Cost of classically simulating quantum sequential measurements on ...

Cost of classically simulating quantum sequential measurements on entangled systems Armin Tavakoli1, 2, ∗ and Adán Cabello3, † 1

Groupe de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland 2 Department of Physics, Stockholm University, S-10691 Stockholm, Sweden 3 Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain

arXiv:1705.07456v2 [quant-ph] 1 Jun 2017

We introduce an ideal experiment with sequential nonprojective measurements on a single entangled system which induce unlimited quantum state transitions on its spacelike separated companion. We show that its classical simulation requires: (i) unlimited rounds of superluminal communication, since the postmeasurement quantum state always violates a Bell inequality, and (ii) unlimited memory, since the sequential measurements would otherwise cause emission of heat in the spacelike separated companion. Our results show that classical simulations of sequential measurements on entangled quantum systems have a significantly higher cost than simulations of standard Bell experiments.

Introduction.—Bell’s theorem [1] states that no local realistic model can simulate the statistical predictions of quantum theory for spacelike separated systems. Therefore, any classical simulation of quantum correlations, obtained from spacelike separated local measurements on an entangled state, needs to be supplemented with superluminal communication. Quantifying the required amount of superluminal communication for successfully simulating quantum theory has received significant attention [2–6]. The amount of such communication shows how quantum protocols outperform their classical counterparts. Examples include the most famous entanglement-based protocols for secure communication [7], reduction of communication complexity [8], and secure private randomness [9]. In a similar spirit, quantum predictions for compatible sequential measurements on a single quantum system do not admit a noncontextual realistic model [10–12]. Again, a classical simluation of the quantum predictions needs to be supplemented with another resource: memory [13]. Memory is the resource of interest also for other sequential scenarios. Both for a classical simulation of many unitaries on a single qubit [14], and for many sequential projective measurements on a single qubit [15], a diverging amount of memory is required to supplement the simulation. What happens if we combine sequential measurements on a single system and entanglement? Such scenarios have been used to show that: (I) quantum nonlocality can be activated [16], (II) quantum nonlocality can arise from sequential measurements violating a local noncontextuality inequality [17], (III) there exists monogamy between certain forms of nonlocality and contextuality [18], (IV) the nonlocality of one part of an entangled system can be shared among multiple observers acting sequentially [19], and (V) unbounded randomness can be certified [20]. Moreover, these works have stimulated experiments with sequential measurements on entangled systems [21–24]. In this Letter, we consider the following question: Which resources are needed to simulate quantum theory’s predictions for sequential measurements on a single entangled system? We present an ideal experiment in which two parties, Alice and Bob, share an entangled state on which Alice performs many sequential local measurements. We show that after any

number of Alice’s measurements, the state she shares with Bob violates a Bell inequality. Thus, a classical simulation always requires more superluminal communication after each of Alice’s measurements. Furthermore, since Alice’s diverging number of measurements induces a diverging number of unique local states of Bob, the simulation also requires infinite memory. Failure to always supply more superluminal communication causes the simulation to fall through on reproducing a Bell inequality violation after each of Alice’s measurements. Failure to supply infinite memory to Bob’s system causes his local memory to be overloaded, necessitating a local erasure of information which, due to Landauer’s principle, casues an emission of heat, which is a form of signaling. Our results show that sequential measurements on entangled quantum systems have a significantly higher cost than in standard Bell experiments. Scenario.—Our scenario is similar to that considered in Refs. [19, 20]. We consider two parties, Alice and Bob, who √ initially share two maximally entangled qubits, |ψ0 i = 1/ 2 (|00i + |11i). At certain times t1 < t2 < . . . < tN , Alice randomly chooses between two measurements, xk and x ¯k , and performs this measurement on her qubit. Each measurement has two possible outcomes denoted 0 and 1. Alice’s measurement at time tk , denoted by jk ∈ {xk , x ¯k }, is a two-outcome positive operator-valued measure (POVM) which has, associated to outcome 0, the POVM element Ekjk = Knˆ jk (µk )Knˆ jk (µk )† , where Knˆ jk (µk ) is the Kraus operator [25] Knˆ jk (µk ) = cos (µk ) |ˆ njk ihˆ njk |+sin (µk ) |−ˆ njk ih−ˆ njk |, (1) where n ˆ jk is a vector on the Bloch sphere that will be specified later. This POVM is a noisy version of the measurement represented by a Pauli matrix along n ˆ jk . The amount of noise is controlled by the value of µk ∈ [0, π/2] and will be specified later. If µk ∈ {0, π/2}, the measurement is projective. If µk = π/4, then Knˆ jk = 11/2, implying a non-interactive measurement. Other values of µk correspond to weak measurements. The time evolution is trivial, that is, the state of Alice’s and Bob’s qubits just after Alice’s measurement at tk is the

