Rigidity of the magic pentagram game
Abstract
A game is rigid if a nearoptimal score guarantees, under the sole assumption of the validity of quantum mechanics, that the players are using an approximately unique quantum strategy. Rigidity has a vital role in quantum cryptography as it permits a strictly classical user to trust behavior in the quantum realm. This property can be traced back as far as 1998 (Mayers and Yao) and has been proved for multiple classes of games. In this paper we prove ridigity for the magic pentagram game, a simple binary constraint satisfaction game involving two players, five clauses and ten variables. We show that all nearoptimal strategies for the pentagram game are approximately equivalent to a unique strategy involving real Pauli measurements on three maximallyentangled qubit pairs.
I Introduction
Quantum rigidity is a strengthening of the guarantee that quantum behavior is taking place. It essentially ascertains that observing certain correlations in a system, for example, correlations that violate Bell inequalities, is sufficient by itself to determine the quantum state and the measurements used to obtain these correlations. This notion was expressed in the work of Mayers and Yao on “selfchecking quantum sources” Mayers and Yao (1998) in 1998, and it can be traced back even earlier Popescu and Rohrlich (1992); Summers and Werner (1987). Rigidity is a central tool for quantum computational protocols that involve untrusted devices, since it allows a user to verify the internal workings of a device based only on its external behavior (see, e.g., Reichardt et al. (2013)).
Since its introduction the notion of rigidity has seen good deal of work, generally focused either on proving rigidity for particular classes of games, or proving that rigid games exist that selftest particular quantum states. Twoplayer games that are known to be rigid include the CHSH game Popescu and Rohrlich (1992); McKague et al. (2012), the magic square game Wu et al. (2016), the chained Bell inequalities Šupić et al. (2016), the MayersYao criterion Mayers and Yao (1998); Magniez et al. (2006), Hardy’s test Rabelo et al. (2012), the Hadamardgraph coloring game Mančinska (2014), and various classes of binary games Miller and Shi (2013); Wang et al. (2016); Bamps and Pironio (2015). New results on rigid games add to the tools available for protocols based on untrusted devices.
In the current paper we prove that the magic pentagram game (see Figure 1) is rigid. This game is a natural one to study: in particular, it was originally proposed alongside the magic square game Mermin (1990), and it shares some of the same properties that make the magic square game useful in cryptography (in particular, it shares the property that an optimal strategy must yield a perfect shared key bit pair between two parties, which was exploited in Jain et al. (2017)). From a resource standpoint, it also offers an improvement over the magic square game: whereas the magic square game requires questions to selftest EPR pairs, we will prove that the magic pentagram game selftests EPR pairs with questions. If we compare the number of bits of randomness needed to generate the questions set to the number EPR pairs tested, the magic square has a ratio of , while the magic pentagram game has a ratio of .
The optimal strategy for the magic pentagram game is shown in Figure 2. Our main result is summarized below, and proved formally in Propositions 9, 10, 12, and Corollary 11.
Theorem 1 (Informal).
Suppose that Alice and Bob have a strategy for the magic pentagram game that wins with probability . Then, after the application of a local isometry on Alice’s and Bob’s systems, the following statements hold.

The shared state is within Euclidean distance from a state of the form , where denotes a Bell state and denotes an arbitrary bipartite state. (Proposition 12.)
Our proof is selfcontained and borrows techniques from previous papers on rigidity McKague et al. (2012); McKague (2016); Wu et al. (2016). One of the challenges for the magic pentagram game is that the first player may associate two different measurements to a single observable — for example, in Figure 1, Alice may use a different measurement for vertex depending on whether the context is or . (This does not occur in the magic square game.) Our early technical work addresses this fact — see Propositions 5–6 and the discussion that follows.
The coefficients of the error terms for Theorem 1 are not given explicitly, and optimizing these coefficients is left as an open problem. (Tracing through the steps of the current proof might yield coefficients in the thousands.)
In the larger picture, the magic square game and the magic pentagram game are examples of binary constraint satisfaction games Cleve and Mittal (2014). Arkhipov Arkhipov (2012) proved that a certain natural subclass of binary constraint satisfaction problems — specifically, those that are based on XOR clauses where every variable is in exactly two clauses — are all in a precise sense reducible to the magic square game and the magic pentagram game. This suggests that our result is a step towards a full classification of winning quantum strategies within this class.
Ii The magic pentagram game
The pentagram game is a binary constraint satisfaction game between two parties, Alice and Bob. Its rules can be defined, as its name suggests, on a pentagram hypergraph, see Fig. 1. The five hyperedges of the pentagram (the clauses or contexts) are labeled , and each contains four vertices. The hyperedges are each assigned a value: , and . The rules of the games are as follows:

A context is chosen and a vertex is chosen (both uniformly at random). The context is given to Alice and the vertex is given to Bob.

