical networks. We make two salient contributions in this paper apropos of this channel. First, we consider a symmetric GAD channel characterized by the parameter $n=1/2,$ and derive its exact classical capacity, by constructing a specific induced classical channel. We show that the Holevo quantity for the GAD channel equals the Shannon capacity of the induced binary symmetric channel, establishing at once the capacity result and that the GAD channel capacity can be achieved without the use of entanglement at the encoder or joint measurements at the decoder. Second, motivated by the inevitable buffering of qubits in quantum networks, we consider a generalized amplitude damping \emph{queue-channel} -- that is, a setting where qubits suffer a waiting time dependent GAD noise as they wait in a buffer to be transmitted. This GAD queue channel is characterized by non-i.i.d. noise due to correlated waiting times of consecutive qubits. We exploit a conditional independence property in conjunction with additivity of the channel model, to obtain a capacity expression for the GAD queue channel in terms of the stationary waiting time in the queue. Our results provide useful insights towards designing practical quantum communication networks, and highlight the need to explicitly model the impact of buffering.

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arXiv2021

Differential phase encoded measurement-device-independent quantum key distribution

We present a measurement-device-independent quantum key distribution (MDI-QKD) using single photons in a linear superposition of three orthogonal time-bin states, for generating the key. The orthogonal states correspond to three distinct paths in the delay line interferometers used by two (trusted) sources. The key information is decoded based on the measurement outcomes obtained by an untrusted third party Charles, who uses a beamsplitter to measure the phase difference between pulses traveling through different paths of the two delay lines. The proposed scheme combines the best of both differential-phase-shift (DPS) QKD and MDI-QKD. It is more robust against phase fluctuations, and also ensures protection against detector side-channel attacks. We prove unconditional security by demonstrating an equivalent protocol involving shared entanglement between the two trusted parties. We show that the secure key rate for our protocol compares well to existing protocols in the asymptotic regime. For the decoy-state variant of our protocol, we evaluate the secure key rate by using a phase-post-selection technique. Finally, we estimate the bit error rate and the phase error rate, in the finite key regime.

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arXiv2020

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power $e_p$ and a complementary quantity, the gate typicality $g_t$. We characterize the boundaries of the set $K_2$ representing all two-qubit gates projected onto the plane $(e_p, g_t)$ showing that the fractional powers of the \textsc{swap} operator form a parabolic boundary of $K_2$, while the other bounds are formed by two straight lines. In this way a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of \textsc{swap} persists in the set $K_N$, for gates of higher dimensions ($N>2$). Furthermore, we study entanglement of bipartite quantum gates applied sequentially $n$ times and analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power a gate applied $n$ times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory (RMT), indicating "thermalization" in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.

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arXiv2020

The Classical Capacity of Additive Quantum Queue-Channels

We consider a setting where a stream of qubits is processed sequentially. We derive fundamental limits on the rate at which classical information can be transmitted using qubits that decohere as they wait to be processed. Specifically, we model the sequential processing of qubits using a single server queue, and derive expressions for the classical capacity of such a quantum `queue-channel.' Focusing on two important noise models, namely the erasure channel and the depolarizing channel, we obtain explicit single-letter capacity formulas in terms of the stationary waiting time of qubits in the queue. Our capacity proof also implies that a `classical' coding/decoding strategy is optimal, i.e., an encoder which uses only orthogonal product states, and a decoder which measures in a fixed product basis, are sufficient to achieve the classical capacity of both queue-channels. Our proof technique for the converse theorem generalizes readily -- in particular, whenever the underlying quantum noise channel is additive, we can obtain a single-letter upper bound on the classical capacity of the corresponding quantum queue-channel. More broadly, our work begins to quantitatively address the impact of decoherence on the performance limits of quantum information processing systems.

cs.IT↗

arXiv2020

Finding good quantum codes using the Cartan form

We present a simple and fast numerical procedure to search for good quantum codes for storing logical qubits in the presence of independent per-qubit noise. In a key departure from past work, we use the worst-case fidelity as the figure of merit for quantifying code performance, a much better indicator of code quality than, say, entanglement fidelity. Yet, our algorithm does not suffer from inefficiencies usually associated with the use of worst-case fidelity. Specifically, using a near-optimal recovery map, we are able to reduce the triple numerical optimization needed for the search to a single optimization over the encoding map. We can further reduce the search space using the Cartan decomposition, focusing our search over the nonlocal degrees of freedom resilient against independent per-qubit noise, while not suffering much in code performance.

