(4,1)-Quantum random access coding does not exist—one qubit is not enough to recover one of four bits

An (n,1,p)-quantum random access (QRA) coding, introduced by Ambainis et al (1999 ACM Symp. Theory of Computing p 376), is the following communication system: the sender which has n-bit information encodes his/her information into one qubit, which is sent to the receiver. The receiver can recover any one bit of the original n bits correctly with probability at least p, through a certain decoding process based on positive operator-valued measures. Actually, Ambainis et al shows the existence of a (2,1,0.85)-QRA coding and also proves the impossibility of its classical counterpart. Chuang immediately extends it to a (3,1,0.79)-QRA coding and whether or not a (4,1,p)-QRA coding such that p > 1/2 exists has been open since then. This paper gives a negative answer to this open question. Moreover, we generalize its negative answer for one-qubit encoding to the case of multiple-qubit encoding


Introduction
The state of n quantum bits (qubits) is represented by a 2 n -dimensional complex-valued unit vector and seems to hold much more information than (classical) n bits. However, due to the famous Holevo bound [1], this is not true information theoretically, i.e., we need n qubits to transmit n-bit information faithfully. As an interesting challenge to this most basic fact in quantum information theory, Ambainis et al introduced the notion of quantum random access (QRA) coding [2]. (The paper [3] includes the contents of [2] and their improvement in [4].) They explored the possibility of using much fewer qubits if the receiver has to recover only partial bits, say one bit out of the n original ones, which are not known by the sender in advance.
As a concrete example, they give (2, 1, 0.85)-QRA coding; the sender having two-bit information sends one qubit and the receiver can recover any one of the two bits with probability at least 0.85. It is also proved that this is not possible classically, i.e., if the sender can transmit one classical bit, then the success probability is at most 1/2. This (2, 1, 0.85)-QRA coding is immediately extended to (3, 1, 0.79)-QRA coding by Chuang (as mentioned in [2]) and it has been open whether we can make a further extension (i.e., whether there is an (n, 1, p)-QRA coding such that n 4 and p > 1/2) since then. This paper gives a negative answer to this open question, namely, we prove there is no (4, 1, p)-QRA coding such that p is strictly greater than 1/2. Our proof is a reduction to the well-known geometric fact that a three-dimensional (3D) ball cannot be divided into 16 nonempty regions by four planes. (Interestingly, the proof for the non-existence of a classical counterpart of (2, 1, p)-QRA coding in [2] uses a similar geometric fact, i.e., a straight line cannot stab all insides of the four quarters of a 2D square.) Our result has nice applications to the analysis of quantum network coding which was introduced very recently [5].
In general, the sender is allowed to send m ( 1) qubits; such a system is denoted by (n, m, p)-QRA coding. Our result can be extended to this general case, namely we can show that (2 2m , m, p)-QRA coding with p > 1/2 does not exist. This is quite tight since we can see that a 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT (2 (m) , m, >1/2)-QRA coding exists by reducing a communication complexity upper bound to the QRA coding [6,7]. This paper is organized as follows. In the rest of section 1, we discuss the related work. Section 2 gives the formal definition of (n, 1, p)-QRA coding and the optimal codings for cases n = 2 and 3. Also, we discuss a possible strategy for n 4. In section 3, we show the non-existence of (4, 1, p)-QRA coding with p > 1/2 and also the limit of (n, m, >1/2)-QRA coding for general m, extending the geometric proof argument for n = 4. Section 4 includes applications of our results to quantum network coding. Finally, section 5 concludes the paper.

Related work
For the relation among these three parameters of (n, m, p)-QRA coding, the following bound is known [4]: m (1 − H(p))n, where H is the binary entropy function, and it is also known [2] that (n, m, p)-QRA coding with m = (1 − H(p))n + O(log n) exists (which is actually classical). Thus, this bound is asymptotically tight and has many applications such as proving the limit of quantum finite automata [2,4], analysing quantum communication complexity [6,8,9], designing locally decodable code [10,11], and so on. However, it says almost nothing for small n and m; if we set n = 4 and p > 1/2, for example, the bound implies only m > 0. This bound neither implies the limit of n for a given m if = p − 1/2 is very small, say = 1/g(n) for rapidly increasing g(n). Our second result says that there does exist a limit of n for any small .
König et al [12] extended the concept of QRA coding to the situation that the receiver wants to compute a (randomly selected) function on the bits the sender has, and applied their limit of its extended concept to the security of the privacy amplification, a primitive of quantum key distribution. The study on QRA coding for more than two parties was done by Aaronson [13], who explored the QRA coding in the setting of the Merlin-Arthur games.

Quantum random access coding
First, we review the notion of QRA coding given in [2]. The (n, m, p)-QRA coding is an m-qubit coding of the sender with n bits so that the receiver can recover any one bit of the n bits with probability at least p. We give the formal definition for m = 1 since we deal with examples of (n, 1, p)-QRA coding in this section (and the definition for general m is similarly given).
Recall that a POVM {E i 0 , E i 1 } has to satisfy the following conditions: (i) E i 0 and E i 1 are both nonnegative Hermitian and (ii) E i 0 + E i 1 = I. It is well-known, since E i 0 and E i 1 are of rank at most 2, that E i 0 can be written as E i 0 = α 1 |u 1 u 1 | + α 2 |u 2 u 2 | for some orthonormal basis {|u 1 , |u 2 }, 0 α 1 1 and 0 α 2 1. Hence, by (ii), For decoding, we use the measurements by the following POVMs (projective measurements, in fact): figure 1. To decode the second bit, for example, we measure the encoding state in the basis {|+ , |− }. The angle between |ϕ(00) and |+ (and also between |ϕ(10) and |+ ) is π/8 and hence the success probability of decoding the value 0 is cos 2 (π/8) > 0.85.
is the same state as the one used in the (3, 1, 0.79)-QRA coding. For decoding, we first apply the universal cloning [15] to the qubit (|ϕ(0x 1 x 2 ) or |ϕ(1x 3 x 4 ) ) and let ρ ρ ρ 1 and ρ ρ ρ 2 be the first and the second clones, respectively. If we want to get x 1 (x 2 , respectively), we apply the decoding process of (3, 1, 0.79)-QRA coding to recover the first bit of ρ ρ ρ 1 . If the result is 0, then by assuming that the transmitted qubit was |ϕ(0x 1 x 2 ) , we recover the second (third, respectively) bit of ρ ρ ρ 2 again by (3, 1, 0.79)-QRA decoding process. Otherwise, i.e., if the result is 1, then by assuming that the transmitted qubit was |ϕ(1x 3 x 4 ) , we output the random bit (0 or 1 with equal probability). Decoding x 3 or x 4 is similar and omitted.
First of all, one should see that the above protocol completely follows Definition 1: the encoding process maps x 1 x 2 x 3 x 4 to a mixed state. The decoding process is a little complicated, but it is well-known (e.g., [14]) that such a physically realizable procedure can be expressed by 6 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT a single POVM. Suppose that the receiver wants to get x 1 or x 2 . Then note that |ϕ(0x 1 x 2 ) is sent with probability 1/2 and if that is the case, the receiver can get a correct result with probability p 0 which is strictly greater than 1/2. Otherwise, i.e., if |ϕ(1x 3 x 4 ) is sent, it seems that we only get a result independent of x 1 or x 2 . (Say, if the first measurement yields the correct answer 1, then the outcome is completely random.) Thus the total success probability might appear to be more than a half. Why is this argument wrong?

Main results
In this section, we give two results on the impossibility of QRA coding. First, we show the non-existence of (4, 1, >1/2)-QRA coding, which means that the (3, 1, 0.79)-QRA coding is the best we can do for the (n, 1, >1/2)-QRA coding.
First let us return to figure 2 to see how (3, 1, 0.79)-QRA coding works. Recall that the measurements for recovering x 1 , x 2 and x 3 are all projective measurements. Now one should observe that each measurement corresponds to a plane in the Bloch sphere which acts as a 'boundary' for the encoding states. For example, the measurement in the basis {|0 , |1 } corresponds to the xy-plane (states |0 and |1 correspond to +z and −z axes, respectively, on the sphere, which means that the measurement determines whether the encoding state lies above or under the xy-plane). Thus, the three planes corresponding to projective measurements of (3, 1, 0.79)-QRA coding divide the Bloch sphere into eight disjoint regions, each of which includes exactly one encoding state. Now suppose that there is a (4, 1, p)-QRA coding whose decoding process is four projective measurements. Then, by a simple extension of the above argument, each measurement corresponds to a plane and the four planes divide the sphere into, say, m regions. On the other hand, by definition we have 16 encoding states and hence m 16. (Otherwise, some two states fall into the same region, meaning the same outcome for those states, a contradiction.) However, it is well-known that a 3D ball cannot be divided into 16 (or more) regions by four planes. Thus, we are done if the decoding process is restricted to projective measurements. Due to the generality of POVMs, one might expect that the above argument does not hold for the case of POVMs. However, as shown in the hereafter, a similar argument applies. 7

