在泛函分析 和量子信息科学 中，正算子值测度 （Positive operator valued measure, POVM ）是一种推广的测度 ，这种测度的值为希尔伯特空间 上半正定算子 。POVM是投影值测度 （Projection valued measure, PVM ） 的推广，相应地，POVM描述的量子测量 是PVM描述的量子测量 (称为投影测量) 的推广。
粗略比喻来说：POVM之于PVM，就如同混合态 之于纯态 一样。 混合态对于刻画一个较大系统的子系统的状态而言是必须的（见量子态的纯化 ）；类似地，POVM的概念则在刻画在较大系统上进行的投影测量对子系统的影响时自然地产生。
POVM是量子力学中最普遍的测量类型，也用于量子场论 。[ 1] 量子信息 领域有着广泛的应用。
设 H {\displaystyle {\mathcal {H}}} 希尔伯特空间 ，而 ( X , M ) {\displaystyle (X,M)} 可测空间 ，其中 M {\displaystyle M} X {\displaystyle X} 博雷尔σ-代数 。若 M {\displaystyle M} F {\displaystyle F} H {\displaystyle {\mathcal {H}}} 正 有界 自伴算子 ，且对于任意 ψ ∈ H {\displaystyle \psi \in {\mathcal {H}}} E ∈ M {\displaystyle E\in M}
E ↦ ⟨ F ( E ) ψ | ψ ⟩ , {\displaystyle E\mapsto \langle F(E)\psi |\psi \rangle ,} 则 F {\displaystyle F} σ-代数 M {\displaystyle M} 可数可加 测度，且总质量 为恆等算子 F ( X ) = I H {\displaystyle F(X)=\operatorname {I} _{\mathcal {H}}} F {\displaystyle F} [ 2]
在最简单的情况下，POVM是有限维希尔伯特空间 H {\displaystyle {\mathcal {H}}} 半正定 埃尔米特矩阵 { F i } {\displaystyle \{F_{i}\}} 单位矩阵 [ 3] :90
∑ i = 1 n F i = I . {\displaystyle \sum _{i=1}^{n}F_{i}=\operatorname {I} .} POVM与投影值测度 的不同之处在于，对于投影值测度， F {\displaystyle F} 正交投影 。
在量子力学 中，POVM的关键性质是它确定了结果空间上的一个概率测度，因此 ⟨ F ( E ) ψ | ψ ⟩ {\displaystyle \langle F(E)\psi |\psi \rangle } 量子态 | ψ ⟩ {\displaystyle |\psi \rangle } E {\displaystyle E} F i {\displaystyle F_{i}} i {\displaystyle i} 量子态 ρ {\displaystyle \rho } 量子测量 时得到它的概率
Prob ( i ) = tr ( ρ F i ) {\displaystyle {\text{Prob}}(i)=\operatorname {tr} (\rho F_{i})} 其中 tr {\displaystyle \operatorname {tr} } 迹 运算。若被测量的量子态是纯态 | ψ ⟩ {\displaystyle |\psi \rangle }
Prob ( i ) = tr ( | ψ ⟩ ⟨ ψ | F i ) = ⟨ ψ | F i | ψ ⟩ {\displaystyle {\text{Prob}}(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle } POVM的最简单情况推广了PVM的最简单情况，即PVM是一组和为恒等矩阵的正交投影 { Π i } {\displaystyle \{\Pi _{i}\}}
∑ i = 1 N Π i = I , Π i Π j = δ i j Π i . {\displaystyle \sum _{i=1}^{N}\Pi _{i}=\operatorname {I} ,\quad \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}.} PVM的概率公式与POVM的概率公式相同。一个重要的区别是POVM的元素不一定正交。因此，POVM元素的数量 n {\displaystyle n} N {\displaystyle N}
奈马克扩张定理 [ 4] [ 5] :285
最简单的情况是，POVM元素作用于一个有限维希尔伯特空间且数目有限。奈马克扩张定理指出，若 { F i } i = 1 n {\displaystyle \{F_{i}\}_{i=1}^{n}} d A {\displaystyle d_{A}} H A {\displaystyle {\mathcal {H}}_{A}} d A ′ {\displaystyle d_{A'}} H A ′ {\displaystyle {\mathcal {H}}_{A'}} { Π i } i = 1 n {\displaystyle \{\Pi _{i}\}_{i=1}^{n}} 等距同构 V : H A → H A ′ {\displaystyle V:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A'}} i {\displaystyle i}
F i = V † Π i V . {\displaystyle F_{i}=V^{\dagger }\Pi _{i}V.} 对于秩 为一的POVM的特殊情况，即存在某（未归一化的）向量 | f i ⟩ {\displaystyle |f_{i}\rangle } F i = | f i ⟩ ⟨ f i | {\displaystyle F_{i}=|f_{i}\rangle \langle f_{i}|} [ 5] :285
V = ∑ i = 1 n | i ⟩ A ′ ⟨ f i | A {\displaystyle V=\sum _{i=1}^{n}|i\rangle _{A'}\langle f_{i}|_{A}} 而PVM则由 Π i = | i ⟩ ⟨ i | A ′ {\displaystyle \Pi _{i}=|i\rangle \langle i|_{A'}} d A ′ = n {\displaystyle d_{A'}=n}
一般情况下，等距同构和PVM可以通过定义[ 6] [ 7] H A ′ = H A ⊗ H B {\displaystyle {\mathcal {H}}_{A'}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}} Π i = I A ⊗ | i ⟩ ⟨ i | B {\displaystyle \Pi _{i}=\operatorname {I} _{A}\otimes |i\rangle \langle i|_{B}}
V = ∑ i = 1 n F i A ⊗ | i ⟩ B {\displaystyle V=\sum _{i=1}^{n}{\sqrt {F_{i}}}_{A}\otimes {|i\rangle }_{B}} 来构造。注意这里 d A ′ = n d A {\displaystyle d_{A'}=nd_{A}}
无论哪种情况，对经过等距映射后的态进行该投影测量得到结果 i {\displaystyle i}
Prob ( i ) = tr ( V ρ A V † Π i ) = tr ( ρ A V † Π i V ) = tr ( ρ A F i ) {\displaystyle {\text{Prob}}(i)=\operatorname {tr} \left(V\rho _{A}V^{\dagger }\Pi _{i}\right)=\operatorname {tr} \left(\rho _{A}V^{\dagger }\Pi _{i}V\right)=\operatorname {tr} (\rho _{A}F_{i})} 通过将该等距同构 V {\displaystyle V} 扩张 为一个幺正算子 U {\displaystyle U} U {\displaystyle U}
∀ 1 ≤ i ≤ d A , V | i ⟩ A = U | i ⟩ A ′ . {\displaystyle \forall 1\leq i\leq d_{A},\quad V|i\rangle _{A}=U|i\rangle _{A'}.} 这总是可以做到。
实现对量子态 ρ {\displaystyle \rho } { F i } i = 1 n {\displaystyle \{F_{i}\}_{i=1}^{n}} [需要解释 ，将 ρ {\displaystyle \rho } H A ′ {\displaystyle {\mathcal {H}}_{A'}} U {\displaystyle U} { Π i } i = 1 n {\displaystyle \{\Pi _{i}\}_{i=1}^{n}}
测量后的状态不是由POVM本身决定的，而是由物理上实现它的PVM来决定。由于相同的POVM有无穷个不同的PVM实现，因此 { F i } i = 1 n {\displaystyle \{F_{i}\}_{i=1}^{n}} W {\displaystyle W}
M i = W F i {\displaystyle M_{i}=W{\sqrt {F_{i}}}} 也满足性质 M i † M i = F i {\displaystyle M_{i}^{\dagger }M_{i}=F_{i}}
V W = ∑ i = 1 n M i A ⊗ | i ⟩ B {\displaystyle V_{W}=\sum _{i=1}^{n}{M_{i}}_{A}\otimes {|i\rangle }_{B}} 也将实现相同的 POVM。在被测状态为纯态 | ψ ⟩ A {\displaystyle |\psi \rangle _{A}} U W {\displaystyle U_{W}}
U W ( | ψ ⟩ A | 0 ⟩ B ) = ∑ i = 1 n M i | ψ ⟩ A | i ⟩ B , {\displaystyle U_{W}(|\psi \rangle _{A}|0\rangle _{B})=\sum _{i=1}^{n}M_{i}|\psi \rangle _{A}|i\rangle _{B},} 当得到的测量结果为 i 0 {\displaystyle i_{0}} | ψ ⟩ A {\displaystyle |\psi \rangle _{A}} [ 3] :84
