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INTRO RANDOM STATES Polarized Ensembles of Random Quantum States Summer School - Randomness in Physics and Mathematics Fabio Deelan Cunden Università degli Studi di Bari Dipartimento di Matematica August 5-17, 2013 INTRO RANDOM STATES Intro Some preliminaries A physical system is specified by the observables 𝐴, 𝐵, 𝐶, ⋅ ⋅ ⋅ ∈ 𝒜 that we can measures. The observables form an algebra 𝒜. In classical mechanics 𝒜 is commutative. In quantum mechanics 𝒜 is not. A state of a system is an assignement of a number 𝜑(𝒜) for each element 𝒜 of the algebra of observables. Such a number 𝜑(𝒜) is the outcome of a measurement of the observable 𝒜. 𝜑:𝐴→ℂ Remark The set of states 𝒮 is convex. 𝜑 linear, 𝜑 ≥ 0, 𝜑(1) = 1. INTRO RANDOM STATES Intro Some preliminaries A physical system is specified by the observables 𝐴, 𝐵, 𝐶, ⋅ ⋅ ⋅ ∈ 𝒜 that we can measures. The observables form an algebra 𝒜. In classical mechanics 𝒜 is commutative. In quantum mechanics 𝒜 is not. A state of a system is an assignement of a number 𝜑(𝒜) for each element 𝒜 of the algebra of observables. Such a number 𝜑(𝒜) is the outcome of a measurement of the observable 𝒜. 𝜑:𝐴→ℂ 𝜑 linear, 𝜑 ≥ 0, 𝜑(1) = 1. Remark The set of states 𝒮 is convex. What happens if the state 𝜑 of the system is random? What about its typical properties? INTRO RANDOM STATES Intro Quantum mechanics In a quantum mechanical setting: ∙ ∙ 𝒜 = ℬ(ℋ), ℋ Hilbert space; 𝜑 = tr(𝜌 ⋅), where 𝜌 = 𝜌† , 𝜌 ≥ 0, tr𝜌 = 1. Examples ℋ = ℂ𝑁 𝜌= 1 𝑁 maximally mixed state 𝜌 = ∣𝜓⟩ ⟨𝜓∣ pure state (a rank-1 projection operator). Remark ∙ ∙ The pure states ∣𝜓⟩ are normalized vectors ∥𝜓∥2 = 1. The set of quantum states 𝒮(ℋ) is convex, the pure states being its extremal points. INTRO RANDOM STATES Composite Quantum Systems 𝑆 =𝐴+𝐵 ℋ𝑆 = ℋ𝐴 ⊗ ℋ 𝐵 dim ℋ𝐴 = 𝑁 dim ℋ𝐵 = 𝑀 “...the quantum theory subscribes to the view that the whole is greater than the sum of its parts.” (H.Weyl) Given a state 𝜑𝐴𝐵 of the whole system, the state of subsystem 𝐴 is given by the reduced state 𝜑𝐴 : Tr 𝜑𝐴𝐵 ∈ 𝒮(ℋ𝐴 ⊗ ℋ𝐵 ) −−−𝐵 −→ 𝜑𝐴 ∈ 𝒮(ℋ𝐴 ) Definition (Entanglement) A pure state 𝜑𝐴𝐵 ∈ 𝒮(ℋ𝐴 ⊗ ℋ𝐵 ) is separable if 𝜑𝐴 ∈ 𝒮(ℋ𝐴 ) is pure. A non separable state is said entangled. INTRO RANDOM STATES The problem Typicality in Hilbert Spaces Let us consider a bipartite quantum system: 𝑆 =𝐴+𝐵 . Let us choose randomly a state ∣𝜓⟩𝐴𝐵 of system 𝑆: What about the typical properties of the subsystem 𝐴? Rephrasing: ∙ Fix an ensemble of random pure quantum states ∣𝜓⟩𝐴𝐵 of a bipartite space ℋ𝑆 = ℋ𝐴 ⊗ ℋ𝐵 . ∙ Consider the consequent ensemble of reduced density matrices 𝜌𝐴 ∈ 𝒮(ℋ𝐴 ). ∙ What are the typical properties of 𝜌𝐴 ? INTRO RANDOM STATES Entanglement characterization Thm (A consequence of SVD) The pure state ∣𝜓⟩𝐴𝐵 is separable iff rank𝜌𝐴 = 1. H0,1,0L Let (𝜆1 , . . . , 𝜆𝑑𝐴 ) = (1, 0, . . . , 0) be the eigenvalues of 𝜌𝐴 . If (𝜆1 , . . . , 𝜆𝑑𝐴 ) = (1, 0, . . . , 0) then ∣𝜓⟩ is separable. If at least two eigenvalues are ∕= 0 the state ∣𝜓⟩ is entangled. N=3 H H1,0,0L How to quantify the non-separability of ∣𝜓⟩? Local Purity: 𝜋𝐴𝐵 (𝜓) = ∑ 𝜆2𝑖 𝑖 The lower the local purity the more entangled is ∣𝜓⟩. 