Quantifying quantum nonlocality

I was a teaching assistant to Alexei Kitaev and Jeff Kimble and learned from both of them. Many people have been part of the Institute for Quantum Information during my time here.

Motivation

Nonlocal correlations

An example
Notation and deﬁnitions
Bell polytopes
The CHSH inequality
Measures of nonlocality

The operation of the machines can be summarized by specifying the conditional joint probability distribution of Alice and Bob's results. This is known as the loophole and can be closed by ensuring that Alice's and Bob's measurement events are spatially separated.

Overview of the thesis

Classical models for the quantum joint correlation

Here we combine the ideas of Chapters 2 and 3 with those of Chapter 5 to show that five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any two-way quantum state.

Monogamy of nonlocal correlations

An LHV model for the full probability distribution is one that gives the same αβ, α, and β values as the quantum theory. An LHV model for the joint correlation is one that gives the same joint correlation αβ, but not necessarily the correct marginals.

Werner states

Furthermore, it turns out that an explicit LHV model stems from Krivine's upper limit on KG(3), which we shall see in the following chapter. If Alice's projective measurements are restricted to a plane in the Poincare sphere, then there is an LHV model for ρWp if and only if p≤1/√.

Generalization to higher dimensions

In other words, there always exists an LHV model for the joint correlation of two traceless observables at ρpforp≤1/KG(2d2) and there exists a state (actually the largest entangled state at log2dqubits) such that the joint correlation is nonlocal forp >1/KG(2log2d+ 1). Then there is an LHV for the entire probability distribution derived from untraceable observations for p≤1/KG(d2−1).

Bell inequalities for Werner states

Conclusions

In section 3.6 we present an LHV model for traceless observabilities on mixtures of the identity with a random state on Cd⊗Cd. The LHV model for Werner states from section 3.5 is actually just a special case of the model in section 3.6: however, we present it separately, because it is the most interesting case and only requires mathematical machines that are probably known to quantum mechanics. theorists.

A primitive for LHV models

However, we have ordered the sections so that the LHV models increase in sophistication, and we recommend that they be read in order. For any r∈ (0,∞), ˆλ byλ =rˆλ does not change the sign of v·λ, so we can draw our shared random vectors from any set with the property that the projection of λ onto the unit sphere is uniform. A particularly convenient choice is to randomly sample each coordinate of λ from a Gaussian distribution with mean 0 and standard deviation 1.

LHV model based on Grothendieck’s upper bound on K G

LHV model based on Krivine’s upper bound on K G

To apply Lemma 3.2.2, we need to verify that A(ˆa) is a unit vector whenever ˆ.

As before, to apply our analysis of Protocol 3.2.1, we must first establish that the function A3 maps unit vectors to unit vectors.

The proof that this protocol yields the correct correlations proceeds in the same way as the proof of Theorem 3.5.2 in the previous section. To determine the value of cn, we require Aˆn(ˆa) to be a unit vector, which is equivalent to the condition.

To prove that protocol 3.7.3 gives the correct correlations, we start by establishing some properties of the function f(t). To prove (iv), we either set Eq. 3.73) into a computer algebra system, or calculate (after Krivine). It is instructive to look through the proof and note where we used each of the four properties of function f(t) listed in Lemma 3.7.4.

Property (iv) ensures that the Dirichlet series D(s), defined in Eq. 3.86), is absolutely convergent for s > 0, which means that we can write 1/D(s) as a Dirichlet series that is itself absolutely convergent.

Figure 3.1: The function f deﬁned by Eq. (3.65)

Discussion

We present the full set of inequalities for M = 2 and the full set of inequalities for the total correlation observable for M = 3. We find that the correlations produced by quantum theory satisfy both sets of inequalities. One bit of communication is therefore sufficient to simulate quantum correlations in both of these scenarios.

In the scenario where two parties choose one of M two-score measures and exchange one bit of information, we present the full set of inequalities for M = 2, and the full set of inequalities for the common observable correlation ForM = 3.

The model

These inequalities provide conditions for the joint probability distribution, which must be satisfied if such correlations can be simulated with joint chance and a fixed amount of communication. Even in this case, however, the amount of two-way communication required to reproduce the joint probability distribution provides a measure of the locality of the correlations. Of particular importance in this regard is the result of Brassard, Cleve and Tapp [52], who demonstrated that correlations produced by projective measurements with two results in an EPR pair can be simulated by a realistic local augmented theory with eight communication bits . .

Surprisingly, a single piece of communication is enough, as we will see in the next chapter.

Bell polytopes

We emphasize that a protocol of this form simulates the joint probability distribution resulting from a set of quantum measurements, but not the quantum measurements themselves: it is not possible to replace local measurements made by two space-like separated parties on an entangled quantum state with classical communication. To each function pair {α, β} there corresponds a deterministic protocol, so the set of all deterministic protocols is a finite collection of such vectors {dζ|ζ= 1, .., K2M}. Therefore, the set of all possible protocols using randomness and no communication is described by a convex sum of the deterministic protocols without communication.

The set of all protocols therefore corresponds to the region ΩMK in RD, which is a polytope because there is a finite number of extremal vectors dζ [20].

