Advantage of Hardy's Nonlocal Correlation in Reverse Zero-Error Channel Coding
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Advantage of Hardy’s Nonlocal Correlation in Reverse Zero-Error Channel Coding
Mir Alimuddin,1 Ananya Chakraborty,1 Govind Lal Sidhardh,1 Ram Krishna Patra,1
Samrat Sen,1 Snehasish Roy Chowdhury,2 Sahil Gopalkrishna Naik,1 and Manik Banik1
1 Department of Physics of Complex Systems, S. N. Bose National Center for Basic Sciences,
Block JD, Sector III, Salt Lake, Kolkata 700106, India.
2 Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 BT Road, Kolkata, India.
Hardy’s argument constitutes an elegant proof of quantum nonlocality as established by the seminal
Bell’s theorem. In this work, we report an exotic application of Hardy’s nonlocal correlations. We
devise a simple communication task and show that the expected payoff of the task cannot be positive
whenever only 1-cbit communication is allowed from the sender to the receiver, who otherwise can
share an unlimited amount of classical correlation. Interestingly, the same classical channel can
arXiv:2303.06848v1 [quant-ph] 13 Mar 2023
ensure a positive payoff when assisted with correlations exhibiting Hardy’s nonlocality. As it turns
out, among all the 2-input-2-output no-signaling correlations, only Hardy’s correlation can ensure a
positive payoff when assisted to 1-cbit channel. This further prompts us to show that in the correlation
assisted reverse zero-error channel coding scenario, where the aim is to simulate a noisy channel
exactly by a noiseless one in assistance with correlations, assistance of non-maximally pure entangled
states – even with vanishingly zero amount of entanglement – could be preferable over the maximal
one.
Introduction.– The pioneering work of J. S. Bell estab- nel. This advantage is striking due to the following
lishes one of the most striking departures of quantum reasons. Firstly, nonlocal correlations are compatible
theory from a classical worldview [1] (see also [2]). Vi- with the no signalling (NS) principle, and hence they
olation of a Bell-type inequality, as demonstrated in by themselves cannot be used for information transfer.
several milestone experiments [3–8], endorses that cer- Second, the advantage reported here is different from
tain correlations obtained from multipartite entangled the nonlocal advantage demonstrated in the communic-
states are not compatible with a local-realistic description ation complexity scenario [13]. In fact, we argue that the
[9]. Apart from its foundational implications, Bell non- scenario considered here is much more elementary than
locality has also been identified as a useful resource for the communication complexity tasks.
several practical tasks, such as, device-independent cryp- The advantage we establish gets reflected in the payoff
tography [10–12], reduction of communication complex- of a guessing game played between two distant players
ity [13], device-independent randomness certification – a sender and a receiver. The expected collaborative
[14–16], and randomness amplification [17]. payoff of this game cannot be positive whenever only
Apart from Bell-type inequalities, another technique 1-bit classical communication is allowed from the sender
popularly known as ’nonlocality without inequality’ to the receiver, who otherwise can share an unlimited
proofs is often used to establish the nonlocal beha- amount of classical correlation between them. Inter-
viour of quantum theory. Unlike the Bell-type inequalit- estingly, the assistance of Hardy’s nonlocal correlation
ies, where statistics of many events are collected, these ensures a strictly positive payoff, establishing a non-
proofs focus on a single event whose occurrence shows trivial utility of Hardy’s correlation in communication
the incompatibility of quantum theory with the notion tasks. Furthermore, we show that Hardy’s nonlocal
of local-realism. While the first proof of this kind for correlations are necessary for ensuring a positive pay-
tripartite quantum systems is due to Greenberger-Horne- off in this game. This motivates us to further show
Zeilinger [18] (see also [19]), for bipartite systems, such that pure non-maximally entangled states are prefer-
a proof was first proposed by Hardy [20], which is con- able over the maximally entangled ones for the game in
sidered to be “simpler and more compelling than the question. This turns out to be the case even when the
arguments that underlie the derivation of Bell-CH in- former state has arbitrarily less amount of entanglement.
