Abstract
A quantum state’s entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.
Highlights
As a consequence of the singular-value decomposition, a normalized state |ψ in a bipartite Hilbert space HA ⊗ HB with finite dimensions dA, dB and ns := min(dA, dB) can be written as ns|ψ = λψi |ψAi ⊗ |ψBi, i=1 (1.1)with Schmidt coefficients λψ1 ≥ · · · ≥ λψns ≥ 0 satisfying i(λψi )2 = 1 and Schmidt vectors |ψAi ∈ HA, |ψBi ∈HB
These demonstrate the scope of the algorithm and motivate why it is often interesting to maximize the norms (1.2): §5a looks at fermionic antisymmetry; later sections discuss more mathematical applications such as absolutely maximally entangled (AME) states and perfect tensors (§5b), varieties of pure quantum states (§5c) and minimal output entropy of channels (§5d)
No other results like this are known, but these are exactly the kind of questions that we can investigate with the algorithm. General statements of this form lead to interesting mathematical inequalities, interesting states, and they allow us to investigate the particle entanglement properties of fermions. These are some of the reasons that there appears to be a renewed interest in these questions: for example, the theorem above was recently rediscovered in the context of hard-core bosons [28]; a weaker version of the well-known bosonic analogue [29,30] of lemma 5.7 was presented as a new result in [31]; part of conjecture 5.10 was stated as a fact in [19]; known N-representability constraints were put to use in quantum algorithms in [32]
Summary
As a consequence of the singular-value decomposition, a normalized state |ψ in a bipartite Hilbert space HA ⊗ HB with finite dimensions dA, dB and ns := min(dA, dB) can be written as ns. Over pure states |ψ in a specified subspace U ⊂ HA ⊗ HB Though they may seem rather abstract, these norms are relevant to a wide range of problems. The first sections of this paper are fully general: §2 states the algorithm and its goal in the most general terms; §3 motivates the relevance of the Schmidt norms by linking them to entanglement;. These demonstrate the scope of the algorithm and motivate why it is often interesting to maximize the norms (1.2): §5a looks at fermionic antisymmetry; later sections discuss more mathematical applications such as absolutely maximally entangled (AME) states and perfect tensors (§5b), varieties of pure quantum states (§5c) and minimal output entropy of channels (§5d). We hope that the reader will select one or more of these topics to see what the algorithm can do in a concrete setting—we refer the reader to the conclusion (§6) for a summary
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