Alice assigns either or to each vertex in the context , and Bob assigns or to .

Alice and Bob can communicate and agree on a strategy prior to the beginning of the game, but are not allowed to communicate once the game has begun.
The game is won if the following two conditions both hold:

The product of the values returned by Alice is equal to the preassigned value .

Alice and Bob return the same value for .
There is no classical strategy to win this game perfectly, as is easily verified. However, it can be won with probability 1 using quantum resources Mermin (1990, 1993). A winning strategy is schematically shown in Fig. 2, with and denoting the Pauli operators , and the identity operator, respectively. They share six qubits, three at Alice’s lab () and three at Bob’s (), prepared in the maximally entangled state
(1) 
where are the eigenbasis of the Pauli operator. (When no confusion arises we drop the tensor product symbol and the subscript labels for Alice and Bob’s subsystems.) Upon receiving a hyperedge label , Alice measures the four Pauli observables associated with the four vertices of on her three qubits, and then assigns to each vertex the value she obtains for the corresponding observable. These observables are reflection operators (i.e., Hermitian operators having eigenvalues in ) such that observables of adjacent vertices (vertices that are connected by the same hyperedge) all commute and thus can be measured simultaneously. Bob measures the observable of his input vertex on his three qubit system and assigns a value to the vertex according to the outcome of his measurement. By construction of this strategy, the winning conditions for this game, as listed above, are fulfilled for every input value and .
We note that in this strategy any two nonadjacent observables anticommute. (This will become important in later proofs.)
Iii Strategies for the magic pentagram game
Our goal is to relate arbitrary strategies for the magic pentagram game to the strategy in Figure 2. The class of strategies that we study are captured in the following definition.
Definition 2.
A projective strategy for the magic pentagram game consists of the following data:

The shared state: Two finitedimensional Hilbert spaces and , and a unit vector .

Alice’s measurements: For each , a projective measurement on , where varies over the set of all functions from to whose parity is equal to .

Bob’s measurements: For vertex , a projective measurement .
The functions obtained from these measurements specify the output values for Alice and Bob. Note that we could have allowed for the shared state to be mixed and for the measurements to be general positiveoperator valued measures (POVMs). However standard techniques imply that any such strategy is a partial trace of one in the above form, so there is no generality lost.
Additionally, we make the following definition.
Definition 3.
A reflection is a Hermitian automorphism whose eigenvalues are contained in . A reflection strategy for the magic pentagram game consists of the following data:

The shared state: Two finitedimensional Hilbert spaces and , and a linear map satisfying .

Alice’s reflections: Reflections
on such that the reflections that belong to any context all commute () and their product is equal to .