equality" [21]. More recently, Hardy-type nonlocality This, in turn, proves that in correlation-assisted reverse
proofs have shown to be useful in several practical tasks zero-error coding scenario [28], where the aim is to sim-
[22–27]. ulate a higher input-output noisy classical channel by a
In this work, we report a novel application of Hardy’s lower input-output identity channel in the presence of
nonlocal correlation in the simplest communication scen- NS correlations, non-maximally entangled states can be
ario. In particular, we show that Hardy’s nonlocal correl- advantageous over the maximally entangled state.
ation shared between two distant parties can empower A two party guessing game.– We start by introducing a
the communication utility of a perfect classical chan- distributed guessing game played between two distant2
players (say) Alice and Bob. There is a Referee (say)
Charlie who, in each run of the game, provides four
closed boxes, numbered 1 to 4, to Bob, who has to open
one of these boxes. Some of these boxes contain a bomb
that will explode upon opening the box. Among the
boxes that don’t contain the bomb, some may be empty,
some may contain a dollar bill, and others may even
prompt Bob to pay a dollar bill to Charlie. In each
run of the game, Charlie randomly picks one among
four different arrangements of these boxes, as shown
in Fig.1. Charlie then informs his choice to Alice, who
then tries to help Bob in picking a box. However, only
1-bit of classical communication is allowed from Alice
to Bob, which may be further assisted with preshared
NS correlations shared between them. From now on, we
will call this the distributed mine-hunting (DMH) game. Figure 1. Distributed mine-hunting (DMH) game. Opening a
General scenario.– The aforesaid game can be formally box with ‘ + $0 assures dollar bill gain for Bob while opening
studied within a more generic set-up. Alice and Bob a box with ‘ − $0 demands him to pay dollar bill to Charlie.
are given some collaborative payoff depending on the A box with a ‘smile’ neither offers nor demands any dollar
classical index z ∈ Z produced by Bob, given that Alice bill, whereas a box with a ‘bomb’ turns out to be fatal to Bob.
received some classical message m ∈ M sampled from a Alice knows which of the arrangements {A-1,· · · ,A-4} Charlie
chooses in a particular run and tries with a limited classical
probability distribution { p(m) | m ∈ M}. Such a game, channel to help Bob to optimize his expected dollar gain.
in fact, is completely specified by the payoff matrix G ≡
( gmz ), where gmz ∈ R is the reward/payoff given when
Bob produced the index ‘z’ provided Alice received the by the players can be represented as a |M| × |Z | matrix
message ‘m’. For instance, the DMH game is specified S ≡ (smz ), where smz denotes the probability of produ-
by the following payoff matrix: cing the output z ∈ Z by Bob given that Alice received
the message m ∈ M. Manifestly, entries of such a mat-
M\Z 1 2 3 4
rix are non-negative with row sum one. In this work,
1 −1 0 0 −∞ we refer to this matrix as a strategy matrix which, up
G DMH ≡ 2 −∞ 0 −∞ 0 (1)
to a transposition, is identical to the notion of ’channel
3 +1 0 − ∞ − ∞ matrix’ used in [30]. Given such a strategy matrix S, the
4 −∞ −∞ 0 0 average payoff can be obtained as
The payoffs quantitatively capture the scenario of DMH
game. A reward of −∞ for the box containing the bomb
P (S) = ∑ p(m) gmz smz . (2)
z,m
captures the notion that choosing such a box must be
avoided at all costs [29]. The reward 0 corresponds to the As it is evident, there will always be a perfect strategy
event where the players survive but do not receive any for such a game if log2 |M| bits of communication are
reward. Events with reward +1 (−1) correspond to the allowed from Alice to Bob. Interesting situations arise
scenario where the players receive (pay) some dollar bill when communication is limited which might further be
from (to) Charlie. The game matrix and the sampling aided by preshared correlations of different types. In the
distribution of Alice’s inputs are common knowledge to next section, we analyze different such cases for DMH
the players. game and present some novel results.
