The study of entanglement through the length of interactive proof systems has been one of the most productive applications of complexity theory to the physical sciences that I know of. Last week Anand Natarajan and John Wright, postdoctoral scholars at Caltech and MIT respectively, added a major stone to this line of work. Anand & John (hereafter “NW”) establish the following wild claim: it is possible for a classical polynomial-time verifier to decide membership in any language in non-deterministic doubly exponential time by asking questions to two infinitely powerful, but untrusted, provers sharing entanglement. In symbols, NEEXP MIP! (The last symbol is for emphasis — no, we don’t have an MIP! class — yet.)
What is amazing about this result is the formidable gap between the complexity of the verifier and the complexity of the language being verified. We know since the 90s that the use of interaction and randomness can greatly expand the power of polynomial-time verifiers, from NP to PSPACE (with a single prover) and NEXP (with two provers). As a result of the work of Natarajan and Wright, we now know that yet an additional ingredient, the use of entanglement between the provers, can be leveraged by the verifier — the same verifier as in the previous results, a classical randomized polynomial-time machine — to obtain an exponential increase in its verification power. Randomness and interaction brought us one exponential; entanglement gives us another.
To gain intuition for the result consider first the structure of a classical two-prover one-round interactive proof system for non-deterministic doubly exponential time, with exponential-time verifier. Cutting some corners, such a protocol can be obtained by “scaling up” a standard two-prover protocol for non-deterministic singly exponential time. In the protocol, the verifier would sample a pair of exponential-length questions , send and to each prover, receive answers and , and perform an exponential-time computation that verifies some predicate about .
How can entanglement help design an exponentially more efficient protocol? At first it may seem like a polynomial-time verifier has no way to even get started: if it can only communicate polynomial-length messages with the provers, how can it leverage their power? And indeed, if the provers are classical, it can’t: it is known that even with a polynomial number of provers, and polynomially many rounds of interaction, a polynomial-time verifier cannot decide any language beyond NEXP.
But the provers in the NW protocol are not classical. They can share entanglement. How can the verifier exploit this to its advantage? The key property that is needed is know as the rigidity of entanglement. In words, rigidity is the idea that by verifying the presence of certain statistical correlations between the provers’ questions and answers the verifier can determine precisely (up to a local change of basis) the quantum state and measurements that the provers must have been using to generate their answers. The most famous example of rigidity is the CHSH game: as already shown by Werner and Summers in 1982, the CHSH game can only be optimally, or even near-optimally, won by measuring a maximally entangled state using two mutually unbiased bases for each player. No other state or measurements will do, unless they trivially imply an EPR pair and mutually unbiased bases (such as a state that is the tensor product of an EPR pair with an additional entangled state).
Rigidity gives the verifier control over the provers’ use of their entanglement. The simplest use of this is for the verifier to force the provers to share a certain number of EPR pairs and measure them to obtain identical uniformly distributed -bit strings. Such a test for EPR pairs can be constructed from CHSH games. In a paper with Natarajan we give a more efficient test that only requires questions and answers of length that is poly-logarithmic in . Interestingly, the test is built on classical machinery — the low-degree test — that plays a central role in the analysis of some classical multi-prover proof systems for NEXP.
At this point we have made an inch of progress: it is possible for a polynomial-time (in ) verifier to “command” two quantum provers sharing entanglement to share EPR pairs, and measure them in identical bases to obtain identical uniformly random -bit strings. What is this useful for? Not much — yet. But here comes the main insight in NW: suppose we could similarly force the provers to generate, not identical uniformly random strings, but a pair of -bit strings that is distributed as a pair of questions from the verifier in the aforementioned interactive proof system for NEEXP with exponential-time (in ) verifier. Then we could use a polynomial-time (in ) verifier to “command” the provers to generate their exponentially-long questions by themselves. The provers would then compute answers as in the NEEXP protocol. Finally, they would prove to the verifier, using a polynomial interaction, that is a valid pair of answers to the pair of questions — indeed, the latter verification is an NEXP problem, hence can be verified using a protocol with polynomial-time verifier.
Sounds crazy? Yes. But they did it! Of course there are many issues with the brief summary above — for example, how does the verifier even know the questions sampled by the provers? The answer is that it doesn’t need to know the entire question; only that it was sampled correctly, and that the quadruple satisfies the verification predicate of the exponential-time verifier. This can be verified using a polynomial-time interactive proof.
Diving in, the most interesting insight in the NW construction is what they call “introspection”. What makes multi-prover proof systems powerful is the ability for the verifier to send correlated questions to the provers, in a way such that each prover has only partial information about the other’s question — informally, the verifier plays a variant of prisonner’s dilemma with the provers. In particular, any interesting distribution will have the property that and are not fully correlated. For a concrete example think of the “planes-vs-lines” distribution, where is a uniformly random plane and a uniformly random line in . The aforementioned test for EPR pairs can be used to force both provers to sample the same uniformly random plane . But how does the verifier ensure that one of the provers “forgets” parts of the plane, to only remember a uniformly random line that is contained in it? NW’s insight is that the information present in a quantum state — such as the prover’s half-EPR pairs — can be “erased” by commanding the prover to perform a measurement in the wrong basis — a basis that is mutually unbiased with the basis used by the other prover to obtain its share of the query. Building on this idea, NW develop a battery of delicate tests that provide the verifier the ability to control precisely what information gets distributed to each prover. This allows a polynomial-time verifier to perfectly simulate the local environment that the exponential-time verifier would have created for the provers in a protocol for NEEXP, thus simulating the latter protocol with exponentially less resources.
One of the aspects of the NW result I like best is that they showed how the “history state barrier” could be overcome. Previous works attempting to establish strong lower bounds on the class MIP, such as the paper by Yuen et al., relies on a compression technique that requires the provers to share a history state of the computation performed by a larger protocol. Unfortunately, history states are very non-robust, and as a result such works only succeeded in developing protocols with vanishing completeness-soundness gap. NW entirely bypass the use of history states, and this allows them to maintain a constant gap.
Seven years ago Tsuyoshi Ito and I showed that MIP contains NEXP. At the time, we thought this may be the end of the story — although it seemed challenging, surely someone would eventually prove a matching upper bound. Natarajan and Wright have defeated this expectation by showing that MIP contains NEEXP. What next? NEEEXP? The halting problem? I hope to make this the topic of a future post.