The Forbidden Quantum Adder

Abstract

Quantum information provides fundamentally different computational resources than classical information. We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces. This is an inherent restriction of quantum physics that is related to the impossibility of copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we demonstrate that a quantum adder, in absence of an ancillary system, is also forbidden for a known orthonormal basis. This allows us to propose an approximate quantum adder that could be implemented in the lab. Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

Introduction

Addition plays a central role in mathematics and physics, while adders are ubiquitous devices in the fields of computation¹ and electronics². In this sense, usual sum operations can be realized by classical Turing machines³ and also, with a suitable algorithm, by quantum Turing machines^4,5. Furthermore, the sum of known state vectors in the same Hilbert space, i.e. quantum superposition, is at the core of quantum physics. In fact, entanglement and the promised exponential speed-up of quantum computing are based on such linear combinations. Here, we consider the existence of a quantum adder, defined as a unitary operation mapping two unknown quantum states encoded in different quantum systems onto their sum codified in a single system. The surprising answer is that this quantum adder is forbidden and it has the quantum cloner as a special case⁶. This no-go result, as other prohibited operations^6,7,8,9, is of fundamental nature and its implications should be further studied. Furthermore, we consider a high-fidelity approximate quantum adder involving ancillary systems. Recently, we have known about a parallel work analyzing a similar problem, in which an optimal approximate quantum adder was found¹⁰.

Let be two quantum states of a finite-dimensional Hilbert space. The conjectured quantum adder, sketched in Fig. 1, would be a mathematical operation defined as the unitary , for every pair of unknown and ancillary vector ,

where the ancillary state may depend on the input states. There are several ways of proving the unphysicality of Eq. (1). The simplest one is to note that the unobservable global phase on its l.h.s. could be distributed in infinite forms on its r.h.s., , with ϕ = ϕ₁ + ϕ₂, yielding an observable relative phase. When the ancillary state does not depend on the input quantum states, the (forbidden) quantum cloner becomes a particular case of this restricted quantum adder. This follows from applying U to two equal state vectors , since the inverse would generate a quantum cloning operation. Therefore, although the general case of the quantum adder is not equivalent to a quantum cloner, it is still forbidden.

Figure 1

Scheme of the conjectured quantum adder.

The inputs are two unknown quantum states, and , while the outputs are proportional to the sum, with an ancillary state .

We consider now a different question, whether a quantum adder may exist for a given orthonormal basis. In this case, as we will see, the global phase does not produce any ambiguity in the equations. Let us consider the action of the unitary operator onto a set of orthonormal vectors: and , with . Hence, as U is a unitary matrix, it imposes some orthogonality conditions on the final vectors, and   , with . The second subspace has dimension d, but these constraints require the existence of at least 2d − 1 orthonormal vectors, which is impossible.

We propose now the use of an ancillary system , which will assure the physicality and experimental feasibility of an approximate quantum adder for arbitrary unknown quantum states, see Fig. 2. This particular adder, U_a, computes the exact sum of the basis elements in qubit systems. Moreover, U_a is extended by linearity to the whole Hilbert space and implements an approximate sum when the input states are superpositions of the basis elements. The adder is given by the following expression in which are orthonormal and .

Figure 2

Fidelity of the proposed approximate quantum adder.

The fidelity is depicted as a function of the parameters of the input states , wherej = 1,2. Here, a) ϕ₁ = ϕ₂ = 0, b) ϕ₁ = ϕ₂ = π/4, c) ϕ₁ = ϕ₂ = π/2 and d) θ₁ = θ₂ = π/4. Note that the diagonal line of each plot corresponds to the approximate quantum cloner that is related to our restricted quantum adder. In this case, the fidelities are the lowest.

Although this approximate quantum adder shows high fidelities, it is not optimal as the one recently found in Ref. 10.

Beyond the sum of quantum states in Eq. (1), we may also consider the statistical addition of density matrices. Here, the input states are the tensor product of any pair of density matrices , while the output state is the statistical sum . The Kraus operators of the quantum channel realizing this adder are given by and , with 1 ≤ i, j ≤ d. These operators straightforwardly perform the sum, i.e., . Moreover, properly modified Kraus operators allow us to extend the previous result to any convex combination of input states. Therefore, the considered addition of density operators is always possible.

Let us compare the adders for state vectors and density operators. By writing the input states in Eq. (1) as density matrices, , both adders yield

By comparing the adders in Eq. (3), we can infer that the one in Eq. (1) would require the knowledge of the sum coherences, which were supposed unknown. The nonexistence of a quantum adder is of fundamental character in quantum physics, comparable and deeply related to the no-cloning theorem.