Alice and Bob are cooperative in nature and aim Results.– We start with the scenario where only 1
to maximize the payoff. Their collaborative strategy bit of classical communication is allowed from Alice
depends on the available resources, which can be to Bob, and they can share an unlimited amount
broadly categorized into two types: (i) correlation shared of share randomness, i.e., classical correlation. By
between them before the game starts and (ii) commu- Ωnc +SR (|M|, |Z |) we denote the set of strategy matrices
nication from Alice to Bob which is allowed even after obtained when n-bits classical communication and un-
the game starts. While correlations can further be clas- limited amount of shared randomness are available.
sified into classical, quantum, or the more general no The set Ωnc +SR (|M|, |Z |) forms a polytope with ex-
signalling types, for direct communication, a classical or treme points Se ’s representing strategy matrices ob-
quantum channel can be used. The strategy employed tained through deterministic encoding M → {0, 1}n3
marginal outcome probabilities depend only on their
own inputs, i.e., p( a| x, y) = p( a| x ) & p(b| x, y) = p(b|y).
Classical correlations that R allow a local-realistic de-
scription, p( a, b| x, y) = Λ µ(λ) p( a, b| x, y, λ)dλ, forms
a strict subset (a sub polytope) of the NS polytope; here
λ ∈ Λ is some classical variable shared between Alice
and Bob and µ(λ) is a probability distribution on Λ [9].
Quite surprisingly, entangled quantum states can lead
to correlations that are not local-realistic and nonlocality
of those correlations can be certified through violation
of some Bell-type inequalities [1]. Given such an NS
correlation (possibly nonlocal) as an assistance to 1-bit
classical communication from Alice to Bob, the general
Figure 2. General strategy to play a game G when the 1-cbit strategy to play a game G is described in Fig.2.
communication channel is assisted with NS correlation. Alice
computes the input x = X (m) to her part of the nonlocal box We will now proceed to analyze the success prob-
based on the message m ∈ M received from Referee. The ability of DMH when a correlation exhibiting Hardy’s
output a of the nonlocal box at her end and the message m nonlocality is provided as assistance to the 1-cbit com-
determines the classical bit c = C ( a, m) sent to Bob. Bob then munication line. When the input and output space of
inputs y = Y (c) into his end of the NS box and obtains the Alice and Bob are binary, i.e., A = B = X = Y ≡ {0, 1},
output b. Finally, he generates his guess as z = Z (b, c).
Hardy come with an elegant argument according to
which any NS correlation {h( a, b| x, y)} satisfying the
at Alice’s end and deterministic decoding {0, 1}n → Z constraints
at Bob’s end [30] (see also [31]). Entries of such extreme
h(00|00) := h0 > 0, h(01|01) := h5 = 0,
strategy matrices can only be 0 or 1, and since only n-
cbits are allowed, such a strategy can have at most 2n h(10|10) := h10 = 0, h(00|11) := h3 = 0, (3)
non-zero columns. Our first technical result is to limit
with, h( a, b| x, y) := h a×23 +b×22 + x×21 +y×20
the optimal success probability of the game in Eq.(1) for
classical strategies. must be nonlocal in nature [20]. Our next result proves a
Theorem 1. The average payoff of the DMH game is up- nontrivial advantage of Hardy correlation while playing
per bounded by zero while following a strategy from the set the DMH game.
Ω1c +SR (4, 4), i.e., P (S) ≤ 0, ∀ S ∈ Ω1c +SR (4, 4).
Theorem 2. A strictly positive average payoff in DMH game
Proof. Since average payoff depends linearly on the can be ensured when a 2-input-2-output Hardy’s nonlocal
strategy [see Eq.(5)], the polytope structure of correlation is available to assist the 1-cbit communication
Ω1c +SR (4, 4) ensures that maximum payoff will be at- channel from Alice to Bob.
tained at one of its vertices [32]. Note that the players’
first priority is to avoid the Bomb at any cost, i.e., the
Proof. Consider the following strategy by Alice and Bob.
−∞ reward in Eq.(1). As it turns out, among all pos-
Alice’s action:
sible vertices, only 5 vertices have a zero probability
• Depending on m ∈ M Alice computes her input in
of choosing any of the Bombs. The strategy matrices
the NS box. For m ∈ {1, 3} she chooses x = 0, otherwise
corresponding to these strategies are listed in Appendix.
she choose x = 1.
A straightforward calculation shows that the average
• Based on the tuple (m, a) ∈ M × A she communic-
payoff is zero for all these strategies. This proves the
ates to Bob. She sends c = 0 to Bob when (m, a) ∈
claim.
{(1, 1), (2, 1), (3, 0), (3, 1)}, else she sends c = 1.
We now consider the scenario where the commu- Bob’s action:
nication line from Alice to Bob is aided with a gen- • The communicated bit from Alice is used as input in
eric input-output correlation { p( a, b| x, y) | p( a, b| x, y) ≥ Bob’s part of the NS box.
0 & ∑ a,b p( a, b| x, y) = 1}, where p( a, b| x, y) denotes the • Depending on the tuple (c, b) ∈ C × B he chooses
probability of obtaining outcome a ∈ A at Alice’s end the box as follows: (0, 0) 7→ 1, (0, 1) 7→ 2, (1, 0) 7→
and b ∈ B at Bob’s end for their respective inputs x ∈ X 3, (1, 1) 7→ 4.
and y ∈ Y . We are interested only in those correlations The above strategy with 1-bit communication and 2-
that satisfy the NS conditions that each of the parties’ input-2-output Hardy’s correlation {h( ab| xy} leads to4
the strategy matrix, This theorem has an interesting implication. It shows
that there exists a communication task wherein a non-
M\Z 1 2 3 4 maximally pure entangled state can be preferable over
1 h8 h12 h1 0 the maximally entangled one even when the entangle-
SH ≡ 2 0 h14 0 h7 (4) ment of the former is vanishingly zero. More formally
3 h0 + h8 h4 + h12 0 0 we can deduce the following corollary.
4 0 0 h11 h7 + h15
Corollary 1. For every non maximally entangled state |ψi ∈
As evident from the payoff matrix (1) and the strategy C2 ⊗ C2 there exists a strategy matrix Sψ such that Sψ ∈
(4), a Box containing Bomb will never be opened. Fur- Ω1c +SR+|ψi (4, 4) but Sψ ∈
/ Ω1c +SR+|φ+ i (4, 4).
thermore, assuming Charlie’s choice to be completely √
random, we have the average payoff Here |φ+ i := (|00i + |11i)/ 2 and Ω1c +SR+|χi (4, 4)
denotes the convex set of strategy matrices simulable
1 T 1
P (S H ) = Tr[G DMH S H ] = h0 > 0. (5) with 1-cbit communication from Alice to Bob when the
4 4 communication channel is further assisted with pre-
This completes the proof. shared quantum state |χi and unlimited shared random-
ness. The Corollary follows when results of Theorems
It is known that a correlation with Hardy success h0
2 & 4 are combined with the fact that all non maximally
yields Clauser-Horne-Shimony-Holt (CHSH) [33] value
pure entangled state exhibits Hardy’s nonlocality[37].
2 + 4 h0 [34]. Since the optimal success of Hardy’s
√ correl-
Discussions.– For a better understanding of the present
ation in quantum theory is known to be (5 5 − 11)/2
work, we briefly analyze a few other quantum advant-
[35], the corresponding √ CHSH value is much less than ages known in the communication scenario. At first, we
the Cirel’son bound 2 2 [36]. Therefore, a natural ques-
recall the seminal quantum superdense coding protocol,
tion is whether other nonlocal quantum correlations can
where it was shown that quantum entanglement, pre-
lead to better success in the DMH game. It can be argued
shared between a sender and a receiver, can increase
that following the strategy discussed in Theorem 2, one
the classical communication capacity of a quantum sys-
can not have a nonzero payoff in the DMH game when
tem [39] (see also [40]). Importantly, in quantum super-
the quantum correlation resulting in the Cirel’son bound
dense coding protocol, a quantum channel is considered,
is in use. However, given a correlation, one can develop
whereas our result shows that quantum entanglement
different strategies to play the DMH game. Therefore
can even empower the communication utility of a perfect
the aforesaid observation is not sufficient to make the
classical channel.