2

(a)

(b)

FIG. 1. (a) From left to right: Alice’s sequential measurements at times t1 < t2 < t3 , respectively. At each tk , Alice performs a measurement; either xk or x ¯k . Each measurement has two possible outcomes: 0 and 1. Alice’s measurements are such that the state of the two qubits after her measurement is always entangled. (b) From left to right: Possible reduced states of Bob’s qubit after Alice’s measurements at t1 < t2 < t3 , respectively. States are represented by nonunit arrows in the equatorial plane of the Bloch sphere. For example, ¯ 1 denotes the state when Alice measured x ¯1 at t1 and obtained outcome 1, ¯ 10 denotes the state when Alice measured x ¯1 at t1 and x2 at t2 and obtained outcomes 1 and 0, respectively. Bob’s states highlighted in purple are those produced in the particular sequence of Alice’s measurements and outcomes shown in (a).

state just before Alice’s measurement at tk+1 , and is determined by Alice’s sequence of measurements and outcomes at {t1 , . . . , tk }. The list of measurements and outcomes of Alice from t1 to tk is denoted lk . Protocol.—Here we describe a protocol in the scenario presented above. One of its features is that, at each tk , for each pair of measurements of Alice, there exist two measurements that Bob could perform such that the outcome statistics of Alice and Bob violates the Clauser-Horne-ShimonyHolt (CHSH) inequality [26]. In a CHSH experiment, Alice and Bob perform measurements Ai and Bj , respectively, for i, j ∈ {0, 1} on shared pairs of systems. The measurement on each system is chosen independently and randomly. Any local realistic model of the outcome statistics must satisfy the CHSH inequality SCHSH ≡ hA0 B0 i + hA0 B1 i + hA1 B0 i − hA1 B1 i ≤ 2, (2) where h·i denotes expectation value. The following Lemma (which is a corollary of the main result of Ref. [27]) is key to understand the protocol. Lemma. Consider any pure entangled state |Ψη i = cos (η) |00i + sin (η) |11i, with η ∈ (0, π/2). For every |Ψη i, Alice can find measurements associated to Kraus operators (1) with n ˆ jk equal to (0, 0, 1) and (1, 0, 0), respectively (i.e., noisy measurements of σz and σx ), for which she can choose a noise parameter µ ∈ / {0, π/2} such that there exists two projective measurements for Bob leading to outcome statistics violating the CHSH inequality (2). Proof. The Bloch vectors associated to the measurements A0 and A1 of Alice are [0, 0, cos (2µ)] and [cos (2µ) , 0, 0] respectively. These are unnormalized for µ ∈ / {0, π/2}.

Let us choose the Bloch vectors representing Bob’s measurements B0 and B1 to be of the form [cos (θ) , 0, sin (θ)] and [− cos (θ) , 0, sin (θ)], respectively, for some θ. These Bloch vectors are normalized and hence correspond to projective measurements. A direct computation of SCHSH in (2) gives SCHSH = 2 cos (2µ) [sin (θ) + sin (2η) cos (θ)] .

