|
|
| Line 1: |
Line 1: |
| {{Orphan|date=December 2012}}
| | I'm Dulcie and I live with my husband and our 3 children in Berghem, in the NB south part. My hobbies are Locksport, Coin collecting and Backpacking.<br>xunjie ジョークを持っていたブランド品業界を掃引します。 |
| | その新規契約パーソンズと中国のスポーツブランドアンタ理解されているパーソンズは - フェイガン交渉が完了し、 |
| | これはケイトですプリンセスファッションの魅力！ウェディングドレスのレースのウェディングドレスのレースのウェディングドレスプリンセスケイトは、 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ gaga milano �rӋ ����] ウルグアイウルグアイ鉱物の変異下旬マグマミネラルが発生した突然変異を経て。 |
| | ガスレンジを構築するために金属製のジッパーパンクロックを使用するので、 |
| | 複雑すぎる色を気にすることの両方の、 [http://bigapplefilmfestival.com/css/shop/Duvetica.php ���`�� �����ȥ�å�] ああするために�<br>��れ |
| | 生産数の産業発展のためのより強固な基盤がある、 |
| | 浙江Hengyi白色ポリエステルFDY 50D/24F 8600 9900呉江新生児白色ポリエステルFDY 63D/24F（薄い）白9100宜興中国·アジアポリエステルFDY 68D/24F 9300呉江新民白色ポリエステル10100呉江のFDY 100D/48F新しいホワイト8700桐郷Fengming白色ポリエステルFDY明るい50D/24F万呉江新民白色ポリエステルFDY明るい50D/24F万呉江遼呉50D/36F白色ポリエステルFDY明るいポリエステルFDY大きな半径白色光68D/24F 9400無錫宏達75D/36FポリエステルFDY明るい白9300無錫ヘン白色ポリエステルFDY明るい108D/48F 8900宜興、[http://www.tstcom.com/Backup/images/jp/shop/toryburch/ �ȥ�`�Щ`�� ؔ�� 2014] ミアは彼女が一目見ただけで美しいではない、 |
| | 黒のレギンスは底打ちしたパンツが、 |
| | 解釈欧州代替ポップを残します。 |
| | 内部の厚いラムウールと足の裏非常に柔らかく、 [http://www.swiss-equity-fair.ch/pagemedia/js/li/bag/coach/ �<br><br>� ؔ�� ���] |
|
| |
|
| A '''Quantum t-design''' is a probability distribution over pure [[quantum states]] which can duplicate properties of the probability distribution over the [[Haar measure]] for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states. These designs are usually unique, and thus almost always calculable. Two particularly important types of t-designs in quantum mechanics are spherical and unitary t-designs.
| | Here is my website; [http://www.swisscrossbow.ch/html/jp/good/gaga.html ガガミラノ 時計 店舗] |
| | |
| Spherical t-designs are designs where points of the design (i.e. the points being used for the averaging process) are points on a [[unit sphere]]. Spherical t-designs and variations thereof have been considered lately and found useful in [[quantum information theory]],<ref>A. Hayashi, T. Hashimoto, M. Horibe. Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A, 72: 032325, 2006. Also quant-ph/0410207.</ref> [[quantum cryptography]] and other related fields.
| |
| | |
| Unitary designs are analogous to spherical designs in that they approximate the entire [[unitary group]] via a finite collection of [[unitary matrices]]. Unitary designs have been found useful in information theory<ref>C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs: constructions and applications, (2006).</ref> and [[quantum computing]]. Unitary designs are especially useful in quantum computing since most operations are represented by unitary operators.
| |
| | |
| == Motivation ==
| |
| In a d-dimensional Hilbert space when averaging over all quantum pure states the natural [[Group (mathematics)|group]] is SU(d), the [[special unitary group]] of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.
| |
| | |
| A particularly widely used example of this is the spin <math>\tfrac{1}{2}</math> system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators. Since every 2x2 unitary operator is a rotation of the [[Bloch sphere]], the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is ''the'' rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
| |
| | |
| Another recent application is the fact that a symmetric informationally complete [[POVM]] is also a spherical 2-design. Also, since a 2-design must have more than <math>d^2</math> elements, a [[SIC-POVM]] is a '''minimal''' 2-design.
