Computational Complexity Measures of Multipartite Quantum Entanglement

We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions— approximability and distinguishability. Built upon the formal treatment of partial separability, we measure the complexity of an entangled quantum state by determining (i) how hard to approximate it from a fixed classical state and (ii) how hard to distinguish it from all partially separable states. We further consider the Kolmogorovian-style descriptive complexity of approximation and distinction of partial entanglement. 1 Computational Aspects of Quantum Entanglement Entanglement is one of the most puzzling notions in the theory of quantum information and computation. A typical example of an entangled quantum state is the Bell state (or the EPR pair) (|00〉+ |11〉)/ √ 2, which played a major role in, e.g., superdense coding [4] and quantum teleportation schemes [1]. Entanglement can be viewed as a physical resource and therefore can be quantified. Today, bipartite pure state entanglement is well-understood with informationtheoretical notions of entanglement measures (see the survey [8]). These measures, nevertheless, do not address computational aspects of the complexity of entangled quantum states. For example, although the Bell state is maximally entangled, it is computationally constructed from the simple classical state |00〉 by an application of the Hadamard and the Controlled-NOT operators. Thus, if the third party gives us a quantum state which is either the Bell state or any separable state, then one can easily tell with reasonable confidence whether the given state is truly the Bell state by reversing the computation since the minimal trace distance between the Bell state and separable states is at least 1/2. This simple fact makes the aforementioned information-theoretical measures unsatisfactory from a computational point of view. We thus need different types of measures to quantify multipartite quantum entanglement. We first need to lay down a mathematical framework for multipartite quantum entanglement and develop a useful terminology to describe a nested structure of entangled quantum states. In this paper, we mainly focus on pure quantum states in the Hilbert space C n of dimension 2. Such a state is called, analogous to a classical string, a quantum string (or qustring, for short) of length n. Any qustring of length n is expressed in terms of the standard basis {|s〉}s∈{0,1}n . Given a qustring |φ〉, let l(|φ〉) denote its length. By Φn we denote the collection of all qustrings of length n and set Φ∞ to be ⋃ n∈N+ Φn, where N + = N − {0}. Ensembles (or series) of qustrings of (possibly) different lengths are of particular interest. We use families of quantum circuits [6,19] as a mathematical model of quantum-mechanical computation. A quantum circuit has input qubits and (possibly) ancilla qubits, where all ancilla qubits are always set to |0〉 at the beginning of computation. We fix a finite universal set of quantum gates, including the identity and the NOT gate. As a special terminology, we say that a property P(n) holds for almost all (or any sufficiently large) n in N if the set {x ∈ N | P(x) does not hold } is finite. All logarithms are conventionally taken to base two. 2 Separability Index and Separability Distance We begin with a technical tool to identify the entanglement structure of an arbitrary quantum state residing in a multipartite quantum system. In a bipartite quantum system, any separable state can be expressed as a tensor product |φ〉⊗|ψ〉 of two qubits |φ〉 and |ψ〉 and thus, any other state has its two qubits entangled with a physical correlation or “bonding.” In a multipartite quantum system, however, all “separable” states may not have such a simple tensor-product form. Rather, various correlations of entangled qubits may be nested—or intertwined over different groups of entangled qubits. For example, consider the qustring |ψ2n〉 = 2−n/2 ∑ x∈{0,1}n |xx〉 of length 2n. For each i ∈ {1, 2, . . . , n}, the ith qubit and the n + ith qubit in |ψ2n〉 are entangled. The reordering of each qubit, nevertheless, unwinds its nested correlations and sorts all the qubits in the blockwise tensor product form |ψ′ 2n〉 = ( 1 2 (|00〉 + |11〉)) ⊗n. Although |ψ2n〉 and |ψ′ 2n〉 are different inputs for a quantum circuit, such a reordering is done at the cost of additional O(n) quantum gates. Thus, the number of those blocks represents the “degree” of the separability of the given qustring. Our first step is to introduce the appropriate terminology that can describe this “nested” bonding structure of a qustring. We introduce the structural notion, separability index, which indicates the maximal number of entangled “blocks” that build up a target qustring of a multipartite quantum system. See [14] also for multipartite separability. Definition 1. 1. For any two qustrings |φ〉 and |ψ〉 of length n, we say that |φ〉 is isotopic to |ψ〉 via a permutation σ on {1, 2, . . . , n} if σ(|φ〉) = |ψ〉. 2. A qustring |φ〉 of length n is called k-separable if |φ〉 is isotopic to |φ1〉 ⊗ |φ2〉 ⊗ · · · ⊗ |φk〉 via a certain permutation σ on {1, 2, . . . , n} for a certain ktuple (|φ1〉, |φ2〉, . . . , |φk〉) of qustrings of length ≥ 1. This permutation σ is said to achieve the k-separability of |φ〉 and the isotopic state |φ1〉 ⊗ |φ2〉 ⊗ · · · ⊗ |φk〉 is said to have a k-unnested form. The series m = (l(|φ1〉), l(|φ2〉), . . . , l(|φk〉)) is called a k-sectioning of |φ〉 by σ. 3. The separability index of |φ〉, denoted sind(|φ〉), is the maximal integer k with 1 ≤ k ≤ n such that |φ〉 is k-separable. 1 Let σ be any permutation on {1, 2, . . . , n} and let |φ〉 be any qustring of length n. The notation σ(|φ〉) denotes the qustring that results from permuting its qubits by σ; that is, σ(|φ〉) = ∑ x αx|xσ(1)xσ(2) · · ·xσ(n)〉 if |φ〉 = ∑ x αx |x1x2 · · ·xn〉, where x = x1x2 · · · xn runs over all binary strings of length n. For any indices n, k ∈ N with k ≤ n, let QSn,k denote the set of all qustrings of length n that have separability index k. For clarity, we re-define the terms “entanglement” and “separability” using the separability indices. These terms are different from the conventional ones. Definition 2. For any qustring |φ〉 of length n, |φ〉 is fully entangled if its separability index equals 1 and |φ〉 is fully separable if it has separability index n. For technicality, we call |φ〉 partially entangled if it is of separability index ≤ n− 1. Similarly, a partially separable qustring is a qustring with separability index ≥ 2. We assume the existence of a quantum source of information; namely, a certain physical process that produces a stream of quantum systems (i.e., qustrings) of (possibly) different lengths. Such a quantum source generates an ensemble (or a series) of qustrings. Of such ensembles, we are particularly interested in the ensembles of partially entangled qustrings. For convenience, we call them entanglement ensembles. Definition 3. Let l be any strictly increasing function from N to N. A series Ξ = {|ξn〉}n∈N is called an entanglement ensemble with size factor l if, for every index n ∈ N, |ξn〉 is a partially entangled qustring of length l(n). How close is a fully entangled state to its nearest partially separable state? Consider the fully entangled qustring |φn〉 = (|0n〉 + |1n〉)/ √ 2 for any n ∈ N. For comparison, let |ψ〉 be any partially separable qustring of length n. By a simple calculation, the L2-norm distance ‖|φn〉 − |ψ〉‖ is shown to be at least