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arXiv:0910.0986 · Physical Review A

Tensor rank of the tripartite state |W〉⊗n

Nengkun Yu, Eric Chitambar, Cheng Guo, Runyao Duan

Published Jan 15, 2010• 32 citations
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Abstract

Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its nonadditivity as an entanglement measure has recently been observed. In this Brief Report, we estimate the tensor rank of multiple copies of the tripartite state $|W\ensuremath{\rangle}=\frac{1}{\sqrt{3}}(|100\ensuremath{\rangle}+|010\ensuremath{\rangle}+|001\ensuremath{\rangle})$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the rank of $|W{\ensuremath{\rangle}}^{\ensuremath{\bigotimes}2}$ is $7$, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between $|W{\ensuremath{\rangle}}^{\ensuremath{\bigotimes}n}$ and multiple copies of the state $|\mathrm{GHZ}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|000\ensuremath{\rangle}+|111\ensuremath{\rangle})$.

DOI
10.1103/physreva.81.014301

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Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its nonadditivity as an entanglement measure has recently been observed. In this Brief Report, we estimate the tensor rank of multiple copies of the tripartite state ∣W\ensuremath⟩=13(∣100\ensuremath⟩+∣010\ensuremath⟩+∣001\ensuremath⟩)|W\ensuremath{\rangle}=\frac{1}{\sqrt{3}}(|100\ensuremath{\rangle}+|010\ensuremath{\rangle}+|001\ensuremath{\rangle})∣W\ensuremath⟩=3​1​(∣100\ensuremath⟩+∣010\ensuremath⟩+∣001\ensuremath⟩). Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the rank of ∣W\ensuremath⟩\ensuremath⨂2|W{\ensuremath{\rangle}}^{\ensuremath{\bigotimes}2}∣W\ensuremath⟩\ensuremath⨂2 is 777, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between ∣W\ensuremath⟩\ensuremath⨂n|W{\ensuremath{\rangle}}^{\ensuremath{\bigotimes}n}∣W\ensuremath⟩\ensuremath⨂n and multiple copies of the state ∣GHZ\ensuremath⟩=12(∣000\ensuremath⟩+∣111\ensuremath⟩)|\mathrm{GHZ}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|000\ensuremath{\rangle}+|111\ensuremath{\rangle})∣GHZ\ensuremath⟩=2​1​(∣000\ensuremath⟩+∣111\ensuremath⟩).
Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its nonadditivity as an entanglement measure has recently been observed. In this Brief Report, we estimate the tensor rank of multiple copies of the tripartite state |W=13(|100+|010+|001). Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the rank of |W 2 is 7, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between |Wn and multiple copies of the state |GHZ=12(|000+|111). © 2010 The American Physical Society.
The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state |W3>=1/√3(|100> + |010> + |001>) and its N-partite generalization |W(N)>. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of |W3> have a rank of either 15 or 16, (ii) two copies of |W(N)> have a rank of 3N - 2, and (iii) n copies of |W(N)> have a rank of O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple-copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states.
A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartite systems, this question encodes some of the most challenging open problems in mathematics and computer science. In particular, we show that there is no easy general criterion to determine the feasibility, and in fact, the problem is NP hard. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the maximum rate to convert the Greenberger-Horne-Zeilinger state to a triangular distribution of three EPR states. Our results are based on connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory.
Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of ∣M⟩⊗k\lvert M\rangle^{\otimes k}∣M⟩⊗k consists of Ω(1.4k)Ω(1.4^k)Ω(1.4k) terms. Here ∣M⟩\lvert M\rangle∣M⟩ is the most standard magic state for fermionic linear optics: the 444 qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the δδδ-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of 1−δ21-δ^21−δ2. Our results on exact fermionic Gaussian rank apply directly to any product of kkk fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an nnn-mode bosonic Fock state with mjm_jmj​ bosons in mode jjj is exactly ∏j(1+mj)\prod_{j}(1+m_j)∏j​(1+mj​), answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.
We construct a lower bound of the tensor rank for a new class of tensors, which we call persistent tensors . We present three specific families of persistent tensors, of which the lower bound is tight. We show that there is a chain of degenerations between these three families of minimal-rank persistent tensors that can be used to study the entanglement transformation between them. In addition, we show that these three families of persistent tensors are indeed different generalizations of multiqubit W states within multiqudit systems and are geometrically in the orbit closure of multiqudit GHZ states. Consequently, we show that one can obtain every one of the generalizations of W state from a multiqudit GHZ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with rate one. Finally, we extend the obtained lower bound of the tensor rank to direct sums with persistent summands and to even more general combinations of tensors, which we call block pyramidal tensors . As a result, we show that the tensor rank is multiplicative under the Kronecker and tensor products of minimal-rank persistent tensors with the GHZ tensor.
Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states ($|{R}_{2n+1}\ensuremath{\rangle}$) to read ${2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the nnn-tangle ${\ensuremath{\tau}}_{n}$.