The Power of Unentanglement

The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. * We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. * We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one

Dimension Independent Disentanglers from Unentanglement and Applications

Quantum entanglement is a key enabling ingredient in diverse applications. However, the presence of unwanted adversarial entanglement also poses challenges in many applications. In this paper, we explore methods to "break" quantum entanglement. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show: For every $d,\ell\ge k$, there is an efficient channel $\Lambda: \mathbb{C}^{d\ell} \otimes \mathbb{C}^{d\ell} \to \mathbb{C}^{dk}$ such that for every bipartite separable state $\rho_1\otimes \rho_2$, the output $\Lambda(\rho_1\otimes\rho_2)$ is close to a k-partite separable state. Concretely, for some distribution $\mu$ on states from $\mathbb{C}^d$, $$\left\|\Lambda(\rho_1 \otimes \rho_2) - \int | \psi \rangle \langle \psi |^{\otimes k} d\mu(\psi)\right\|_1 \le \tilde O \left(\left(\frac{k^{3}}{\ell}\right)^{1/4}\right).$$ Moreover, $\Lambda(| \psi \rangle \langle \psi |^{\otimes \ell}\otimes | \psi \rangle \langle \psi |^{\otimes \ell}) = | \psi \rangle \langle \psi |^{\otimes k}$. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible. Leveraging our disentanglers, we show that unentangled quantum proofs of almost general real amplitudes capture NEXP, greatly relaxing the nonnegative amplitudes assumption in the recent work of QMA^+(2)=NEXP. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form $| \psi \rangle = \sqrt{a} | \psi_+ \rangle + \sqrt{1-a} | \psi_- \rangle$ where $| \psi_+ \rangle$ has non-negative amplitudes, $| \psi_- \rangle$ only has negative amplitudes and $| a-(1-a) | \ge 1/poly(n)$ with $a \in [0,1]$. Additionally, we present a protocol achieving an almost largest possible gap before obtaining QMA^R(k)=NEXP$, namely, a 1/poly(n) additive improvement to the gap results in this equality.Comment: 28 page

Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement

Savitch's theorem states that NPSPACE computations can be simulated in PSPACE. We initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), where an exponentially long classical proof is streamed to a poly-space quantum verifier. Besides two main results, we also show that a quantum analogue of Savitch's theorem is unlikely to hold, as SQCMASPACE=NEXP. For completeness, we introduce Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE=QMA_EXP (quantum analogue of NEXP). Our first main result shows, in contrast to the classical setting, the solution space of a quantum constraint satisfaction problem (i.e. a local Hamiltonian) is always connected when exponentially long proofs are permitted. For this, we show how to simulate any Lipschitz continuous path on the unit hypersphere via a sequence of local unitary gates, at the expense of blowing up the circuit size. This shows quantum error-correcting codes can be unable to detect one codeword erroneously evolving to another if the evolution happens sufficiently slowly, and answers an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State Connectivity problem. Our second main result is that any SQCMASPACE computation can be embedded into "unentanglement", i.e. into a quantum constraint satisfaction problem with unentangled provers. Formally, we show how to embed SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux, Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of scaling the promise gap with the streamed proof size. As a corollary, we obtain the first systematic construction for obtaining QMA(2)-type upper bounds on arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap scales exponentially with the number of bits of communication in the interactive proof.Comment: 60 pages, 4 figure

Holography beyond AdS/CFT

Physicists have long sought to fully understand how gravity can be fully formulated within a quantum mechanical framework. A promising avenue of research in this direction was born from the idea of holography - that gravitational physics can be recast as a different theory living in fewer dimensions. Evidence of this phenomena was first observed in the arena of black hole physics, where the entropy of a black hole was calculated to scale with its area, not its volume. The advent of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence provided an explicit realization of holography for gravitational theories in AdS space. This brought a flurry of activity dedicated to dissecting this correspondence. Ultimately, however, it will be necessary to move beyond the confines of AdS/CFT in order to understand our universe, as we live in a de Sitter type universe. The research presented in this dissertation attempts to broaden the scope of holgraphic theories to include more phenomenologically relevant universes. To do so, we utilize a top-down approach and take results from AdS/CFT that appear to be general holographic results and see how they can be applied in spacetimes other than AdS. In particular, we take the Ryu-Takayanagi (RT) formula, along with its related results, and investigate what we can learn by applying it to general spacetimes. Doing so naturally forces us to utilize holographic screens, as these are the largest such surfaces where the RT formula can be self-consistently applied. This approach allows us to examine properties of the purported boundary theory for general spacetimes, including the entanglement structure and propagation speeds of excitations dual to bulk excitations. This is done in chapters 2 and 3. In chapters 4 and 5, we use this lens of generalized holography to elucidate the nature of the relationship between entanglement and emergent geometry. Finally, in chapter 6, we revisit one the important underlying assumption that the RT formula is general and demonstrate the validity of this assumption

Quantum Merlin-Arthur and proofs without relative phase

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) $\subseteq$ NEXP.Comment: 18 pages, 2 figure

Improved Soundness for QMA with Multiple Provers

We present three contributions to the understanding of QMA with multiple provers: 1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a PCP). 2) We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a monolithic protocol where Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers k and a soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.) 3) We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to sublinear multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature - namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom