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Phase conjugate of quantum states in finite-dimensional Hilbert space

Xiang-Fa Zhou, Zhang, Y S, Guang‐Can Guo

Published Mar 26, 2006• 0 citations
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Abstract

We show that, for $N$ parallel input states, an anti-linear map with respect to a specific basis is essentially a classical operator. We also consider the information contained in phase-conjugate pairs $|\\phi > |\\phi^*>$, and prove that there is more information about a quantum state encoded in phase-conjugate pairs than in parallel pairs.

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We show that, for NNN parallel input states, an anti-linear map with respect to a specific basis is essentially a classical operator. We also consider the information contained in phase-conjugate pairs ∣phi>∣phi∗>|\\phi > |\\phi^*>∣phi>∣phi∗>, and prove that there is more information about a quantum state encoded in phase-conjugate pairs than in parallel pairs.
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
Phase-space representations can be used to reveal resources for quantum computational advantage. In odd-dimensional qudit systems, states and operations admitting non-negative quasi-probability representations are efficiently classically simulable, and analogous conclusions hold for bosonic systems, usually represented in quadrature phase space. However, when quantum information is encoded in bosonic systems, the link between physical phase-space representations and computational resources becomes nontrivial: large negativities may arise in the physical space even when the encoded model remains efficiently simulable. We establish a general framework connecting physical phase spaces of bosonic systems to the effective computational phase spaces associated with arbitrary encodings. We show that bosonic Hilbert spaces with finitely many modes can always be exactly encoded on a quantum computer, without truncation or approximations, and that for any encoding there exists an adapted physical phase-space representation in which all provably classically simulable states and protocols are non-negative. Conversely, negativity can always persist in physical representations for arbitrary encodings. Our results formalize the operational meaning of phase-space negativity in bosonic quantum information and its relation with computational resources.
We show that, for NNN parallel input states, an antilinear map with respect to a specific basis is essentially a classical operation. We also consider the information contained in a phase-conjugate pair $\ensuremath{\mid}\ensuremath{\phi}⟩\ensuremath{\mid}{\ensuremath{\phi}}^{*}⟩$, and prove that there is more information about a quantum state encoded in a phase-conjugate pair than in a parallel pair.
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two nnn-dimensional Hilbert spaces, n≥3n\ge 3n≥3, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) nnn-tuples of lines in an nnn-dimensional Hilbert space VVV by F⊥(V)\mathbb{F}^{\perp}(V)F⊥(V) (respectively F(V)\mathbb{F}(V)F(V)) and, for partitions πππ of the set {1..n}\{1..n\}{1..n}, call two tuples πππ-linked if the spans along πππ-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps F⊥(Cn)→F(Cn)\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)F⊥(Cn)→F(Cn), n≥3n\ge 3n≥3 respecting πππ-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the F(Cn)\mathbb{F}(\mathbb{C}^n)F(Cn)-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple (ℓi)i(\ell_i)_i(ℓi​)i​ of lines into ((⨁j≠iℓj)⊥)i\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i((⨁j=i​ℓj​)⊥)i​.