Quantum Interactive Proofs and the Complexity of Separability Testing
TL;DR: In this article, it was shown that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm.
Abstract: We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.
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TL;DR: This survey describes recent results obtained for quantum property testing and surveys known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc.
Abstract: The area of property testing tries to design algorithms that can efficiently
handle very large amounts of data: given a large object that either has a
certain property or is somehow "far" from having that property, a tester should
efficiently distinguish between these two cases. In this survey we describe
recent results obtained for quantum property testing. This area naturally falls
into three parts. First, we may consider quantum testers for properties of
classical objects. We survey the main examples known where quantum testers can
be much (sometimes exponentially) more efficient than classical testers.
Second, we may consider classical testers of quantum objects. This is the
situation that arises for instance when one is trying to determine if quantum
states or operations do what they are supposed to do, based only on classical
input-output behavior. Finally, we may also consider quantum testers for
properties of quantum objects, such as states or operations. We survey known
bounds on testing various natural properties, such as whether two states are
equal, whether a state is separable, whether two operations commute, etc. We
also highlight connections to other areas of quantum information theory and
mention a number of open questions.
133 citations
Book•
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TL;DR: An overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define is provided.
Abstract: Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class NP, known as QMA, in which a quantum state plays the role of a proof also called a certificate or witness, and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model.In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.
48 citations
Posted Content•
TL;DR: In this article, the authors present a survey of quantum testers for properties of quantum objects, such as states or operations. But their focus is on properties of classical objects and not on quantum objects.
Abstract: The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow "far" from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.
24 citations
08 Feb 2021
TL;DR: In this article, the controlled SWAP test is adapted to an efficient and useful test for entanglement of a pure state, and it is shown that the test can evidence the presence of entanglements (and further, genuine n-qubit entenglement) and can distinguish entomblement classes, and that concurrence of a two-qu bit state is related to the test's output probabilities.
Abstract: Quantum entanglement is essential to the development of quantum computation, communications, and technology. The controlled SWAP test, widely used for state comparison, can be adapted to an efficient and useful test for entanglement of a pure state. Here we show that the test can evidence the presence of entanglement (and further, genuine n-qubit entanglement), can distinguish entanglement classes, and that the concurrence of a two-qubit state is related to the test's output probabilities. We also propose a multipartite measure of entanglement that acts similarly for n-qubit states. The number of copies required to detect entanglement decreases for larger systems, to four on average for many (n ≥ 8) qubits for maximally entangled states. For non-maximally entangled states, the average number of copies of the test state required to detect entanglement increases with decreasing entanglement. Furthermore, the results are robust to second order when typical small errors are introduced to the state under investigation.
23 citations
TL;DR: In this article, the authors introduced a family of entanglement measures called concentratable entanglements, which is a general framework for quantifying and characterizing multipartite entangements.
Abstract: Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state $|\ensuremath{\psi}⟩$ for these applications is often directly related to the degree or type of entanglement present in $|\ensuremath{\psi}⟩$. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.
12 citations
References
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Book•
01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index
25,929 citations
TL;DR: An unknown quantum state \ensuremath{\Vert}\ensure Math{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations.
Abstract: An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, ``Alice,'' and the receiver, ``Bob,'' must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state \ensuremath{\Vert}\ensuremath{\varphi}〉 which Alice destroyed.
11,600 citations
TL;DR: Practical application of the generalized Bells theorem in the so-called key distribution process in cryptography is reported, based on the Bohms version of the Einstein-Podolsky-Rosen gedanken experiment andBells theorem is used to test for eavesdropping.
Abstract: Practical application of the generalized Bells theorem in the so-called key distribution process in cryptography is reported. The proposed scheme is based on the Bohms version of the Einstein-Podolsky-Rosen gedanken experiment and Bells theorem is used to test for eavesdropping. © 1991 The American Physical Society.
9,259 citations
03 May 1971
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.
Abstract: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology. Here “reduced” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed.
6,675 citations
IBM1
TL;DR: The set of states accessible from an initial EPR state by one-particle operations are characterized and it is shown that in a sense they allow two bits to be encoded reliably in one spin-1/2 particle.
Abstract: As is well known, operations on one particle of an Einstein-Podolsky-Rosen (EPR) pair cannot influence the marginal statistics of measurements on the other particle. We characterize the set of states accessible from an initial EPR state by one-particle operations and show that in a sense they allow two bits to be encoded reliably in one spin-1/2 particle: One party, ``Alice,'' prepares an EPR pair and sends one of the particles to another party, ``Bob,'' who applies one of four unitary operators to the particle, and then returns it to Alice. By measuring the two particles jointly, Alice can now reliably learn which operator Bob used.
4,780 citations