(3)

We choose θ so that SCHSH is maximal, i.e., we solve the equation ∂SCHSH /∂θ = 0. The solution of our interest is θ = arctan [1/ sin (2η)], which is independent of µ. Inserting this in Eq. (3) we find p SCHSH = 6 − 2 cos (4η) cos (2µ) . (4) The minimal value of the square root is 2 and is achieved for product states. Therefore, for every entangled state corresponding to η ∈ / {0, π/2}, the square root is larger than 2. Hence, if Alice chooses her noise parameter µ such that " # 1 2 0 < µ < arccos p ≡ F (η), (5) 2 6 − 2 cos (4η) then the outcome statistics of Alice and Bob will violate the CHSH inequality (2). Let us now describe the protocol. Just before t1 , Alice and Bob share the maximally entangled state |ψ0 i. Alice chooses some nonzero µ1 < F (π/4) = π/8. She then randomly chooses j1 ∈ {x1 , x ¯1 } associated to Bloch vectors (0, 0, 1) and (1, 0, 0), respectively, and performs the measurement {E1j1 , 11 − E1j1 }, knowing that, if Bob were to randomly choose between two suitable measurements, their outcome statistics would violate the CHSH inequality.

3 From her observed outcome, Alice calculates the postmeasurement state |ψ1l1 i of the two qubits. This state is necessarily pure and entangled, and can thus be written in the form |ψ1l1 i = UAl1 ⊗ UBl1 cos θl1 |00i + sin θl1 |11i , (6) where θl1 ∈ / {0, π/2}, and UAl1 and UBl1 are unitary operators. † Then, Alice applies on her qubit the unitary UAl1 , which cancels the unitary UAl1 in (6). After Alice’s actions at t1 , the reduced state of Bob’s qubit is one of four possible states (see Fig. 1). At t2 , Alice again chooses some positive µ2 < F (θl1 ). She makes a random choice of measurement j2 ∈ {x2 , x ¯2 } associated to Bloch vectors (0, 0, 1) and (1, 0, 0), respectively, and performs the measurement {E2j2 , 11 − E2j2 }. Again, the Lemma assures that, if Bob were to measure in suitable bases, their outcome statistics would violate the CHSH inequality. Alice observes the outcome of her measurement and uses it to calculate the new postmeasurement state |ψ2l2 i of the two qubits. Just as in the previous step, Alice rotates her reduced state back to the computational basis by applying a suitable unitary. After Alice’s actions at t2 , the reduced state of Bob’s qubit is one of 16 possible states (see Fig. 1). Alice continues this process indefinitely: Just after tk−1 , lk−1 Alice uses her list lk−1 to calculate the global state |ψk−1 i. lk−1 At tk , she chooses some positive µk < F θ , randomly choses a measurement jk ∈ {xk , x ¯k } associated to Bloch vectors (0, 0, 1) and (1, 0, 0), respectively, and performs the measurement {Ekjk , 11 − Ekjk }. The Lemma guarantees that Bob can perform some measurements so that the CHSH inequality is violated. Alice observes her measurement outcome and uses it to calculate the postmeasurement state |ψklk i which takes the form |ψklk i = UAlk ⊗ UBlk cos θlk |00i + sin θlk |11i , (7) for some angle θlk ∈ / {0, π/2}. Subsequently, she undoes the † rotation of her local state by applying UAlk . This renders the reduced state of Bob’s qubit in one of 4k possible states (see Fig. 1). At each time tk , Alice’s alternative measurements are both nonprojective, depend on Alice’s previous choices of measurements, and also on the outcomes of the previous measurements. This way, the initial entanglement is never consumed regardless of Alice’s performed measurements and observed outcomes, and Alice’s two measurement options always violate the CHSH inequality. To illustrate the properties, in Table I we display data from the first few steps of one possible execution of the protocol. There we can see that at each time step, the measurement of Alice becomes stronger without ever becoming projective. Furthermore, the entanglement, quantified by the negativity [28], remains nonzero. From t2 and onwards, not all of the 4k possible states just after tk contain the same amount of entanglement, and therefore we must consider the weakest possible