| ψ ′ ⟩ A = M i 0 | ψ ⟩ ⟨ ψ | M i 0 † M i 0 | ψ ⟩ . {\displaystyle |\psi '\rangle _{A}={\frac {M_{i_{0}}|\psi \rangle }{\sqrt {\langle \psi |M_{i_{0}}^{\dagger }M_{i_{0}}|\psi \rangle }}}.} 若被测态的密度矩阵为 ρ A {\displaystyle \rho _{A}}
ρ A ′ = M i 0 ρ M i 0 † t r ( M i 0 ρ M i 0 † ) . {\displaystyle \rho '_{A}={M_{i_{0}}\rho M_{i_{0}}^{\dagger } \over {\rm {tr}}(M_{i_{0}}\rho M_{i_{0}}^{\dagger })}.} 因此可见，测量后状态明确依赖于幺正算子 W {\displaystyle W} M i † M i = F i {\displaystyle M_{i}^{\dagger }M_{i}=F_{i}} M i {\displaystyle M_{i}}
正算子值测量与投影测量的另一个区别在于它通常是不可重复的。若第一次测量得到结果 i 0 {\displaystyle i_{0}} i 1 {\displaystyle i_{1}}
Prob ( i 1 | i 0 ) = tr ( M i 1 M i 0 ρ M i 0 † M i 1 † ) t r ( M i 0 ρ M i 0 † ) {\displaystyle {\text{Prob}}(i_{1}|i_{0})={\operatorname {tr} (M_{i_{1}}M_{i_{0}}\rho M_{i_{0}}^{\dagger }M_{i_{1}}^{\dagger }) \over {\rm {tr}}(M_{i_{0}}\rho M_{i_{0}}^{\dagger })}} 它在 M i 0 {\displaystyle M_{i_{0}}} M i 1 {\displaystyle M_{i_{1}}}
态的布洛赫球 表示（蓝色）与对态 | ψ ⟩ = | 0 ⟩ {\displaystyle |\psi \rangle =|0\rangle } | φ ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) {\displaystyle |\varphi \rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )} 假定有一量子系统对应的希尔伯特空间为二维的，其态要么是 | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle } | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle } { | ψ ⟩ ⟨ ψ | , | φ ⟩ ⟨ φ | } {\displaystyle \{|\psi \rangle \langle \psi |,|\varphi \rangle \langle \varphi |\}} | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle } [ 3] :87 完美辨认非正交态的不可实现性是各种量子信息协议（如量子密碼學 、量子硬币翻转 量子貨幣 ）的基础。
退而求其次，实际能做到的最好效果是所谓无歧义量子态分辨（Unambigious quantum state discrimination, UQSD）：在系统处于 | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle } [ 8] | ψ ⊥ ⟩ {\displaystyle |\psi ^{\perp }\rangle } | ψ ⟩ {\displaystyle |\psi \rangle } { | ψ ⟩ ⟨ ψ | , | ψ ⊥ ⟩ ⟨ ψ ⊥ | } {\displaystyle \{|\psi \rangle \langle \psi |,|\psi ^{\perp }\rangle \langle \psi ^{\perp }|\}} | ψ ⊥ ⟩ ⟨ ψ ⊥ | {\displaystyle |\psi ^{\perp }\rangle \langle \psi ^{\perp }|} | φ ⟩ {\displaystyle |\varphi \rangle } | ψ ⟩ ⟨ ψ | {\displaystyle |\psi \rangle \langle \psi |} { | φ ⟩ ⟨ φ | , | φ ⊥ ⟩ ⟨ φ ⊥ | } {\displaystyle \{|\varphi \rangle \langle \varphi |,|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|\}} | φ ⊥ ⟩ {\displaystyle |\varphi ^{\perp }\rangle } | φ ⟩ {\displaystyle |\varphi \rangle }
然而这并不足以令人满意，这种方法无法用单一测量来探测 | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle } [ 8] [ 9]
F ψ = 1 1 + | ⟨ φ | ψ ⟩ | | φ ⊥ ⟩ ⟨ φ ⊥ | {\displaystyle F_{\psi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|} F φ = 1 1 + | ⟨ φ | ψ ⟩ | | ψ ⊥ ⟩ ⟨ ψ ⊥ | {\displaystyle F_{\varphi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\psi ^{\perp }\rangle \langle \psi ^{\perp }|} F ? = I − F ψ − F φ = 2 | ⟨ φ | ψ ⟩ | 1 + | ⟨ φ | ψ ⟩ | | γ ⟩ ⟨ γ | , {\displaystyle F_{?}=\operatorname {I} -F_{\psi }-F_{\varphi }={\frac {2|\langle \varphi |\psi \rangle |}{1+|\langle \varphi |\psi \rangle |}}|\gamma \rangle \langle \gamma |,} 其中
| γ ⟩ = 1 2 ( 1 + | ⟨ φ | ψ ⟩ | ) ( | ψ ⟩ + e i arg ( ⟨ φ | ψ ⟩ ) | φ ⟩ ) . {\displaystyle |\gamma \rangle ={\frac {1}{\sqrt {2(1+|\langle \varphi |\psi \rangle |)}}}(|\psi \rangle +e^{i\arg(\langle \varphi |\psi \rangle )}|\varphi \rangle ).} 注意到 tr ( | φ ⟩ ⟨ φ | F ψ ) = tr ( | ψ ⟩ ⟨ ψ | F φ ) = 0 {\displaystyle \operatorname {tr} (|\varphi \rangle \langle \varphi |F_{\psi })=\operatorname {tr} (|\psi \rangle \langle \psi |F_{\varphi })=0} ψ {\displaystyle \psi } | ψ ⟩ {\displaystyle |\psi \rangle } φ {\displaystyle \varphi } | φ ⟩ {\displaystyle |\varphi \rangle }
当系统处于 | ψ ⟩ {\displaystyle |\psi \rangle } | φ ⟩ {\displaystyle |\varphi \rangle }
1 − | ⟨ φ | ψ ⟩ | . {\displaystyle 1-|\langle \varphi |\psi \rangle |.} 这一结果称为 Ivanović-Dieks-Peres 极限，冠名于UQSD研究的先驱者.[ 10] [ 11] [ 12]
由于这些POVM是秩为一的，可以看到通过前文叙述的构造投影测量的方法，便可在物理上实现这些POVM。将扩大后的希尔伯特空间中的三种可能的态分别标记为 | result ψ ⟩ {\displaystyle |{\text{result ψ}}\rangle } | result φ ⟩ {\displaystyle |{\text{result φ}}\rangle } | result ? ⟩ {\displaystyle |{\text{result ?}}\rangle } U UQSD {\displaystyle U_{\text{UQSD}}} | ψ ⟩ {\displaystyle |\psi \rangle }
U UQSD | ψ ⟩ = 1 − | ⟨ φ | ψ ⟩ | | result ψ ⟩ + | ⟨ φ | ψ ⟩ | | result ? ⟩ , {\displaystyle U_{\text{UQSD}}|\psi \rangle ={\sqrt {1-|\langle \varphi |\psi \rangle |}}|{\text{result ψ}}\rangle +{\sqrt {|\langle \varphi |\psi \rangle |}}|{\text{result ?}}\rangle ,} 类似地，它在 | φ ⟩ {\displaystyle |\varphi \rangle }
U UQSD | φ ⟩ = 1 − | ⟨ φ | ψ ⟩ | | result φ ⟩ + e − i arg ( ⟨ φ | ψ ⟩ ) | ⟨ φ | ψ ⟩ | | result ? ⟩ . {\displaystyle U_{\text{UQSD}}|\varphi \rangle ={\sqrt {1-|\langle \varphi |\psi \rangle |}}|{\text{result φ}}\rangle +e^{-i\arg(\langle \varphi |\psi \rangle )}{\sqrt {|\langle \varphi |\psi \rangle |}}|{\text{result ?}}\rangle .} 于是投影测量便可以相同于POVM的概率来得到所要的结果。
这种POVM已在实验上用于区分一个光子的非正交偏振态。不过实现该POVM的投影测量与此处阐述的有轻微的不同。[ 13] [ 14]
