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### Research Interests

Quantum computing, quantum information theory, quantum error correction, quantum computational complexity theory, quantum stochastic resonance, and quantum biology are my current research interests.

### Quantum Communication

#### Quantum Shannon Theory

Quantum Shannon theory is the study of the ultimate performance limits of quantum communication. It is the quantum generalization of Shannon's theory of classical information (See interesting historical video). Quantum Shannon theory has had many breakthroughs in the past ten years.

Researchers continue to make breakthroughs in this field. One notable breakthrough shows that two quantum communication devices that individually cannot transmit quantum information can combine to transmit quantum information reliably. Dave Bacon has written a summary article that describes this effect.

Min-Hsiu Hsieh and I determined the capacity of an entanglement-assisted quantum channel for communication of classical and quantum information. The region consists of rate triples, where each rate triple corresponds to an achievable protocol that consumes entanglement in order to generate classical and quantum communication. Our solution unifies many prior results in quantum Shannon theory, representing one of the most general settings in quantum communication. To solve part of the problem, we use techniques of Devetak and Shor that give a method for "piggybacking" classical information along with quantum information. We then determined the full triple trade-off between classical communication, quantum communication, and entanglement when two parties share a state or a noisy channel connects a sender to a receiver. The results here are a further generalization of the above contribution. We have talked about this work at some length in a few articles, and later, along with Patrick Hayden and Saikat Guha, I showed that bosonic channels have a very strong trade-off if enough photons are available at the channel input.

#### Network Quantum Information Theory

The goal of network information theory is to extend the methods of Shannon to settings beyond the single-sender, single-receiver channel. Several important channel models arise in the network setting, such as the multiple access channel (many senders and one receiver), the broadcast channel (one sender and many receivers), the interference channel (many senders and many receivers), and the relay channel (a sender and receiver connected by a relay station in between).

We have contributed to the growing area of network quantum information theory. Specifically, we are studying the communication of classical data over quantum channels. An important reason for doing so is that a quantum receiver acting over many channel outputs can outperform a classical receiver (such as homodyne or heterodyne detection) for the practical class of bosonic channels (which can model free-space or fiber-optic links). Omar Fawzi, Patrick Hayden, Pranab Sen, Ivan Savov, and I studied the quantum interference channel, and we proved that a new receiver (called a “quantum simultaneous decoder”), combined with a particular coding strategy, can achieve non-trivial communication rates over the quantum interference channel. Saikat Guha, Ivan, and I later applied this decoder to a particular bosonic interference channel, and we found that this quantum decoder can outperform a classical strategy. Ivan and I later pushed quantum network information theory further, by applying some of the techniques from the above paper to the quantum broadcast channel and the quantum relay channel (the latter with Mai Vu as well).

#### Polar Codes for Classical, Private, and Quantum Communication

Channel polarization is a phenomenon discovered by Arikan in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels become near perfect for data transmission and the other fraction become near useless for this task. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and “frozen” bits through the useless ones.

Along with Saikat Guha, I showed how to leverage several known results from the quantum information literature to demonstrate that the channel polarization effect takes hold for channels with classical inputs and quantum outputs. We constructed linear polar codes based on this effect, and we also demonstrated that a quantum successive cancellation decoder works well, by exploiting Pranab Sen's “non-commutative union bound” that holds for a sequence of projectors applied to a quantum state.

Mahdavifar and Vardy exploited the channel polarization phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. Saikat Guha and I built on their work and demonstrated how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we were also able to construct quantum polar codes by operating these quantum wiretap polar codes in superposition. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we showed that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving.

#### Entanglement-assisted Quantum Convolutional Coding

The main contribution of my Ph.D. work was to quantum error correction. This theory is necessary for a quantum computer or quantum communication device to operate reliably.

In particular, Todd Brun and I developed a theory of entanglement-assisted quantum convolutional codes. A quantum convolutional code is one that operates on a potentially infinite stream of quantum information and uses a periodic encoding circuit to perform the encoding of the quantum stream. An entanglement-assisted quantum error-correcting code is one that exploits entanglement shared between a sender and receiver before quantum communication begins. Todd Brun's and my theory of entanglement-assisted quantum convolutional coding applies to both quantum error correction and distillation of entanglement.

It turns out that the Viterbi algorithm, an efficient algorithm for decoding classical convolutional codes, is useful for decoding an entanglement-assisted quantum convolutional code. Other researchers have shown how to apply the Viterbi algorithm to a quantum convolutional code that does not use entanglement. We have shown how the Viterbi algorithm is useful in entanglement-assisted quantum convolutional coding and received some attention from USC for doing so because of the school's connection to Andrew Viterbi.

#### Continuous-Variable Quantum Information

It is possible to encode quantum information in an analog or continuous-variable quantum state. The physical implementation for such an analog quantum state is typically as a mode of the electromagnetic field. One advantage for analog quantum communication is that it is a little more straightforward for experimentalists to implement quantum protocols with continuous-variable systems. For example, experimentalists have implemented a two-mode entangling quantum gate in a continuous-variable system.

Todd Brun and I (and some others) have made a few contributions to the theory of continuous-variable quantum information processing. In particular, we have shown how to perform both entanglement-assisted quantum error correction and operator quantum error correction in a continuous-variable system. This technique might be useful in a future quantum processor if the errors that occur are larger than the uncertainty in the measurement of the errors. We have also shown how to perform coherent communication with continuous-variable quantum processors. Our work on coherent communication shows how to perform coherent teleportation and coherent superdense coding with continuous variables and shows the sense in which these protocols are dual to each other.

### Quantum Computational Complexity Theory

The quantum separability problem has long been a problem of theoretical interest in quantum information science. The main question is, “Given some description of a two-party quantum state, is this state
separable or entangled?” Such a question is of fundamental importance for
experimentalists producing quantum states who wish to verify whether the state they have
generated is entangled.

One can frame the question in different ways. The traditional approach has been
to consider the description of the state as a matrix and to ask about the computational
complexity of this question as the dimension of the matrix becomes large.
Several researchers have proved that this question is NP-hard if there is a promise
that the state is either separable or at a distance from the set of separable states that is no smaller than an inverse polynomial in the dimension of the matrix. If the distance
is promised to be just a constant, then there is a quasi-polynomial time algorithm that can decide the problem.

In recent work, Patrick Hayden, Kevin Milner, and
I considered a variant of the problem, where instead the description of the state is given as a description of a quantum circuit that can generate the two-party state. This very well could be the more relevant phrasing of the problem when large-scale quantum computation is available. We found that it is possible to solve this problem if one is allowed to interact with a computationally unbounded “prover”, by sending one quantum message to him and receiving one quantum message back from him. This result also sheds some light on the class of problems that can be solved in such a way (QIP(2)), as a prior paper had remarked that this class is somewhat “mysterious”. We also showed that this problem is hard for a class known as “quantum statistical zero knowledge” and it is NP-hard as well.

### Quantum Biology

The field of quantum biology is an interdisciplinary area of research that has attracted members of both the quantum chemistry community and the quantum information community. Could quantum mechanical effects, such as coherence or entanglement, be contributing to evolutionary processes? Could these quantum effects be important for molecular functionality? These questions are interesting because many classical models are successful in modeling some of these processes, and it would be surprising if quantum effects
contribute to an increased performance of a biological system. Furthermore,
if we improved our understanding of these processes, we might be able to
take advantage of this, for example, by increasing the efficiency of solar cells.

James McCracken, Ari Mizel, and I
contributed to this field,
by studying the “quantumness” of the transfer of an exciton to a reaction center in a
light harvesting complex (a molecule relevant for photosynthesis). We numerically simulated a test of non-classicality, called the
Leggett-Garg test, and showed that the light harvesting complex may potentially exhibit non-classical effects even at room temperature.

### Quantum Stochastic Resonance

Stochastic resonance is a phenomenon that occurs in a wide variety of nonlinear systems. It occurs when a system performs better with noise than without noise.

Bart Kosko and collaborators have published quite a few articles on the occurrence of stochastic resonance in neural processing. The primary content of these articles is a “forbidden-interval” theorem that gives necessary and sufficient conditions for the stochastic resonance noise benefit to occur. Bart Kosko has also published a trade book called *Noise* that discusses the phenomenon and has resulted in media coverage and an award.

Bart Kosko and I extended the forbidden-interval theorems to a quantum communication scenario. It exploits the quantum squeezing of light and homodyne detection for the measurement of information. The result is that a quantum stochastic resonance effect occurs in this system and may have applications to continuous-variable quantum key distribution. I have also developed the idea of a stochastic resonance occurring in quantum teleportation.

Cosmo Lupo, Stefano Mancini, and I have explored the stochastic resonance phenomenon in several settings including channel discrimination and quantum communication.

*Last modified: February 22, 2013.*