Time

µ

t0 t1 π/9 t2 π/12 t3 π/40 t4 π/500

Number of possible states of Bob’s qubit

Smallest negativity

1 4 16 64 256

0.5 0.3214 0.0966 0.0077 0.00005

Largest Largest Smallest negativity value of SCHSH value of SCHSH

0.5 0.3214 2.1667 0.4774 2.0590 0.4887 2.0119 0.4902 2.00008

2.1667 2.0590 2.7313 2.7965

TABLE I. Data from the first four time steps in one possible execution of the quantum protocol: choices of the noise parameter for Alice’s measurement at tk , the number of different local states of Bob just after tk , the smallest and largest negativity of the 4k possible global states just after tk , and the smallest and largest values of SCHSH achieved with the 4k−1 possible states. The choices of µ carry no special significance other than that they satisfy the relation 0 < µk < F (θlk−1 ) for all k.

entanglement. As displayed in Table I, the negativity of the weakest entangled state quickly decreases. However, some entanglement is always present. When choosing her noise parameter µk , Alice ensures that even the weakest entangled state violates the CHSH inequality (2). That this is indeed the case can be seen from the corresponding smallest values of SCHSH in Table I. In contrast, from the largest possible negativity we see that the protocol sometimes, albeit with small probability, acts as a probabilistic entanglement amplification scheme [29, 30]. The classical simulation requires unlimited rounds of superluminal communication.—Since the postmeasurement state just after tk and the two measurement options Alice has at time tk+1 always violate the CHSH inequality, any local realist model aiming to simulate the quantum predictions after tk has to be supplemented with some superluminal communication. Following previous literature [2–6], we assume that, in these models, after the measurement performed by Alice, there is a round of communication between Alice and Bob. In this round, depending on her measurement and the resulting local realistic state of her system, Alice communicates some information to Bob. The critical observation is that the communication required to simulate the quantum predictions just after tk will not be enough for reproducing the quantum predictions after tk+1 . The reason is that, at tk+1 , the new quantum predictions will again violate the CHSH inequality. Thus, regardless of Alice’s measurement choice and observed outcome, any simulation of the predictions of quantum theory for the experiment based on a local realistic model complemented with superluminal communication requires unlimited rounds of superluminal communication. Although the classical simulation requires unlimited rounds of superluminal communication, the total amount of communication is finite. To show this, we use that the average amount of communication C required to simulate a nonsignaling probability distribution achieving the value SCHSH is given by C = SCHSH /2 − 1 [5]. Let us denote the average communication over all possible postmeasurement states at time tk by Ck . The total amount of communication is finite if