Bell inequalities with auxiliary communication

Complete set of Bell inequalities with auxiliary communication

The above inequalities completely describe the range of probability distributions that can be created with one piece of communication. It is easy to check whether any probability distribution that satisfies the no-signaling conditions satisfies the inequalities. Cf. This probability distribution violates the no-signaling conditions (in both directions), but satisfies Eq. 4.4), indicating that these disparities are strictly speaking stronger than no signaling.

Complete set of Bell inequalities for the joint observable

That the inequalities are trivial also follows from the equations. 4.3) and (4.4), because in this scenario all possible joint observers can be obtained from probability distributions satisfying the no-signaling conditions [17, 18]. Let us show that quantum theory satisfies all of the above Bell's common observable inequalities with auxiliary communication. We do this for one of the inequalities, and all other inequalities follow a similar argument.

Since|X+Y| ≤ |X|+|Y|and|P| ≤1 for each productPofAi,Bi, and I, it follows that |Tk| is less than or equal to the sum of the absolute value of the coefficients in the polynomial expansion of Tk.

Conclusion

Naively, Alice can just tell Bob the direction of her measurement ˆa (or vice versa), but this requires an infinite amount of communication. The question of whether a simulation can be done with a limited amount of communication was raised independently by Maudlin [53], Brassard, Cleve, Tapp [52] and Steiner [55]. Steiner's model is weaker because the amount of communication can be unbounded in the worst case, although such cases occur with zero probability.).

In light of the previous chapter, where we could prove no non-trivial lower bound, the only lower bound on the amount of communication is given by Bell's theorem: At least some communication is needed.

Simulation protocol

Their approaches differ in how the communication cost of the simulation is determined: Brassard et al. take the cost to be the number of bits sent in the worst case, Steiner, the average. Then, if ˆλ1·λˆ2 = cose, Alice sends −1 with probability η/π and 1 with probability 1−η/π, so that the communication can be compressed to π/2. The common unit vectors ˆλ1 and ˆλ2 described in the text divide the Bloch sphere into four quadrants, as shown.

The communication: Alice sends c= +1 if her measuring axis is in the N or S quadrant, and. c) Bob's output: this depends on the bit received from Alice.

Figure 5.1: The protocol: The shared unit vectors ˆ λ 1 and ˆ λ 2 described in the text divide the Bloch sphere into four quadrants, as shown

Application to teleportation experiments

If we allow Bob to measure a qubit using elements of a positive operator-valued measure, then there may not be a local description of the hidden variables that respects the two-bit classical communication limit. Finally, using the classical teleportation protocol, we obtain a (not necessarily optimal) protocol for simulating joint projective measurements on the partially complex states of two qubits, which uses two bits of communication: Alice first simulates her measurement and determines the state after measuring Bob's qubit; Alice and Bob then perform the classic teleportation protocol.

Conclusion

In this chapter, we show how to precisely simulate the joint quantum correlation of two-outcome measurements using five communication bits. Suppose that Alice and Bob share a state Cd⊗Cd, in which they perform local measurements with two results. Let ˆa ∈ Rn be the vector associated with Alice's measurement in accordance with Lemma 2.3.1 and let ˆb∈Rn be the vector associated with Bob's measurement.

A communication primitive

Simulation of the joint correlation using ﬁve bits of communication

Then each term in the sum is non-negative, from which it follows thatdk ≥0 for allk. The trick is choosing an appropriate functionC. The range of C is a direct sum of tensor products of Rn.

Discussion

Two parties, A and B, share a quantum stateρ, and each chooses one of two observables to measure their part of the state. The results of local measurements of an entangled quantum state are therefore described by a probability distribution without signaling. Maximization over probability distributions without signaling can be represented as a linear program, and so we obtain an upper bound by solving this linear program.

For the CHSH inequality, there is a non-signaling probability distribution such that BCHSH NS = 4, so this technique is not useful [17]).

Framework

The CHSH game

Since local measurements on spatially separated components of a multipart quantum system can be performed simultaneously, such measurements cannot be used to send a signal from one party to another. We use this technique to (i) bound the trade-off between (A and B) and (A and C)'s violation of the CHSH inequality, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating the odd cycle Clock inequality of Ref.

Main technique

Applications

An analogue of the CKW theorem for nonlocal quantum correlations

Suppose that three parties, A, B, and C, share no-signal correlations, each choosing to measure one of two observables. In particular, CHSH correlations are monogamous: if AB violates the CHSH inequality, then AC cannot, as has been shown independently for no-signal correlations by Masanes, Ac´ın and Gisin [76]. In fact, Theorem 7.4.1 can be easily obtained from Result 3 in Ref. [76] by symmetrization of B and C.).

For no-signal probability distributions, we also have an inverse of Theorem 7.4.1: For any pair (BABCHSH). 7.2), there is a no-signal probability distribution with these expectation values.

Classical correlation restricts Bell inequality violation

Then, the maximum quantum value of the Bell inequality is obtained by a state that has support on one qubit at each site. Roger, “Experimental realization of the Einstein-Podolsky-Rosen-Bohm thought experiment: A new violation of Bell's inequalities,” Phys. Unger, “Bound for non-locality in any world where communication complexity is not trivial,” Phys.

Barrett, “Nonessential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality,” Phys. Wootters, “Teleportation of an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Popescu, “Experimental realization of teleportation of an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys.