arXiv subjects

John A. Smolin

Publications and source records attributed to John A. Smolin.

At least 55 records · Page 3Linked to original sources

arXiv2008

Codeword stabilized quantum codes: algorithm and structure

The codeword stabilized ("CWS") quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we prove three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank (arXiv:cs/0610159).

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arXiv2007

Additive Extensions of a Quantum Channel

We study extensions of a quantum channel whose one-way capacities are described by a single-letter formula. This provides a simple technique for generating powerful upper bounds on the capacities of a general quantum channel. We apply this technique to two qubit channels of particular interest--the depolarizing channel and the channel with independent phase and amplitude noise. Our study of the latter demonstrates that the key rate of BB84 with one-way post-processing and quantum bit error rate q cannot exceed H(1/2-2q(1-q)) - H(2q(1-q)).

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arXiv2007

Codeword Stabilized Quantum Codes

We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, we find ((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding circuits for all codes within our framework.

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arXiv2007

A simple family of nonadditive quantum codes

Most known quantum codes are additive, meaning the codespace can be described as the simultaneous eigenspace of an abelian subgroup of the Pauli group. While in some scenarios such codes are strictly suboptimal, very little is understood about how to construct nonadditive codes with good performance. Here we present a family of nonadditive quantum codes for all odd blocklengths, n, that has a particularly simple form. Our codes correct single qubit erasures while encoding a higher dimensional space than is possible with an additive code or, for n of 11 or greater, any previous codes.

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arXiv2006

Degenerate Quantum Codes for Pauli Channels

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.

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arXiv2005

The entangling and disentangling power of unitary transformations are unequal

We consider two capacity quantities associated with bipartite unitary gates: the entangling and the disentangling power. For two-qubit unitaries these two capacities are always the same. Here we prove that these capacities are different in general. We do so by constructing an explicit example of a qubit-qutrit unitary whose entangling power is maximal (2 ebits), but whose disentangling power is strictly less. A corollary is that there can be no unique ordering for unitary gates in terms of their ability to perform non-local tasks. Finally we show that in large dimensions, almost all bipartite unitaries have entangling and disentangling capacities close to the maximal possible (and thus in high dimensions the difference in these capacities is small for almost all unitaries).

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arXiv2005

The Entanglement of Superpositions

Given a bipartite quantum state (in arbitrary dimension) and a decomposition of it as a superposition of two others, we find bounds on the entanglement of the superposition state in terms of the entanglement of the states being superposed. In the case that the two states being superposed are bi-orthogonal, the answer is simple, and, for example, the entanglement of the superposition cannot be more than one e-bit more than the average of the entanglement of the two states being superposed. However for more general states, the situation is very different.

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arXiv2005

Entanglement of assistance and multipartite state distillation

We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal EPR pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and GHZ states; in fact, under a restricted class of protocols, which we call "one-way broadcasting", the GHZ-rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalisations to m parties, and show (for m=4) that the maximal amount of entanglement that can be localised between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localisable entanglement, and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries, and any general channel to a mixture of partial isometries.

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arXiv2004

The Continuous Variable Quantum Teleportation Controversy

I argue that the objections by Rudolph and Sanders \cite{RS} to performing continuous variable quantum teleportation experiments using lasers, as well as the various rebuttals to their paper, are based on a misunderstanding of the Partition Ensemble Fallacy.

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arXiv2004

Inequalities and separations among assisted capacities of quantum channels

We exhibit discrete memoryless quantum channels whose quantum capacity assisted by two-way classical communication, $Q_2$, exceeds their unassisted one-shot Holevo capacity $C_H$. These channels may be thought of as having a data input and output, along with a control input that partly influences, and a control output that partly reveals, which of a set of unitary evolutions the data undergoes en route from input to output. The channel is designed so that the data's evolution can be exactly inferred by a classically coordinated processing of 1) the control output, and 2) a reference system entangled with the control input, but not from either of these resources alone. Thus a two-way classical side channel allows the otherwise noisy evolution of the data to be corrected, greatly increasing the capacity. The same family of channels provides examples where the classical capacity assisted by classical feedback, $C_B$, and the quantum capacity assisted by classical feedback $Q_B$, both exceed $C_H$. A related channel, whose data input undergoes dephasing before interacting with the control input, has a classical capacity $C=C_H$ strictly less than its $C_2$, the classical capacity assisted by independent classical communication.

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arXiv2003

Can Quantum Cryptography Imply Quantum Mechanics?