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Denoting the Bloch vectors of ρ ρ ρ w and |u i u i | as r w and u i , respectively, (1) is rewritten as by fact (ii) on the Bloch sphere described before example 2. (Note that the Bloch vector for becomes the following simple linear inequalities for the fixed s i s. Now, let B be the set of all Bloch vectors. Let also D (0) s i and D (1) s i be the subsets of R 3 defined by D (0) must not be empty. These subsets are the 16 non-empty regions of the ball divided by the four planes { r | r · s i = c i }.
Lemma 1 contradicts the following well-known geometric fact, which completes the proof of theorem 1.

Lemma 2. A ball cannot be divided into 16 non-empty regions by four planes.
Our second result is a generalization of theorem 1. By using the notion of Bloch vectors of n-qubit states, we have the following generalization.
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT such that {|u i j } 2 m j=1 is an orthonormal basis. Thus, for all i ∈ {1, . . . , 2 2m }, the following must be satisfied: Denoting the Bloch vectors of ρ ρ ρ w and |u i j u i j | as r w and u i j (which are (2 2m − 1)-dimensional real vectors), respectively, (5) is rewritten as by (4). (Notice that an m-qubit state can be identified with a 2 m -level quantum state.) If we let (6) is simplified as follows: Let B be the set of all Bloch vectors for m-qubit states. Note that B ⊆ R 2 2m −1 . Let also D (0) s i and D (1) must not be empty. These subsets are included into non-empty regions of R 2 2m −1 divided by the 2 2m hyperplanes { r | r · s i = c i }. Now, the following geometric fact (see, e.g., [20]) completes the proof of theorem 2. Lemma 4. R 2 2m −1 cannot be divided into 2 2 2m non-empty regions by 2 2m hyperplanes.

Applications to network coding
Network coding, introduced in [21], is nicely explained by using the so-called Butterfly network as shown in figure 3. The capacity of each directed link is all one and there are two source-sink pairs s 1 to t 1 and s 2 to t 2 . Notice that both paths have to use the single link from s 0 to t 0 and hence the total amount of flow in both paths is bounded by one, say, 1/2 for each. Interestingly, this max-flow min-cut theorem no longer applies for 'digital information flow.' As shown in the figure, we can transmit two bits, x and y, on the two paths simultaneously.
Protocol X3C3C: Input x 1 x 2 x 3 at s 1 , y 1 y 2 y 3 at s 2 ; Output Out 1 at t 1 , Out 2 at t 2 .
Step 4. (Decoding the j-th bit at t 1 and t 2 ) By our result in this paper, we can prove a kind of optimality of the result (iii). Firstly, we cannot extend the above three bits to four bits, that is, there exists no X4C4C. The reason is easy: if we could then we would get a (4, 1, >1/2)-QRA coding for the s 1 -t 1 path by fixing the state at s 2 to say |0 . Secondly, we can prove that the two side links (s 1 to t 2 and s 2 to t 1 ) which are unusable in the conventional multicommodity flow are in fact useful; if we remove them even for the two bits case, then the network can be viewed as a (4, 1, p)-QRA coding system, which cannot achieve p > 1/2.

Concluding remarks
This paper showed the limit of (n, m, >1/2)-QRA coding using the reduction to well-known geometric facts. In particular, our result is completely tight for m = 1. The result for general m is quite good since we can observe that a (2 (m) , m, >1/2)-QRA coding exists by a reduction from a result of 'unbounded-error' communication complexity by Klauck [7] to the QRA coding (such a reduction can be done in a similar way to [6]). However, the case of general m is not completely tight: we have not shown whether (2 2m − 1, m, >1/2)-QRA coding exists. An interesting open question is the possibility of (n, 2, >1/2)-QRA coding. (6, 2, 0.79)-QRA coding is obvious since we can use two (3, 1, 0.79)-QRA codings independently. For n = 7, there is the following simple construction. We can also regard the task of (n, 1, >1/2)-QRA coding as a special case of the following task: the receiver is required to compute the inner product between the two n bits of the sender and the receiver by using one qubit from the sender. (In the QRA coding, the receiver has one of only n candidates 100 · · · 0, 010 · · · 0, and 000 · · · 1.) In case that the sender's input is taken from {0, 1} n , we can show that n = 3 is impossible by the reduction to a simple geometric fact again while n = 2 is possible with success probability ≈0.79 using the four states that sit in the vertices of the tetrahedron inscribed in the Bloch sphere.