claim that
√ the quantum correlation leading to CHSH It is also known that the assistance of entanglement
value 2 2 is not good for playing the DMH game. We
can increase the zero-error communication capacity of
thus proceed to state our next result.
a noisy classical channel [41]. In the zero-error scen-
Theorem 3. Any 2-input-2-output NS correlation providing ario, the aim is to simulate a perfect classical channel
a strictly positive payoff in DMH game as an assistance to the with a lower number of inputs & outputs with a noisy
1-cbit channel must exhibit Hardy’s nonlocality. channel having a higher number of inputs & outputs.
The assistance of nonlocal correlations arising from en-
Proof of this theorem is given in the Appendix. Al-
tangled quantum states turns out to be advantageous in
though it is known that two-qubit maximally entangled
such a simulation task. The present advantage can be
state does not exhibit Hardy’s nonlocality [37, 38], still
seen in the correlation-assisted reverse zero-error cod-
Theorem 3 is not sufficient to make a claim that such
ing scenario, where the aim is to simulate exactly a
a state shared between Alice and Bob can not lead
higher input-output noisy classical channel with a lower
to a strategy yielding strictly positive payoff in DMH
input-output perfect classical channel in the presence of
game. Increasing the cardinality of the input-output sets
pre-shared entanglement between sender and receiver.
X , Y , A, B Alice and Bob can generate a more general
While such an advantage of quantum entanglement is
NS correlation and then try to utilize this correlation to
already known (see Proposition 21 in [28]), the full pic-
assist the 1-cbit channel to obtain a nonzero payoff in
ture of entanglement assistance is not well understood.
DMH game. However, our next result (proof provided
Our Theorem 4 proves a nontrivial result in this direction
in the appendix) proves a no-go to this aim.
as it shows that in the correlation-assisted reverse zero-
Theorem 4. Two qubit maximally entangled state together error coding scenario, non-maximally entangled states,
with 1-cbit channel from Alice to Bob does not result in a with less amount of entanglement, might be preferable
strategy ensuring strictly positive average payoff in DMH over a state having more entanglement. In particular,
game. the noisy channel in Eq.(4) corresponding to the Hardy’s5
correlation (3) can be perfectly simulated by 1-cbit com- (Grant no: I-HUB/PDF/2021-22/008). MB acknow-
munication assisted with a non-maximally entangled ledges support through the research grant of INSPIRE
state producing the Hardy correlation (3). However, it Faculty fellowship from the Department of Science
can not be simulated by 1-cbit communication when and Technology, Government of India and the start-up
the sender and receiver share a two-qubit maximally research grant from SERB, Department of Science and
entangled state as assistance. In the resource theory Technology (Grant no: SRG/2021/000267).
of quantum entanglement [42], where local operations
and classical communication (LOCC) are considered
to be free, a maximally entangled state is more useful
than a non-maximally entangled one, and a determin- Appendix A: Detailed proof of Theorem 1
istic LOCC transformation is always possible from the
former to the later [43]. On the other hand, our result
shows that entanglement comparisons among different As already mentioned, the entries of an extreme
states are not immediate when classical communication strategy matrix Se of the set Ω1c +SR (4, 4) are either zero
is regarded as a costly resource. or one with all row sum one and can not have more than
two nonzero columns. However, all such extreme points
A nonlocal advantage is also known in a variant of the
are no good for playing the game G DMH . For instance,
communication scenario known as the communication
consider the extreme strategy matrix,
complexity problem [13]. In such a scenario, Bob’s goal
is not to determine Alice’s data M, but to determine
some information that is a function of M in a way that 1 0 0 0
may depend on other data N that resides with Bob while e
0 0 1 0
Sns :=
1
. (6)
N is unknown to Alice. In that sense, our scenario is 0 0 0
closer to the standard framework of Shannon [44] (and 0 0 1 0
considered by Holevo in quantum set up [45]), where
at Bob’s end, no further data set N is considered. How- Comparing with Eq.(1), it is evident that if the players
ever, unlike Shannon’s and Holevo’s framework, which execute a strategy corresponding to this strategy mat-
provides a set-up to analyze utilities of noisy channels rix, then there is a nonzero probability that the bomb
in the asymptotic limit, we only consider the single- will be triggered (for m = 2). Having a nonzero non-
shot use of a perfect classical channel in the presence of survival probability, the players would certainly avoid
preshared correlations (classical as well as quantum). this strategy. On the other hand, for any of the following
Conclusion and outlook.– In conclusion, the present five strategies
work establishes a novel use of quantum entanglement
in a communication scenario. In particular, we show that
1 0 00
0 1 0
0
quantum correlations exhibiting Hardy’s nonlocality can
0 0 01 0 1 0 0
e , Se
Ss1 := := ,
empower the communication utility of a perfect clas- s2
1 0 00 0 1 0 0
sical communication channel. Our work also motivates
0 0 01 0 0 1 0
many questions for future study. For instance, it would
0 1 0 0 0 1 0 0
be interesting to see whether any nonlocal correlation
0 1 0 0 0 0 0 1
e , Se
can be made useful as a communication resource in the Ss3 :=
0 s4 := , (7)
1 0 0 0 1 0 0
sense discussed here. It will also be interesting to see
0 0 0 1 0 0 0 1
whether maximally entangled states of higher dimen-
sions provide some advantage in the DMH game. More 0 0 1 0
generally, characterizing the set Ωnc +SR+χ (|M|, |Z |) for 0 1 0 0
e
Ss5 :=
an arbitrary quantum state |χi would be very interest-
0 1 0 0
ing. 0 0 1 0
We thankfully acknowledge discussions with
Ashutosh Rai. GLS acknowledges support from the CSIR bomb will never be triggered. Furthermore, we have
project 09/0575(15830)/2022-EMR-I. SGN acknowledges e ) = 0, ∀ K ∈ {1, · · · , 5}. Moreover, these turn
P (SsK
support from the CSIR project 09/0575(15951)/2022- out to be the only extremal strategy matrices of Ω1c +SR
EMR-I. MA and MB acknowledge funding from the that do not trigger the bomb. Since all the deterministic
National Mission in Interdisciplinary Cyber-Physical strategies result in either zero or −∞ payoff, therefore
systems from the Department of Science and Technology the optimal payoff with 1 bit classical communication
through the I-HUB Quantum Technology Foundation and shared randomness is simply zero.6
Appendix B: Proof of Theorem 3 received. This leads to the strategy matrix with elements
1 1
Proof. The set of strategy matrices simulable by 1-cbit
h i
∑ smzc := ∑ Tr
(m) (c)
smz = |φ+ ihφ+ | Ec ⊗ Nz ,
communication with the assistance of a given NS correl- c =0 c =0
ation P ≡ { p( ab| xy) | a, b, x, y ∈ {0, 1}} and unlimited 1 h i
∗(c)
∑ Tr
(m)