It has been suggested that the ability of quantum mechanics to allow secure distribution of secret key together with its inability to allow bit commitment or communicate superluminally might be sufficient to imply the rest of quantum mechanics. I argue using a toy theory as a counterexample that this is not the case. I further discuss whether an additional axiom (key storage) brings back the quantum nature of the theory.

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arXiv2002

The Power of LOCCq State Transformations

Reversible state transformations under entanglement non-increasing operations give rise to entanglement measures. It is well known that asymptotic local operations and classical communication (LOCC) are required to get a simple operational measure of bipartite pure state entanglement. For bipartite mixed states and multipartite pure states it is likely that a more powerful class of operations will be needed. To this end \cite{BPRST01} have defined more powerful versions of state transformations (or reducibilities), namely LOCCq (asymptotic LOCC with a sublinear amount of quantum communication) and CLOCC (asymptotic LOCC with catalysis). In this paper we show that {\em LOCCq state transformations are only as powerful as asymptotic LOCC state transformations} for multipartite pure states. We first generalize the concept of entanglement gambling from two parties to multiple parties: any pure multipartite entangled state can be transformed to an EPR pair shared by some pair of parties and that any irreducible $m$ $(m\ge 2)$ party pure state can be used to create any other state (pure or mixed), using only local operations and classical communication (LOCC). We then use this tool to prove the result. We mention some applications of multipartite entanglement gambling to multipartite distillability and to characterizations of multipartite minimal entanglement generating sets. Finally we discuss generalizations of this result to mixed states by defining the class of {\em cat distillable states}.

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arXiv2002

Superactivation of Bound Entanglement

We show that, in a multi-party setting, two non-distillable (bound-entangled) states tensored together can make a distillable state. This is an example of true superadditivity of distillable entanglement. We also show that unlockable bound-entangled states cannot be asymptotically unentangled, providing the first proof that some states are truly bound-entangled in the sense of being both non-distillable and non-separable asymptotically.

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arXiv2002

Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem

The entanglement-assisted classical capacity of a noisy quantum channel is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that this capacity is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs $ρ$, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of $ρ$ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement assisted communication--communication over a discrete memoryless channel (DMC) between parties who share prior random information--we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMC's of equal capacity can simulate one another with unit asymptotic efficiency.

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arXiv2002

Trust enhancement by multiple random beacons

Random beacons-information sources that broadcast a stream of random digits unknown by anyone beforehand-are useful for various cryptographic purposes. But such beacons can be easily and undetectably sabotaged, so that their output is known beforehand by a dishonest party, who can use this information to defeat the cryptographic protocols supposedly protected by the beacon. We explore a strategy to reduce this hazard by combining the outputs from several noninteracting (eg spacelike-separated) beacons by XORing them together to produce a single digit stream which is more trustworthy than any individual beacon, being random and unpredictable if at least one of the contributing beacons is honest. If the contributing beacons are not spacelike separated, so that a dishonest beacon can overhear and adapt to earlier outputs of other beacons, the beacons' trustworthiness can still be enhanced to a lesser extent by a time sharing strategy. We point out some disadvantages of alternative trust amplification methods based on one-way hash functions.

cs.CR↗

arXiv2001

Rank Two Bipartite Bound Entangled States Do Not Exist

We explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement. We show a relation between the rank, marginal ranks, and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n \times n Hilbert space. A direct consequence of this result is that there are no bipartite bound entangled states of rank two. We also show that a separability condition in terms of a quantum entropy inequality is associated with the above results. We explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a Schmidt rank n pure state and provide a lower bound of n-1. We also prove that a mixture of a non-zero amount of any pure entangled state with a pure product state is distillable.

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arXiv2001

Remote State Preparation

Quantum teleportation uses prior entanglement and forward classical communication to transmit one instance of an unknown quantum state. Remote state preparation (RSP) has the same goal, but the sender knows classically what state is to be transmitted. We show that the asymptotic classical communication cost of RSP is one bit per qubit - half that of teleportation - and becomes even less when transmitting part of a known entangled state. We explore the tradeoff between entanglement and classical communication required for RSP, and discuss RSP capacities of general quantum channels.

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arXiv2001

More Is Not Necessarily Easier

In this brief comment, we consider the exact, deterministic, and nonasymptotic transformation of multiple copies of pure states under LOCC. It was conjectured in quant-ph/0103131 that, if $k$ copies of $|ψ\>$ can be transformed to $k$ copies of $|ϕ\>$, the same holds for all $r \geq k$. We present counterexamples to the above conjecture.

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