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arXiv2019

Qubits through Queues: The Capacity of Channels with Waiting Time Dependent Errors

We consider a setting where qubits are processed sequentially, and derive fundamental limits on the rate at which classical information can be transmitted using quantum states that decohere in time. Specifically, we model the sequential processing of qubits using a single server queue, and derive explicit expressions for the capacity of such a `queue-channel.' We also demonstrate a sweet-spot phenomenon with respect to the arrival rate to the queue, i.e., we show that there exists a value of the arrival rate of the qubits at which the rate of information transmission (in bits/sec) through the queue-channel is maximized. Next, we consider a setting where the average rate of processing qubits is fixed, and show that the capacity of the queue-channel is maximized when the processing time is deterministic. We also discuss design implications of these results on quantum information processing systems.

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arXiv2018

Pretty good state transfer via adaptive quantum error correction

We examine the role of quantum error correction (QEC) in achieving pretty good quantum state transfer over a class of $1$-d spin Hamiltonians. Recasting the problem of state transfer as one of information transmission over an underlying quantum channel, we identify an adaptive QEC protocol that achieves pretty good state transfer. Using an adaptive recovery and approximate QEC code, we obtain explicit analytical and numerical results for the fidelity of transfer over ideal and disordered $1$-d Heisenberg chains. In the case of a disordered chain, we study the distribution of the transition amplitude, which in turn quantifies the stochastic noise in the underlying quantum channel. Our analysis helps us to suitably modify the QEC protocol so as to ensure pretty good state transfer for small disorder strengths and indicates a threshold beyond which QEC does not help in improving the fidelity of state transfer.

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arXiv2016

Can local dynamics enhance entangling power?

It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate-typicality and study its properties. Analyzing multiple applications of any entangling operator interlaced with random local gates, we prove that both investigated quantities approach their asymptotic values in a simple exponential form. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.

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arXiv2015

Quantifying incompatibility beyond entropic uncertainty

We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsallis ($T_{2}$) entropy. We explicitly evaluate the incompatibility of a pair of qubit observables in both operational scenarios. We also obtain an efficiently computable lower bound for the mutually incompatibility of a general set of observables.

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arXiv2015

Unextendible mutually unbiased bases in prime-squared dimensions

A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1$ MUBs that saturate both a Shannon ($H_{1}$) and a collision ($H_{2}$) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.

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arXiv2015

The Functional Analysis of Quantum Information Theory

This book is a compilation of notes from a two-week international workshop on the "The Functional Analysis of Quantum Information Theory" that was held at the Institute of Mathematical Sciences during 26/12/2011-06/01/2012. The workshop was devoted to the mathematical framework of quantized functional analysis (QFA), and aimed at illustrating its applications to problems in quantum communication. The lectures were given by Gilles Pisier (Pierre and Marie Curie University and Texas A&M), K.R. Parthasarathy (ISI Delhi), Vern Paulsen (University of Houston), and Andreas Winter (Universitat Autonoma de Barcelona). Topics discussed include Operator Spaces and Completely bounded maps, Schmidt number and Schmidt rank of bipartite entangled states, Operator Systems and Completely Positive Maps, and, Operator Methods in Quantum Information.

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arXiv2014

A disturbance tradeoff principle for quantum measurements

We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy ($T_{2}$). We also investigate the disturbances associated with the class of non-projective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.

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arXiv2014

Measures of disturbance and incompatibility for quantum measurements

We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables $A$ and $B$ with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of $A$ followed by a measurement of $B$ on a given state, and the other obtained from a measurement of $B$ alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables $A$ and $B$, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally non-degenerate and associated with mutually unbiased bases. In the process, we also establish an important connection between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of non-projective measurements. In particular, we obtain a non-trivial upper bound on the incompatibility of one Lüders instrument with another.

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arXiv2014

Unextendible Mutually Unbiased Bases from Pauli Classes

We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.

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arXiv2013

Operational measure of incompatibility of noncommuting observables

Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the non-orthogonality of their eigenstates. We prove that this measure has all the desired properties: it is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any N noncommuting observables and compute it exactly when the observables are mutually unbiased.

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