| |
| | |
| == Spherical Designs ==
| |
| Complex projective (t,t)-designs have been studied in [[quantum information theory]] as quantum 2-designs, and in t-designs of vectors in the unit sphere in <math>\mathbb{R}^N</math> which, when transformed to vectors in <math>\mathbb{C}^{N/2}</math> become complex projective (t/2,t/2)-designs.
| |
| | |
| Formally, we define<ref name=autogenerated1>[http://arxiv.org/abs/quant-ph/0701126 [quant-ph/0701126] Quantum t-designs: t-wise independence in the quantum world]</ref> a complex projective (t,t)-design as a probability distribution over quantum states <math>(p_i,|\phi_i\rangle)</math> if
| |
| | |
| <math>\sum_i p_i (|\phi_i\rangle \langle \phi_i|)^{\otimes t} = \int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi</math>
| |
| | |
| Here, the integral over states is taken over the Haar measure on the unit sphere in <math>C^N</math>
| |
| | |
| Exact t-designs over quantum states cannot be distinguished from the uniform probability distribution over all states when using t copies of a state from the probability distribution. However in practice even t-designs may be difficult to compute. For this reason approximate t-designs are useful.
| |
| | |
| Approximate (t,t)-designs are most useful due to their ability to be efficiently implemented. i.e. it is possible to generate a quantum state <math>|\phi\rangle</math> distributed according to the probability distribution <math>p_i |\phi_i\rangle</math> in <math>O(\log^c N)</math> time.
| |
| This efficient construction also implies that the [[POVM]] of the operators <math>Np_i |\phi_i\rangle\langle\phi_i|</math> can be implemented in <math>O(\log^c N)</math> time.
| |
| | |
| The technical definition of an approximate (t,t)-design is:
| |
| | |
| If <math>\sum_i p_i |\phi_i\rangle \langle \phi_i| = \int_{\psi}|\psi\rangle \langle \psi|d\psi</math>
| |
| | |
| and <math>(1-\epsilon)\int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi \leq \sum_i p_i (|\phi_i\rangle \langle \phi_i|)^{\otimes t} \leq (1+\epsilon)\int_{\psi}(|\psi\rangle \langle \psi|)^{\otimes t}d\psi</math>
| |
| | |
| then <math>(p_i,|\phi_i\rangle)</math> is an <math>\epsilon</math>-approximate (t,t)-design.
| |
| | |
| It is possible, though perhaps inefficient, to find an <math>\epsilon</math>-approximate (t,t) design consisting of quantum pure states for a fixed t.
| |
| | |
| ===Construction===
| |
| For convenience N is assumed to be a power of 2.
| |
| | |
| Using the fact that for any N there exists a set of <math>N^d</math> functions {0,...,N-1} <math>\rightarrow</math> {0,...,N-1} such that for any distinct <math>k_1, ..., k_d \in</math> {0,...,N-1} the image under f, where f is chosen at random from S, is exactly the uniform distribution over tuples of d elements of {0,...,N-1}.
| |
| | |
| Let <math>|\psi\rangle = \sum_{i=1}^N \alpha |i\rangle</math> be drawn from the Haar measure. Let <math>P_n</math> be the probability distribution of <math>\alpha_n</math> and let <math>P= \lim_{N\rightarrow \infty} \sqrt{N} P_N</math>. Finally let <math>\alpha</math> be drawn from P. If we define <math>X = |\alpha|</math> with probability <math>\tfrac12</math> and <math>X = -|\alpha|</math> with probability <math>\tfrac{1}{2}</math> then:
| |
| <math>E[X^j] = 0</math> for odd j and <math>E[X^j] = (\tfrac{j}{2})!</math> for even j.
| |
| | |
| Using this and [[Gaussian quadrature]] we can construct <math>p_{f,g} = \frac{\sum_{i=1}^N a_{f,i}^2}{|S_1| |S_2|}</math> so that <math>p_{f,g}|\psi_{f,g}\rangle</math> is an approximate (t,t)-design.
| |
| | |
| == Unitary Designs ==
| |
| Elements of the unitary design are elements of the unitary group, U(d), the group of <math>d \times d</math> unitary matrices. A t-design of unitary operators will generate a t-design of states.
