Tensor rank of the tripartite state

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 $|\mathrm{GHZ}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|000\ensuremath{\rangle}+|111\ensuremath{\rangle})$.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 $|\mathrm{GHZ}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|000\ensuremath{\rangle}+|111\ensuremath{\rangle})$.

arXiv:0910.0986Physical Review AJan 15, 201032 citations

Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States

Lin Chen, Eric Chitambar, Runyao Duan +2 more

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.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.

arXiv:1003.3059Physical Review LettersNov 8, 201097 citations

Tripartite Entanglement Transformations and Tensor Rank

Eric Chitambar, Runyao Duan, Yaoyun Shi

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.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.

arXiv:0805.2977Physical Review LettersOct 2, 200874 citations

Persistent Tensors and Multiqudit Entanglement Transformation

Masoud Gharahi, Vladimir Lysikov

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.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.

arXiv:2211.00652QuantumJan 31, 20242 citations

Exponential lower bounds on the fermionic Gaussian rank of magic states and the bosonic coherent state rank of Fock states

Oliver Reardon-Smith

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 $\lvert M\rangle^{\otimes k}$ consists of $Ω(1.4^k)$ terms. Here $\lvert M\rangle$ is the most standard magic state for fermionic linear optics: the $4$ qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the $δ$ -approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of $1-δ^2$ . Our results on exact fermionic Gaussian rank apply directly to any product of $k$ 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 $n$ -mode bosonic Fock state with $m_j$ bosons in mode $j$ is exactly $\prod_{j}(1+m_j)$ , 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.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 $\lvert M\rangle^{\otimes k}$ consists of $Ω(1.4^k)$ terms. Here $\lvert M\rangle$ is the most standard magic state for fermionic linear optics: the $4$ qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the $δ$-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of $1-δ^2$. Our results on exact fermionic Gaussian rank apply directly to any product of $k$ 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 $n$-mode bosonic Fock state with $m_j$ bosons in mode $j$ is exactly $\prod_{j}(1+m_j)$, 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.

quant-pharXiv:2610.03699Oct 5, 20260 citations

Rates of Multipartite Entanglement Transformations

Alexander Streltsov, Clément Meignant, Jens Eisert

The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: the rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multipartite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multipartite quantum networks. In this Letter, we report substantial progress by deriving simple upper and lower bounds on the rates that can be achieved in asymptotic multipartite entanglement transformations. These bounds are based on ideas of entanglement combing and state merging. We identify cases where the bounds coincide and hence provide the exact rates. As an example, we bound rates at which resource states for the cryptographic scheme of quantum secret sharing can be distilled from arbitrary pure tripartite quantum states. This result provides further scope for quantum internet applications, supplying tools to study the implementation of multipartite protocols over quantum networks.The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: the rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multipartite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multipartite quantum networks. In this Letter, we report substantial progress by deriving simple upper and lower bounds on the rates that can be achieved in asymptotic multipartite entanglement transformations. These bounds are based on ideas of entanglement combing and state merging. We identify cases where the bounds coincide and hence provide the exact rates. As an example, we bound rates at which resource states for the cryptographic scheme of quantum secret sharing can be distilled from arbitrary pure tripartite quantum states. This result provides further scope for quantum internet applications, supplying tools to study the implementation of multipartite protocols over quantum networks.

arXiv:1709.09693Physical Review LettersAug 17, 202017 citations

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 $|\mathrm{GHZ}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|000\ensuremath{\rangle}+|111\ensuremath{\rangle})$.

\lvert M\rangle^{\otimes k}

consists of

Ω(1.4^k)

terms. Here

\lvert M\rangle

is the most standard magic state for fermionic linear optics: the

4

qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the

δ

-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of

1-δ^2

. Our results on exact fermionic Gaussian rank apply directly to any product of

k

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

n

-mode bosonic Fock state with

m_j

bosons in mode

j

is exactly

\prod_{j}(1+m_j)

, 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.