Bob’s reflections: Reflections on .
Note that any projective strategy can be converted into a reflection strategy, and vice versa, via the relations
(2)  
(3)  
(4) 
where on .
The probability distribution obtained from a projective measurement on is given by , and the probability distribution obtained from a projective measurement on is given by . For any context and any vertex , the probability that Alice and Bob will assign different values to the vertex in a given reflection strategy is given by
Thus the losing probability (that is, one minus the expected score) for the reflection strategy is given by
Thus we have the following.
Proposition 4.
Let be a reflection strategy for the magic pentagram game which achieves winning probability . Then, for any context and vertex ,
(5) 
Next we prove a series of properties for nearoptimal strategies, all of which are consequences of Proposition 4.
Proposition 5 (Changing contexts).
Let
be a reflection strategy with expected score . Let be a sequence of vertices and and be sequences of contexts such that for all . Then,
Proof.
Applying Proposition 4 inductively, we find that and are both within Euclidean distance from . ∎
The next two propositions certify the relation between reflection operators in a strategy with expected score . For convenience, hereafter we refer to sequences of matrices satisfying as approximate sequences.
Proposition 6 (Approximate commutativity).
Let be a reflection strategy with expected score . Let and be adjacent vertices, such that , and let be the other hyperedge which contains . Then,
(6)  
(7) 
Proof.
The desired result follows easily by applications of Proposition 4. ∎
Each vertex has two reflection operators for Alice ( and , where ). It is helpful for some of the proofs that follow to single out one distinguished reflection operator for each vertex. We therefore make the following (arbitrary) assignments,
(8) 
Proposition 7 (Approximate anticommutativity).
Let be a reflection strategy with expected score , and let and be nonadjacent vertices (i.e., vertices that never occur in the same context). Then,
(9)  
(10) 
Proof.
By Proposition 5, it suffices to prove these relations with replaced by . We give a proof for , which generalizes to cover all other cases by symmetry. The proof is inspired by the proof of rigidity for the magic square game Wu et al. (2016). Applying the rules for Alice’s measurements from Definition 3 and the foregoing propositions, we find that the following sequence is an approximate sequence:
and relation (10) follows similarly. ∎
Proposition 8.
Let be a sequence of vertices and . Then,
where if are adjacent and if are nonadjacent.
Iv Rigidity
In this section, we will use the following notation: will denote qubit registers (each with a fixed isomorphism to ). The linear maps denote the Hadamard maps , and the linear maps denote the Pauli operators. For any reflection on , and , let the map
(11) 
denote the controlled operation . Note that these maps interact as follows:
(12)  
(13) 
The next theorem asserts that some of the reflections in a nearoptimal strategy for the magic pentagram game can be simulated by Pauli operators. Let
(14) 
where the s are given in Eq. (8). These operators are chosen so that for (and similarly for ) the pairs , belong to nonadjacent vertices, while all the other pairs of operators belong to adjacent vertices. Thus the approximate commutativity conditions and anticommutativity conditions are what one would expect for the corresponding Pauli operators. We note that the particular choice of the s and s here is not unique. The following results will hold for any choice of s and s as long as they satisfy the required approximate commutation relations.
Proposition 9.
Let be a reflection strategy with expected score . Then, there exists an isometry from to such that for all ,
(15)  
(16) 
Proof.
Our construction of the isometries follows previous papers on rigidity (e.g., McKague (2016)). For each define
(17) 
by
(18) 
Then, the following is an approximate sequence:
Thus,
Additionally, the following is an approximate sequence:
Thus,
Also, if with , then by Proposition 6, the following is a approximate sequence:
Therefore
(19) 
and by similar reasoning,
(20) 
Define by the same expression used to define , except with the operators replaced with :
(21) 
Then, by Proposition 4. Let
(22) 
Then, the following is an approximate sequence:
Therefore,
(23) 
The desired result for follows by similar reasoning. ∎
Likewise, we have the following.
Proposition 10.
Let be a reflection strategy with expected score . Then, there exists an isometry from to such that for all ,
(24)  
(25) 
Proof.
Note that Propositions 9 and 10 easily generalize to sequences of measurements — for example, the following is an approximate sequence:
(26)  
(27)  
(28)  
(29)  
(30) 
Applying this method inductively, we have the following corollary.
Corollary 11.
Finally, we prove the following proposition, which addresses the image of the under the isometry . For each , let
(31) 
be defined by
(32) 
(This is a matrix expression for an EPR pair.) Let
(33)  
(34)  
(35) 
Proof.
Let . By the score assumption,
(37)  
(38) 
for , therefore by Propositions 9 and 10,
(39)  
(40) 
Note that , while the other Bell states fail significantly to satisfy the same equalities:
(41)  
(42)  
(43) 
Write
(44) 
where varies over . Conditions (39) and (40) imply that all components except must have Euclidean norm less than . The desired result follows. ∎
V Summary and Conclusions
Quantum rigidity allows a classical user to certify manipulations of quantum systems, thus enabling quantum cryptography in a scenario in which the user does not trust her quantum apparatus (deviceindependent quantum cryptography). In this paper we have expanded the toolbox for the deviceindependent setting by showing that the magic pentagram game is rigid. In particular, this means that it is possible to certify the existence of ebits using a game that consists of only questions.
In our style of proof we have reduced some of the arguments for rigidity to bare manipulations of sequences of operators (see the proofs in section IV). This style in particular allows us to cleanly handle conditions such as approximate commutativity and anticommutativity. Such an approach could be useful for proving more general results.
A natural next step would be to try to parallelize our result (following Reichardt et al. (2013); Ostrev (2015); Ostrev and Vidick (2016); Coladangelo (2016); Natarajan and Vidick (2017); McKague (2016, 2017); Chao et al. (2016); Coudron and Natarajan (2016)) to show that parallel copies of the magic pentagram game can be used to certify a maximally entangled state of arbitrary size. Then, we could try to choose a small subset of the questions from the parallelized game and prove that that subset is adequate to achieve rigidity.
The magic pentagram game is an example of a binary constraint satisfaction XOR game in which every variable appears in exactly two contexts. This class of games was studied in Arkhipov (2012), and the author proved that any game in the class that exhibits pseudotelepathy must in a sense contain either the magic square game or the magic pentagram game (as topological minors of its relational graph). An interesting further direction would be to explore further the consequences for our rigidity result (and Wu et al. (2016)) for the class from Arkhipov (2012).
Acknowledgements.
The authors would like to thank Cedric Lin for bringing Ref. Arkhipov (2012) to our attention, and Matthew McKague for helpful technical discussions about our proofs. AK is funded by the US Department of Defense.References
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