| |
| | |
| Suppose <math>{U_k}</math> is your unitary design (i.e. a set of unitary operators). Then for ''any'' pure state <math>|\psi\rangle</math> let <math>|\psi_k\rangle = U_k|\psi\rangle</math>. Then <math>{|\psi_k\rangle}</math> will always be a t-design for states.
| |
| | |
| Formally define<ref>[http://arxiv.org/abs/0809.3813 [0809.3813] Unitary designs and codes]</ref> a ''unitary t-design'', X, if
| |
| | |
| <math>\frac{1}{|X|}\sum_{U \in X} U^{\otimes t}\otimes (U^{*})^{\otimes t} = \int_{U(d)} U^{\otimes t}\otimes (U^{*})^{\otimes t}dU</math>
| |
| | |
| Observe that the space linearly spanned by the matrices <math>U^{\otimes r}\otimes (U^{*})^{\otimes s}dU</math> over all choices of U is identical to the restriction <math>U \in X</math> and <math>r + s = t</math> This observation leads to a conclusion about the duality between unitary designs and unitary codes.
| |
| | |
| Using the permutation maps it is possible<ref name=autogenerated1 /> to verify directly that a set of unitary matrices forms a t-design.<ref>B. Collins and P. ´ Sniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys.,264 (2006), 773–795.</ref>
| |
| | |
| One direct result of this is that for any finite <math>X \subseteq U(d)</math>
| |
| | |
| <math>\frac{1}{|X|^2} \sum_{U,V \in X}|tr(U*V)|^{2t} \geq \int_{U(d)}|tr(U*V)|^{2t}dU</math>
| |
| | |
| With equality if and only if X is a t-design.
| |
| | |
| 1 and 2-designs have been examined in some detail and absolute bounds for the dimension of X, |X|, have been derived.<ref>D. Gross, K. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys., 48 (2007),052104, 22.</ref>
| |
| | |
| ===Bounds for unitary designs===
| |
| Define <math>Hom(U(d),t,t)</math> as the set of functions homogeneous of degree t in <math>U</math> and homogeneous of degree t in <math>U^{*}</math>, then if for every <math>f \in Hom(U(d),t,t)</math>:
| |
| | |
| <math>\frac{1}{|X|} \sum_{U \in X} f(U) = \int_{U(d)}f(U) dU</math>
| |
| | |
| then X is a unitary t-design.
| |
| | |
| We further define the inner product for functions <math>f</math> and <math>g</math> on <math>U(d)</math> as the average value of <math>\bar{f}g</math> as:
| |
| | |
| <math>\langle f,g\rangle := \int_{U(d)}\bar{f(U)}g(U) dX</math>
| |
| | |
| and <math>\langle f,g\rangle_X</math> as the average value of <math>\bar{f}g</math> over any finite subset <math>X \subset U(d)</math>.
| |
| | |
| it follows that X is a unitary t-design iff <math>\langle 1,f\rangle_X = \langle 1,f\rangle \quad\forall f</math>.
| |
| | |
| From the above it is demonstrable that if X is a t-design then <math>|X| \geq dim(Hom(U(d),\left\lceil\tfrac{t}2\right\rceil,\left\lfloor\tfrac{t}2\right\rfloor))</math> is an '''absolute bound''' for the design. This imposes an upper bound on the size of a unitary design. This bound is '''absolute''' meaning it depends only on the strength of the design or the degree of the code, and not the distances in the subset, X.
| |
| | |
| ----
| |
| | |
| A unitary code is a finite subset of the unitary group in which a few inner product values occur between elements. Specifically, a unitary code is defined as a finite subset <math>X \subset U(d)</math> if for all <math>U \neq M</math> in X <math>|tr(U^*M)|^2</math> takes only distinct values.
| |
| | |
| It follows that <math>|X| \leq dim(Hom(U(d),s,s))</math> and if U and M are orthogonal: <math>|X| \leq dim(Hom(U(d),s,s-1))</math>
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| [[Category:Quantum mechanics]]
| |
| [[Category:Quantum information science]]
| |
| [[Category:Information theory]]
| |
| [[Category:Quantum information theory]]
| |