4 P∞ ¯ C¯ ≡ k=1 Ck is finite. To show that C is finite, we consider the states |ψη i which are unitarily equivalent to the states shared by Alice and Bob. Applying the Horodecki criterion [31] to |ψη i, one finds the maximal value of SCHSH at time tk . It is a straightforward calculation to show that this quantity is an upper bound on the sum of the CHSH value (4) of |ψη i, as obtained when applying a noisy measurement in our protocol at time tk , and the average maximal CHSH value, obtained from applying the Horodecki criterion to the four possible postmeasurement states at tk+1 weighted by the respective probability of obtaining each state. This argument can√be repeated throughout the protocol, and consequently C¯ < 2−1, which is the communication cost of simluating a maximal violation of the CHSH inequality achieved with |ψ0 i. The classical simulation requires unbounded memory.—At each time step in the protocol, Alice choses with uniform probability between two measurement options. After each measurement of Alice, the number of possible reduced states of Bob’s qubit quadruples. Any classical simulation must account for this exponentially increasing number of possible states. Since each of Alice’s measurement choices is random, any classical simulation requires to have at least the same number of local realist states as the number of pure quantum states achieved during the experiment. This follows from an argument previously used in [15] which can be adapted to our protocol as follows. A stochastic process is a one-dimensional chain of discrete random variables that attains values in a finite or countably infinite alphabet. An input-output process [32] is a collection of stochastic processes in which each such process corresponds to all possible output sequences given a particular infinite input sequence. The experiment is an example of an inputoutput process. It has input alphabet {xk , x ¯k } and output alphabet {0, 1}. As shown in Ref. [32], for any input-output process there is a unique finite-state machine, i.e., an abstract machine that can be in exactly one of a finite number of states at any given time, with the following property: it has minimal entropy over the state probability distribution and maximal mutual information with the future output of the process given the past choices of inputs and past observed outputs, and the future input of the process. This machine is called the εtransducer [32] of the input-output process. It consists of the input and output alphabets, a set of causal states, and the set of conditional transition probabilities between the causal states. Each causal state is associated to the set of input-output pasts producing the same probabilities for all possible input-output futures. Thus, the causal states constitute equivalence classes for the set of input-output pasts. A causal state stores all the information about the past needed to predict the future output, but as little as possible of the remaining information overhead contained in the past. The Shannon entropy over the stationary distribution of the causal states represents the minimum internal entropy needed to be stored to optimally compute future outputs. It depends on how Alice’s measurements are chosen; here we have assumed that they are selected from a uniform probability distribution with entropy one bit at each time step.

The number of causal states of the ε-transducer corresponding to our experiment is infinite. This implies that the classical system that simulates the experiment has to store new information in its memory. This leads to two possibilities: either the memory is infinite and additional information can always be stored without need of erasing previous information, or the memory is finite and the system has to erase a part of it to allocate new information. However, due to Landauer’s principle [33], the erasure of information has a thermodynamical cost. Landauer’s principle states that the erasure of information in an information-carrying degree of freedom is acompanied by an associated increase of entropy in some non-information carrying degree of freedom. There is strong evidence supporting the validity of Landauer’s principle in both the classical and quantum domains [34–40]. Since we are assuming a local realistic model supplemented by communication, the memory should be allocated in the local systems. Since Bob’s quantum state and its classical counterpart (represented by a causal state of the ε-transducer) are changing after each of Alice’s measurements, this implies that there should be some information erasure in the local memory associated to Bob’s system. Therefore, after sufficiently many measurements of Alice, Bob’s system begins to emit heat. Such heating at a distance is a form of signaling. Conclusions.— Since Brassard, Cleve, and Tapp pointed out the interest of the question “to what extent does the behavior of an entangled quantum system provide savings compared with classical systems?” [2], the amount of superluminal communication required for simulating quantum correlations in Bell experiments has been a subject of much attention. Here we have shown that no amount of superluminal communication is enough to simulate the predictions of quantum theory in the case of experiments combining entanglement and sequential nonprojective measurements. In addition to infinite rounds of superluminal communication, classical systems also need to have access to infinite memory. This shows that the cost of simulating these new experiments is much higher than the cost of simulating the Bell experiments at the core of current quantum information processing protocols. This indicates that new experiments combining entanglement and sequential nonprojective measurements will offer more possibilities and motivates further efforts to identify applications demanding these quantum resources. Acknowledgements.—We thank Nicolas Brunner, Nicolas Gisin, Mile Gu, and Matthias Kleinmann for their comments on the manuscript, Gustavo Cañas for his help with Fig. 1, Jim Crutchfield for discussions, and Matthias Kleinmann for checking all the calculations. This work was supported by the project “Photonic Quantum Information” (Knut and Alice Wallenberg Foundation, Sweden), Project No. FIS2014-60843-P, “Advanced Quantum Information” (MINECO, Spain), with FEDER funds, and the FQXi Large Grant “The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory.” A.T. acknowledges financial support from the Swiss National Science Foundation (Starting Grant DIAQ).

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