^ Peres, Asher ; Terno, Daniel R. Quantum information and relativity theory. Reviews of Modern Physics . 2004, 76 (1): 93–123. Bibcode:2004RvMP...76...93P S2CID 7481797 arXiv:quant-ph/0212023 doi:10.1103/RevModPhys.76.93 ^ Davies, Edward Brian. Quantum Theory of Open Systems. London: Acad. Press. 1976: 35. ISBN 978-0-12-206150-9 ^ 3.0 3.1 3.2 M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, (2000) ^ I. M. Gelfand and M. A. Neumark, On the embedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213. ^ 5.0 5.1 A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993. ^ J. Preskill, Lecture Notes for Physics: Quantum Information and Computation, Chapter 3, http://theory.caltech.edu/~preskill/ph229/index.html （页面存档备份 ，存于互联网档案馆 ） ^ J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 2.3, https://cs.uwaterloo.ca/~watrous/TQI/ （页面存档备份 ，存于互联网档案馆 ） ^ 8.0 8.1 J.A. Bergou; U. Herzog; M. Hillery. Discrimination of Quantum States. M. Paris; J. Řeháček (编). Quantum State Estimation 417 –465. ISBN 978-3-540-44481-7doi:10.1007/978-3-540-44481-7_11 ^ Chefles, Anthony. Quantum state discrimination. Contemporary Physics (Informa UK Limited). 2000, 41 (6): 401–424. Bibcode:2000ConPh..41..401C ISSN 0010-7514 S2CID 119340381 arXiv:quant-ph/0010114v1 doi:10.1080/00107510010002599 ^ Ivanovic, I.D. How to differentiate between non-orthogonal states . Physics Letters A (Elsevier BV). 1987, 123 (6): 257–259. Bibcode:1987PhLA..123..257I ISSN 0375-9601 doi:10.1016/0375-9601(87)90222-2 ^ Dieks, D. Overlap and distinguishability of quantum states. Physics Letters A (Elsevier BV). 1988, 126 (5–6): 303–306. Bibcode:1988PhLA..126..303D ISSN 0375-9601 doi:10.1016/0375-9601(88)90840-7 ^ Peres, Asher. How to differentiate between non-orthogonal states. Physics Letters A (Elsevier BV). 1988, 128 (1–2): 19. Bibcode:1988PhLA..128...19P ISSN 0375-9601 doi:10.1016/0375-9601(88)91034-1 ^ B. Huttner; A. Muller; J. D. Gautier; H. Zbinden; N. Gisin. Unambiguous quantum measurement of nonorthogonal states. Physical Review A (APS). 1996, 54 (5): 3783–3789. Bibcode:1996PhRvA..54.3783H PMID 9913923 doi:10.1103/PhysRevA.54.3783 ^ R. B. M. Clarke; A. Chefles; S. M. Barnett; E. Riis. Experimental demonstration of optimal unambiguous state discrimination. Physical Review A (APS). 2001, 63 (4): 040305(R). Bibcode:2001PhRvA..63d0305C S2CID 39481893 arXiv:quant-ph/0007063 doi:10.1103/PhysRevA.63.040305 
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. doi:10.1103/RevModPhys.76.93. 
. Springer. 2004: 417–465. ISBN 978-3-540-44481-7. doi:10.1007/978-3-540-44481-7_11. 