1/𝑑𝐴 ≤ 𝜋𝐴𝐵 ≤ 1 1 1 1 , , L 3 3 3 H0,0,1L INTRO RANDOM STATES Random Pure States Unbiased ensemble There exists a “uniform” measure 𝜇Haar on pure states (from topological reasons) maximal simmetry in the Hilbert space ℋ ⇕ minimal prior knowledge of system 𝑆. This ensemble has been intensively investigated (lot of RMT technology). Induced measure on density matrices The unitarily invariant measure on pure states 𝜇Haar induces a measure 𝜈𝐴 on the reduced density matrices (the push-forward measure of 𝜇Haar w.r.t. the map Tr𝐵 ): 𝜈𝐴 = (Tr𝐵 )∗ 𝜇Haar = 𝑓 × 𝜇 INTRO RANDOM STATES Joint pdf of the eigenvalues The eigenvalues are distribuited according to the following density: 𝑓𝑁,𝑀 (𝜆1 , . . . , 𝜆𝑁 ) = 𝐶𝑁,𝑀 ∏ 𝑖<𝑗 (𝜆𝑖 − 𝜆𝑗 )2 ∏ −𝑁 𝜆𝑀 𝑙 𝑙 dim ℋ𝐴 = 𝑁 dim ℋ𝐵 = 𝑀 ∏ 𝐶𝑁,𝑀 : normalization constant 2 : levels repulsion −𝑁 𝜆𝑀 𝑙 : subsystem 𝐴 is strongly correlated with 𝐵 (𝜆𝑖 − 𝜆𝑗 ) 𝑖<𝑗 ∏ 𝑙 The pdf is supported in the domain defined by ∑ 𝜆𝑖 ≥ 0 , 𝜆𝑘 = 1 . INTRO RANDOM STATES Want to study different physically motivated measures on the space of pure states. The setup A bipartite quantum system ℋ = ℋ𝐴 ⊗ ℋ𝐵 , with 𝑑𝐴 = dim ℋ𝐴 ≤ dim ℋ𝐵 = 𝑑𝐵 . Suppose that the state of the quantum system has the form (up to normalization) ∣𝜓⟩ = 𝛼 ∣𝜓1 ⟩ + 𝛽 ∣𝜓2 ⟩ , where ∣𝜓1 ⟩ , ∣𝜓2 ⟩ ∈ ℋ are random states. Once the probability distributions of ∣𝜓1 ⟩ and ∣𝜓2 ⟩ are specified, the random state ∣𝜓⟩ is characterized by a well defined distribution. (1) INTRO RANDOM STATES ∣𝜓⟩ = superposition of a fixed pure state and an unbiased random one. We get the following one-parameter ensemble [ ] √ ∣𝜓⟩ = 𝜖1𝐴𝐵 + 1 − 𝜖2 𝑈𝐴𝐵 ∣𝜙0 ⟩ (2) where: ∙ the normalized state ∣𝜙0 ⟩ ∈ ℋ is fixed; ∙ 𝜖 ∈ [0, 1] is a tunable parameter; ∙ 1𝐴𝐵 is the identity operator; ∙ 𝑈𝐴𝐵 ∈ 𝒰 (ℋ) is a random unitary (Haar distributed). ∣𝜓⟩ = 𝜖 ∣𝜙0 ⟩ + √ 1 − 𝜖2 ∣𝜙⟩ (3) INTRO RANDOM STATES Typical local purity We study the local purity of one of its party 𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] . The lower the local purity the more entangled is ∣𝜓⟩. (4) INTRO RANDOM STATES Typical local purity We study the local purity of one of its party 𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] . (4) The lower the local purity the more entangled is ∣𝜓⟩. The typical purity of the random state (2) is the average value of (4): ( ) 𝔼[𝜋𝐴𝐵 (𝜓)] = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb + 𝑂 where 𝜋unb = 𝑑𝐴 +𝑑𝐵 𝑑𝐴 𝑑𝐵 ( 1 𝑑𝐴 𝑑𝐵 ) , 𝜋0 = 𝜋𝐴𝐵 (∣𝜙0 ⟩) is the local purity of the bias ∣𝜙0 ⟩. (5) INTRO RANDOM STATES Typical local purity We study the local purity of one of its party 𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] . (4) The lower the local purity the more entangled is ∣𝜓⟩. The typical purity of the random state (2) is the average value of (4): ( ) 𝔼[𝜋𝐴𝐵 (𝜓)] = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb + 𝑂 where 𝜋unb = 𝑑𝐴 +𝑑𝐵 𝑑𝐴 𝑑𝐵 ( 1 𝑑𝐴 𝑑𝐵 ) (5) , 𝜋0 = 𝜋𝐴𝐵 (∣𝜙0 ⟩) is the local purity of the bias ∣𝜙0 ⟩. The average value (5) is typical (concentration of measure phenomenon): ( ) { } 𝑑𝐴 𝑑𝐵 𝛼2 Pr 𝜋𝐴𝐵 (𝜓) − 𝔼[𝜋𝐴𝐵 (𝜓)] > 𝛼 ≤ 2 exp − 32(1 − 𝜖2 ) (6) INTRO RANDOM STATES Isopurity manifolds [ ∣𝜓⟩ = 𝜖𝑈𝐴 ⊗ 𝑈𝐵 + √ ] 1 − 𝜖2 𝑈𝐴𝐵 ∣𝜙0 ⟩ (7) 4 ( 4 ∣𝜙0 ⟩ = ∣𝜙sep ⟩ ⇒ 𝜋0 = 1 (8) ⇒ 𝜋0 = 1/𝑑𝐴 (9) 𝜋𝐴𝐵 ≃ 𝜖 𝜋0 + 1 − 𝜖 ) 𝜋unb Separable bias: Maximally entangled bias: ∣𝜙0 ⟩ = ∣𝜙ent ⟩𝐴𝐵 Π AB N=30 M=30 1.0 0.8 0.6 0.4 Maximally Entangled Φ ent > 0.2 0.0 1 0.5 Separable Φ sep > 0 Ε 4 0.5 1 INTRO RANDOM STATES Generation of random pure states with fixed purity An inexpensive strategy for generating random pure states with fixed value of 𝜋𝐴𝐵 : 1. Choose 𝜖 ∈ [0, 1] such that ( ) 𝜋𝐴𝐵 = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb , (10) where 𝜋0 = 1 or 𝜋0 = 1/𝑁 if the desired value of 𝜋𝐴𝐵 is, respectively, larger or smaller than the unbiased typical value 𝜋Haar . 2. Generate a pure state ∣𝜓⟩ by superposition √ ∣𝜓⟩ = 𝜖 ∣𝜙0 ⟩ + 1 − 𝜖2 ∣𝜙⟩ , (11) where ∣𝜙⟩ ∼ 𝜇Haar and ∣𝜙0 ⟩ is a random separable or maximally entangled pure state sampled, according to the value of 𝜋0 chosen in (10). INTRO RANDOM STATES Proposals ∙ New ensembles? INTRO RANDOM STATES Proposals ∙ New ensembles? We can think to ensembles with fixed energy ⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . . INTRO RANDOM STATES Proposals ∙ ∙ New ensembles? We can think to ensembles with fixed energy ⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . . Typicality in multipartite systems?? INTRO RANDOM STATES Proposals ∙ ∙ ∙ New ensembles? We can think to ensembles with fixed energy ⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . . Typicality in multipartite systems?? What is the role of the entanglement of random states in Statistical Mechanics? INTRO RANDOM STATES Bengtsson I and Życzkowski K 2006 Geometry of quantum states (Cambridge University Press) Hayden P, Leung D W and Winter A 2006 Comm. Math. Phys. 265 95 Cunden F D, Facchi P, Florio G and Pascazio S, Eur. Phys. J. Plus 128 48, (2013) (arXiv:1303.4209 [math-ph]). Cunden F D, Facchi P and Florio G, J. Phys. A: Math. Theor. 46 315306, (2013) (arXiv:1304.6219v2 [math-ph]). INTRO RANDOM STATES Robustness of separability Application: stability of separability of quantum states with respect to random additive perturbations. If the state of a bipartite system ∣𝜉0 ⟩𝐴 ⊗ ∣𝜒0 ⟩𝐵 is separable, how much noise 𝜂 is necessary to make the its reduced state 𝜌𝐴 distinguishable from a pure state? √ ∣𝜓⟩ = 1 − 𝜂 2 ∣𝜉0 ⟩𝐴 ⊗ ∣𝜒0 ⟩𝐵 + 𝜂 ∣𝜙⟩ , 0 ≤ 𝜂 ≤ 1, (12) where ∙ ∣𝜙⟩ ∼ 𝜇𝑁 𝑀 is an unbiased random perturbation; ∙ 𝜂 measures the strength of the noise. In the limit of large system sizes, 𝑑𝐴 , 𝑑𝐵 → ∞, the threshold critical value becomes ( ) ( ) 1 1 +𝑂 , (13) 𝜂★2 = 1 − √ 𝑑𝐴 2 As long as the state ∣𝜓⟩ of the large quantum system has the form (12) with √ 1 𝜂 < 1 − √ ≃ 0.54, 2 the effective dimension is 𝑑eff (𝜌𝐴 ) < 2, and separability will be (approximately) preserved. (14)