SR forms a polytope. We denote this set by Ω1c +SR+ P . = Ec Nz . (9)
Vertices of this polytope correspond to strategies where c =0
the players follow an encoding and decoding scheme (m) ∗(c)
characterised by deterministic functions of the form The aim is to find POVM elements Ec and Nz such
x = X (m), c = C (m, a), y = Y (c), z = Z (c, b). The lin- that the resulting strategy yields a positive payoff in
earity of the payoff function again implies that the max- DMH game. For simplicity, we drop the notation ‘∗’
∗(c)
imum payoff occurs at one of the vertices of Ω1c +SR+ P . in Nz keeping in mind that if we do indeed find
Elementary counting shows that there are 24 × 28 × a solution for Eq.(9), we need to complex conjugate
(c) (c)
2 × 44 such deterministic strategies. Furthermore, ele-
2
the matrices Nz . We also note that if Nz forms a
ments of the strategy matrix is related linearly to the NS ∗(c)
measurement so does Nz . Therefore a strictly positive
correlation P = { p( a, b| x, y)}, payoff in DMH game demands following conditions to
be satisfied:
smz = ∑ δx,X (m) × δc,C(m,a) × δy,Y (c) h i h i
x,a,c,b,y∈{0,1} (1) (0) (1) (1)
Tr E0 N4 = 0 = Tr E1 N4 , (10a)
× δz,Z(c,b) × p( a, b| x, y). (8) h
(2) (0)
i h
(2) (1)
i
Tr E0 N1 = 0 = Tr E1 N1 , (10b)
For a non-negative payoff, the game matrix enforces h
(2) (0)
i h
(2) (1)
i
some of the entries of S ≡ (smz ) to be zero. Since Eq.(8) Tr E0 N3 = 0 = Tr E1 N3 , (10c)
is linear, one can solve these equality constraints to see h
(3) (0)
i h
(3) (1)
i
Tr E0 N3 = 0 = Tr E1 N3 , (10d)
what restrictions are imposed on the NS correlation
P( a, b| x, y). By brute-forcing through all the determin-
h i h i
(3) (0) (3) (1)
Tr E0 N4 = 0 = Tr E1 N4 , (10e)
istic strategies and by symbolic programming, we were h i h i
(4) (0) (4) (1)
able to verify that for each vertex of Ω1c +SR+ P , the pos- Tr E0 N1 = 0 = Tr E1 N1 , (10f)
itivity of average payoff imposes the conditions in Eq.(3) h
(4) (0)
i h
(4) (1)
i
(or its local reversible relabelling) to the NS correla- Tr E0 N2 = 0 = Tr E1 N2 , (10g)
tion { p( a, b| x, y)}. This proves the claim that among 1 h i 1 h i
∑ Tr ∑ Tr
(3) (c) (1) (c)
all 2-input-2-output correlations, only Hardy’s nonlocal Ec N1 > Ec N1 . (11)
correlations can provide a positive payoff in the DMH c =0 c =0
game.
The inequality (11) can can further be written as
h i h i
(3) (0) (1) (1) (0) (1)
Appendix C: Proof of Theorem 4 Tr E0 N1 − N1 > Tr E0 N1 − N1 .
(12)
Proof. Let Alice and Bob share the two-qubit maximally
entangled state |φ+ i = √1 (|00i + |11i). Note that the Note that, if a solution of POVMs exist satisfying
2 (10) and (12), then there also exists a solution with
existence of a strategy involving SR that yields an ad- (m) (m)
vantage in winning the game necessarily implies the ex- { E0 , E1 } being projective measurements ∀ m ∈
istence of a strategy without involving any SR. We now {1, · · · , 4}. This can be argued easily by expanding
(m) (m)
prove that such a strategy does not exist. Without loss of all the operators { E0 , E1 } in the spectral form. For
generality, we can assume that Alice does a 2 outcome example in inequality (12) on the LHS, we can choose
measurement depending on the classical message m she (3)
the projector formed by the eigenvector of E0 giving
(m) (m) (m) (m)
receives, i.e., she performs { E0 , E1
| = I}
E0 + E1 the maximum value of the trace. Similarly on the RHS,
for m ∈ {1 · · · 4}. Alice then communicates c ∈ {0, 1} we can choose the projector formed by the eigenvector
(m) (1)
if the outcome Ec clicks. Based on the communic- of E0 giving the lowest value of trace. Moreover, no-
ated bit c, Bob performs a 4 outcome measurement, tice than for Eqs.(10) all the projectors corresponding
and based on the measurement outcome, he chooses a (m) (m)
to different eigenvalues of { E0 , E1 } must satisfy the
(c)
box. Bob’s measurement is denoted by { Nz }4z=1 with Eqs.(10) individually. Thus we start by assuming all
(c) (m) (m)
∑4z=1 Nz = I when the communication c ∈ {0, 1} is measurements{ E0 , E1 } are projective. In particular,7
let
(3) (3)
E0 = |ψihψ|, E1 = |ψ⊥ ihψ⊥ |,
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